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• Published: 22 July 2024

Neural general circulation models for weather and climate

• Dmitrii Kochkov   ORCID: orcid.org/0000-0003-3846-4911 1   na1 ,
• Janni Yuval   ORCID: orcid.org/0000-0001-7519-0118 1   na1 ,
• Ian Langmore 1   na1 ,
• Peter Norgaard 1   na1 ,
• Jamie Smith 1   na1 ,
• Griffin Mooers 1 ,
• Milan Klöwer 2 ,
• James Lottes 1 ,
• Stephan Rasp 1 ,
• Peter Düben   ORCID: orcid.org/0000-0002-4610-3326 3 ,
• Sam Hatfield 3 ,
• Peter Battaglia 4 ,
• Alvaro Sanchez-Gonzalez 4 ,
• Matthew Willson   ORCID: orcid.org/0000-0002-8730-1927 4 ,
• Michael P. Brenner 1 , 5 &
• Stephan Hoyer   ORCID: orcid.org/0000-0002-5207-0380 1   na1  

Nature volume  632 ,  pages 1060–1066 ( 2024 ) Cite this article

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• Atmospheric dynamics
• Climate and Earth system modelling
• Computational science

General circulation models (GCMs) are the foundation of weather and climate prediction 1 , 2 . GCMs are physics-based simulators that combine a numerical solver for large-scale dynamics with tuned representations for small-scale processes such as cloud formation. Recently, machine-learning models trained on reanalysis data have achieved comparable or better skill than GCMs for deterministic weather forecasting 3 , 4 . However, these models have not demonstrated improved ensemble forecasts, or shown sufficient stability for long-term weather and climate simulations. Here we present a GCM that combines a differentiable solver for atmospheric dynamics with machine-learning components and show that it can generate forecasts of deterministic weather, ensemble weather and climate on par with the best machine-learning and physics-based methods. NeuralGCM is competitive with machine-learning models for one- to ten-day forecasts, and with the European Centre for Medium-Range Weather Forecasts ensemble prediction for one- to fifteen-day forecasts. With prescribed sea surface temperature, NeuralGCM can accurately track climate metrics for multiple decades, and climate forecasts with 140-kilometre resolution show emergent phenomena such as realistic frequency and trajectories of tropical cyclones. For both weather and climate, our approach offers orders of magnitude computational savings over conventional GCMs, although our model does not extrapolate to substantially different future climates. Our results show that end-to-end deep learning is compatible with tasks performed by conventional GCMs and can enhance the large-scale physical simulations that are essential for understanding and predicting the Earth system.

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Solving the equations for Earth’s atmosphere with general circulation models (GCMs) is the basis of weather and climate prediction 1 , 2 . Over the past 70 years, GCMs have been steadily improved with better numerical methods and more detailed physical models, while exploiting faster computers to run at higher resolution. Inside GCMs, the unresolved physical processes such as clouds, radiation and precipitation are represented by semi-empirical parameterizations. Tuning GCMs to match historical data remains a manual process 5 , and GCMs retain many persistent errors and biases 6 , 7 , 8 . The difficulty of reducing uncertainty in long-term climate projections 9 and estimating distributions of extreme weather events 10 presents major challenges for climate mitigation and adaptation 11 .

Recent advances in machine learning have presented an alternative for weather forecasting 3 , 4 , 12 , 13 . These models rely solely on machine-learning techniques, using roughly 40 years of historical data from the European Center for Medium-Range Weather Forecasts (ECMWF) reanalysis v5 (ERA5) 14 for model training and forecast initialization. Machine-learning methods have been remarkably successful, demonstrating state-of-the-art deterministic forecasts for 1- to 10-day weather prediction at a fraction of the computational cost of traditional models 3 , 4 . Machine-learning atmospheric models also require considerably less code, for example GraphCast 3 has 5,417 lines versus 376,578 lines for the National Oceanic and Atmospheric Administration’s FV3 atmospheric model 15 (see Supplementary Information section  A for details).

Nevertheless, machine-learning approaches have noteworthy limitations compared with GCMs. Existing machine-learning models have focused on deterministic prediction, and surpass deterministic numerical weather prediction in terms of the aggregate metrics for which they are trained 3 , 4 . However, they do not produce calibrated uncertainty estimates 4 , which is essential for useful weather forecasts 1 . Deterministic machine-learning models using a mean-squared-error loss are rewarded for averaging over uncertainty, producing unrealistically blurry predictions when optimized for multi-day forecasts 3 , 13 . Unlike physical models, machine-learning models misrepresent derived (diagnostic) variables such as geostrophic wind 16 . Furthermore, although there has been some success in using machine-learning approaches on longer timescales 17 , 18 , these models have not demonstrated the ability to outperform existing GCMs.

Hybrid models that combine GCMs with machine learning are appealing because they build on the interpretability, extensibility and successful track record of traditional atmospheric models 19 , 20 . In the hybrid model approach, a machine-learning component replaces or corrects the traditional physical parameterizations of a GCM. Until now, the machine-learning component in such models has been trained ‘offline’, by learning parameterizations independently of their interaction with dynamics. These components are then inserted into an existing GCM. The lack of coupling between machine-learning components and the governing equations during training potentially causes serious problems, such as instability and climate drift 21 . So far, hybrid models have mostly been limited to idealized scenarios such as aquaplanets 22 , 23 . Under realistic conditions, machine-learning corrections have reduced some biases of very coarse GCMs 24 , 25 , 26 , but performance remains considerably worse than state-of-the-art models.

Here we present NeuralGCM, a fully differentiable hybrid GCM of Earth’s atmosphere. NeuralGCM is trained on forecasting up to 5-day weather trajectories sampled from ERA5. Differentiability enables end-to-end ‘online training’ 27 , with machine-learning components optimized in the context of interactions with the governing equations for large-scale dynamics, which we find enables accurate and stable forecasts. NeuralGCM produces physically consistent forecasts with accuracy comparable to best-in-class models across a range of timescales, from 1- to 15-day weather to decadal climate prediction.

Neural GCMs

A schematic of NeuralGCM is shown in Fig. 1 . The two key components of NeuralGCM are a differentiable dynamical core for solving the discretized governing dynamical equations and a learned physics module that parameterizes physical processes with a neural network, described in full detail in Methods , Supplementary Information sections  B and C , and Supplementary Table 1 . The dynamical core simulates large-scale fluid motion and thermodynamics under the influence of gravity and the Coriolis force. The learned physics module (Supplementary Fig. 1 ) predicts the effect of unresolved processes, such as cloud formation, radiative transport, precipitation and subgrid-scale dynamics, on the simulated fields using a neural network.

a , Overall model structure, showing how forcings F t , noise z t (for stochastic models) and inputs y t are encoded into the model state x t . The model state is fed into the dynamical core, and alongside forcings and noise into the learned physics module. This produces tendencies (rates of change) used by an implicit–explicit ordinary differential equation (ODE) solver to advance the state in time. The new model state x t +1 can then be fed back into another time step, or decoded into model predictions. b , The learned physics module, which feeds data for individual columns of the atmosphere into a neural network used to produce physics tendencies in that vertical column.

The differentiable dynamical core in NeuralGCM allows an end-to-end training approach, whereby we advance the model multiple time steps before employing stochastic gradient descent to minimize discrepancies between model predictions and reanalysis (Supplementary Information section  G.2 ). We gradually increase the rollout length from 6 hours to 5 days (Supplementary Information section  G and Supplementary Table 5 ), which we found to be critical because our models are not accurate for multi-day prediction or stable for long rollouts early in training (Supplementary Information section  H.6.2 and Supplementary Fig. 23 ). The extended back-propagation through hundreds of simulation steps enables our neural networks to take into account interactions between the learned physics and the dynamical core. We train deterministic and stochastic NeuralGCM models, each of which uses a distinct training protocol, described in full detail in Methods and Supplementary Table 4 .

We train a range of NeuralGCM models at horizontal resolutions with grid spacing of 2.8°, 1.4° and 0.7° (Supplementary Fig. 7 ). We evaluate the performance of NeuralGCM at a range of timescales appropriate for weather forecasting and climate simulation. For weather, we compare against the best-in-class conventional physics-based weather models, ECMWF’s high-resolution model (ECMWF-HRES) and ensemble prediction system (ECMWF-ENS), and two of the recent machine-learning-based approaches, GraphCast 3 and Pangu 4 . For climate, we compare against a global cloud-resolving model and Atmospheric Model Intercomparison Project (AMIP) runs.

Medium-range weather forecasting

Our evaluation set-up focuses on quantifying accuracy and physical consistency, following WeatherBench2 12 . We regrid all forecasts to a 1.5° grid using conservative regridding, and average over all 732 forecasts made at noon and midnight UTC in the year 2020, which was held-out from training data for all machine-learning models. NeuralGCM, GraphCast and Pangu compare with ERA5 as the ground truth, whereas ECMWF-ENS and ECMWF-HRES compare with the ECMWF operational analysis (that is, HRES at 0-hour lead time), to avoid penalizing the operational forecasts for different biases than ERA5.

Model accuracy

We use ECMWF’s ensemble (ENS) model as a reference baseline as it achieves the best performance across the majority of lead times 12 . We assess accuracy using (1) root-mean-squared error (RMSE), (2) root-mean-squared bias (RMSB), (3) continuous ranked probability score (CRPS) and (4) spread-skill ratio, with the results shown in Fig. 2 . We provide more in-depth evaluations including scorecards, metrics for additional variables and levels and maps in Extended Data Figs. 1 and 2 , Supplementary Information section  H and Supplementary Figs. 9 – 22 .

a , c , RMSE ( a ) and RMSB ( c ) for ECMWF-ENS, ECMWF-HRES, NeuralGCM-0.7°, NeuralGCM-ENS, GraphCast 3 and Pangu 4 on headline WeatherBench2 variables, as a percentage of the error of ECMWF-ENS. Deterministic and stochastic models are shown in solid and dashed lines respectively. e , g , CRPS relative to ECMWF-ENS ( e ) and spread-skill ratio for the ENS and NeuralGCM-ENS models ( g ). b , d , f , h , Spatial distributions of RMSE ( b ), bias ( d ), CRPS ( f ) and spread-skill ratio ( h ) for NeuralGCM-ENS and ECMWF-ENS models for 10-day forecasts of specific humidity at 700 hPa. Spatial plots of RMSE and CRPS show skill relative to a probabilistic climatology 12 with an ensemble member for each of the years 1990–2019. The grey areas indicate regions where climatological surface pressure on average is below 700 hPa.

Deterministic models that produce a single weather forecast for given initial conditions can be compared effectively using RMSE skill at short lead times. For the first 1–3 days, depending on the atmospheric variable, RMSE is minimized by forecasts that accurately track the evolution of weather patterns. At this timescale we find that NeuralGCM-0.7° and GraphCast achieve best results, with slight variations across different variables (Fig. 2a ). At longer lead times, RMSE rapidly increases owing to chaotic divergence of nearby weather trajectories, making RMSE less informative for deterministic models. RMSB calculates persistent errors over time, which provides an indication of how models would perform at much longer lead times. Here NeuralGCM models also compare favourably against previous approaches (Fig. 2c ), with notably much less bias for specific humidity in the tropics (Fig. 2d ).

Ensembles are essential for capturing intrinsic uncertainty of weather forecasts, especially at longer lead times. Beyond about 7 days, the ensemble means of ECMWF-ENS and NeuralGCM-ENS forecasts have considerably lower RMSE than the deterministic models, indicating that these models better capture the average of possible weather. A better metric for ensemble models is CRPS, which is a proper scoring rule that is sensitive to full marginal probability distributions 28 . Our stochastic model (NeuralGCM-ENS) running at 1.4° resolution has lower error compared with ECMWF-ENS across almost all variables, lead times and vertical levels for ensemble-mean RMSE, RSMB and CRPS (Fig. 2a,c,e and Supplementary Information section  H ), with similar spatial patterns of skill (Fig. 2b,f ). Like ECMWF-ENS, NeuralGCM-ENS has a spread-skill ratio of approximately one (Fig. 2d ), which is a necessary condition for calibrated forecasts 29 .

An important characteristic of forecasts is their resemblance to realistic weather patterns. Figure 3 shows a case study that illustrates the performance of NeuralGCM on three types of important weather phenomenon: tropical cyclones, atmospheric rivers and the Intertropical Convergence Zone. Figure 3a shows that all the machine-learning models make significantly blurrier forecasts than the source data ERA5 and physics-based ECMWF-HRES forecast, but NeuralCGM-0.7° outperforms the pure machine-learning models, despite its coarser resolution (0.7° versus 0.25° for GraphCast and Pangu). Blurry forecasts correspond to physically inconsistent atmospheric conditions and misrepresent extreme weather. Similar trends hold for other derived variables of meteorological interest (Supplementary Information section  H.2 ). Ensemble-mean predictions, from both NeuralGCM and ECMWF, are closer to ERA5 in an average sense, and thus are inherently smooth at long lead times. In contrast, as shown in Fig. 3 and in Supplementary Information section  H.3 , individual realizations from the ECMWF and NeuralGCM ensembles remain sharp, even at long lead times. Like ECMWF-ENS, NeuralGCM-ENS produces a statistically representative range of future weather scenarios for each weather phenomenon, despite its eight-times-coarser resolution.

All forecasts are initialized at 2020-08-22T12z, chosen to highlight Hurricane Laura, the most damaging Atlantic hurricane of 2020. a , Specific humidity at 700 hPa for 1-day, 5-day and 10-day forecasts over North America and the Northeast Pacific Ocean from ERA5 14 , ECMWF-HRES, NeuralGCM-0.7°, ECMWF-ENS (mean), NeuralGCM-ENS (mean), GraphCast 3 and Pangu 4 . b , Forecasts from individual ensemble members from ECMWF-ENS and NeuralGCM-ENS over regions of interest, including predicted tracks of Hurricane Laura from each of the 50 ensemble members (Supplementary Information section  I.2 ). The track from ERA5 is plotted in black.

We can quantify the blurriness of different forecast models via their power spectra. Supplementary Figs. 17 and 18 show that the power spectra of NeuralCGM-0.7° is consistently closer to ERA5 than the other machine-learning forecast methods, but is still blurrier than ECMWF’s physical forecasts. The spectra of NeuralGCM forecasts is also roughly constant over the forecast period, in stark contrast to GraphCast, which worsens with lead time. The spectrum of NeuralGCM becomes more accurate with increased resolution (Supplementary Fig. 22 ), which suggests the potential for further improvements of NeuralGCM models trained at higher resolutions.

Water budget

In NeuralGCM, advection is handled by the dynamical core, while the machine-learning parameterization models local processes within vertical columns of the atmosphere. Thus, unlike pure machine-learning methods, local sources and sinks can be isolated from tendencies owing to horizontal transport and other resolved dynamics (Supplementary Fig. 3 ). This makes our results more interpretable and facilitates the diagnosis of the water budget. Specifically, we diagnose precipitation minus evaporation (Supplementary Information section  H.5 ) rather than directly predicting these as in machine-learning-based approaches 3 . For short weather forecasts, the mean of precipitation minus evaporation has a realistic spatial distribution that is very close to ERA5 data (Extended Data Fig. 4c–e ). The precipitation-minus-evaporation rate distribution of NeuralGCM-0.7° closely matches the ERA5 distribution in the extratropics (Extended Data Fig. 4b ), although it underestimates extreme events in the tropics (Extended Data Fig. 4a ). It is noted that the current version of NeuralGCM directly predicts tendencies for an atmospheric column, and thus cannot distinguish between precipitation and evaporation.

Geostrophic wind balance

We examined the extent to which NeuralGCM, GraphCast and ECMWF-HRES capture the geostrophic wind balance, the near-equilibrium between the dominant forces that drive large-scale dynamics in the mid-latitudes 30 . A recent study 16 highlighted that Pangu misrepresents the vertical structure of the geostrophic and ageostrophic winds and noted a deterioration at longer lead times. Similarly, we observe that GraphCast shows an error that worsens with lead time. In contrast, NeuralGCM more accurately depicts the vertical structure of the geostrophic and ageostrophic winds, as well as their ratio, compared with GraphCast across various rollouts, when compared against ERA5 data (Extended Data Fig. 3 ). However, ECMWF-HRES still shows a slightly closer alignment to ERA5 data than NeuralGCM does. Within NeuralGCM, the representation of the geostrophic wind’s vertical structure only slightly degrades in the initial few days, showing no noticeable changes thereafter, particularly beyond day 5.

Generalizing to unseen data

Physically consistent weather models should still perform well for weather conditions for which they were not trained. We expect that NeuralGCM may generalize better than machine-learning-only atmospheric models, because NeuralGCM employs neural networks that act locally in space, on individual vertical columns of the atmosphere. To explore this hypothesis, we compare versions of NeuralCGM-0.7° and GraphCast trained to 2017 on 5 years of weather forecasts beyond the training period (2018–2022) in Supplementary Fig. 36 . Unlike GraphCast, NeuralGCM does not show a clear trend of increasing error when initialized further into the future from the training data. To extend this test beyond 5 years, we trained a NeuralGCM-2.8° model using only data before 2000, and tested its skill for over 21 unseen years (Supplementary Fig. 35 ).

Climate simulations

Although our deterministic NeuralGCM models are trained to predict weather up to 3 days ahead, they are generally capable of simulating the atmosphere far beyond medium-range weather timescales. For extended climate simulations, we prescribe historical sea surface temperature (SST) and sea-ice concentration. These simulations feature many emergent phenomena of the atmosphere on timescales from months to decades.

For climate simulations with NeuralGCM, we use 2.8° and 1.4° deterministic models, which are relatively inexpensive to train (Supplementary Information section  G.7 ) and allow us to explore a larger parameter space to find stable models. Previous studies found that running extended simulations with hybrid models is challenging due to numerical instabilities and climate drift 21 . To quantify stability in our selected models, we run multiple initial conditions and report how many of them finish without instability.

Seasonal cycle and emergent phenomena

To assess the capability of NeuralGCM to simulate various aspects of the seasonal cycle, we run 2-year simulations with NeuralGCM-1.4°. for 37 different initial conditions spaced every 10 days for the year 2019. Out of these 37 initial conditions, 35 successfully complete the full 2 years without instability; for case studies of instability, see Supplementary Information section  H.7 , and Supplementary Figs. 26 and 27 . We compare results from NeuralGCM-1.4° for 2020 with ERA5 data and with outputs from the X-SHiELD global cloud-resolving model, which is coupled to an ocean model nudged towards reanalysis 31 . This X-SHiELD run has been used as a target for training machine-learning climate models 24 . For comparison, we evaluate models after regridding predictions to 1.4° resolution. This comparison slightly favours NeuralGCM because NeuralGCM was tuned to match ERA5, but the discrepancy between ERA5 and the actual atmosphere is small relative to model error.

Figure 4a shows the temporal variation of the global mean temperature to 2020, as captured by 35 simulations from NeuralGCM, in comparison with the ERA5 reanalysis and standard climatology benchmarks. The seasonality and variability of the global mean temperature from NeuralGCM are quantitatively similar to those observed in ERA5. The ensemble-mean temperature RMSE for NeuralGCM stands at 0.16 K when benchmarked against ERA5, which is a significant improvement over the climatology’s RMSE of 0.45 K. We find that NeuralGCM accurately simulates the seasonal cycle, as evidenced by metrics such as the annual cycle of the global precipitable water (Supplementary Fig. 30a ) and global total kinetic energy (Supplementary Fig. 30b ). Furthermore, the model captures essential atmospheric dynamics, including the Hadley circulation and the zonal-mean zonal wind (Supplementary Fig. 28 ), as well as the spatial patterns of eddy kinetic energy in different seasons (Supplementary Fig. 31 ), and the distinctive seasonal behaviours of monsoon circulation (Supplementary Fig. 29 ; additional details are provided in Supplementary Information section  I.1 ).

a , Global mean temperature for ERA5 14 (orange), 1990–2019 climatology (black) and NeuralGCM-1.4° (blue) for 2020 using 35 simulations initialized every 10 days during 2019 (thick line, ensemble mean; thin lines, different initial conditions). b , Yearly global mean temperature for ERA5 (orange), mean over 22 CMIP6 AMIP experiments 34 (violet; model details are in Supplementary Information section  I.3 ) and NeuralGCM-2.8° for 22 AMIP-like simulations with prescribed SST initialized every 10 days during 1980 (thick line, ensemble mean; thin lines, different initial conditions). c , The RMSB of the 850-hPa temperature averaged between 1981 and 2014 for 22 NeuralGCM-2.8° AMIP runs (labelled NGCM), 22 CMIP6 AMIP experiments (labelled AMIP) and debiased 22 CMIP6 AMIP experiments (labelled AMIP*; bias was removed by removing the 850-hPa global temperature bias). In the box plots, the red line represents the median. The box delineates the first to third quartiles; the whiskers extend to 1.5 times the interquartile range (Q1 − 1.5IQR and Q3 + 1.5IQR), and outliers are shown as individual dots. d , Vertical profiles of tropical (20° S–20° N) temperature trends for 1981–2014. Orange, ERA5; black dots, Radiosonde Observation Correction using Reanalyses (RAOBCORE) 41 ; blue dots, mean trends for NeuralGCM; purple dots, mean trends from CMIP6 AMIP runs (grey and black whiskers, 25th and 75th percentiles for NeuralGCM and CMIP6 AMIP runs, respectively). e – g , Tropical cyclone tracks for ERA5 ( e ), NeuralGCM-1.4° ( f ) and X-SHiELD 31 ( g ). h – k , Mean precipitable water for ERA5 ( h ) and the precipitable water bias in NeuralGCM-1.4° ( i ), initialized 90 days before mid-January 2020 similarly to X-SHiELD, X-SHiELD ( j ) and climatology ( k ; averaged between 1990 and 2019). In d – i , quantities are calculated between mid-January 2020 and mid-January 2021 and all models were regridded to a 256 × 128 Gaussian grid before computation and tracking.

Next, we compare the annual biases of a single NeuralGCM realization with a single realization of X-SHiELD (the only one available), both initiated in mid-October 2019. We consider 19 January 2020 to 17 January 2021, the time frame for which X-SHiELD data are available. Global cloud-resolving models, such as X-SHiELD, are considered state of the art, especially for simulating the hydrological cycle, owing to their resolution being capable of resolving deep convection 32 . The annual bias in precipitable water for NeuralGCM (RMSE of 1.09 mm) is substantially smaller than the biases of both X-SHiELD (RMSE of 1.74 mm) and climatology (RMSE of 1.36 mm; Fig. 4i–k ). Moreover, NeuralGCM shows a lower temperature bias in the upper and lower troposphere than X-SHiELD (Extended Data Fig. 6 ). We also indirectly compare precipitation bias in X-SHiELD with precipitation-minus-evaporation bias in NeuralGCM-1.4°, which shows slightly larger bias and grid-scale artefacts for NeuralGCM (Extended Data Fig. 5 ).

Finally, to assess the capability of NeuralGCM to generate tropical cyclones in an annual model integration, we use the tropical cyclone tracker TempestExtremes 33 , as described in Supplementary Information section   I.2 , Supplementary Fig. 34 and Supplementary Table 6 . Figure 4e–g shows that NeuralGCM, even at a coarse resolution of 1.4°, produces realistic trajectories and counts of tropical cyclone (83 versus 86 in ERA5 for the corresponding period), whereas X-SHiELD, when regridded to 1.4° resolution, substantially underestimates the tropical cyclone count (40). Additional statistical analyses of tropical cyclones can be found in Extended Data Figs. 7 and 8 .

Decadal simulations

To assess the capability of NeuralGCM to simulate historical temperature trends, we conduct AMIP-like simulations over a duration of 40 years with NeuralGCM-2.8°. Out of 37 different runs with initial conditions spaced every 10 days during the year 1980, 22 simulations were stable for the entire 40-year period, and our analysis focuses on these results. We compare with 22 simulations run with prescribed SST from the Coupled Model Intercomparison Project Phase 6 (CMIP6) 34 , listed in Supplementary Information section  I.3 .

We find that all 40-year simulations of NeuralGCM, as well as the mean of the 22 AMIP runs, accurately capture the global warming trends observed in ERA5 data (Fig. 4b ). There is a strong correlation in the year-to-year temperature trends with ERA5 data, suggesting that NeuralGCM effectively captures the impact of SST forcing on climate. When comparing spatial biases averaged over 1981–2014, we find that all 22 NeuralGCM-2.8° runs have smaller bias than the CMIP6 AMIP runs, and this result remains even when removing the global temperature bias in CMIP6 AMIP runs (Fig. 4c and Supplementary Figs. 32 and 33 ).

Next, we investigated the vertical structure of tropical warming trends, which climate models tend to overestimate in the upper troposphere 35 . As shown in Fig. 4d , the trends, calculated by linear regression, of NeuralGCM are closer to ERA5 than those of AMIP runs. In particular, the bias in the upper troposphere is reduced. However, NeuralGCM does show a wider spread in its predictions than the AMIP runs, even at levels near the surface where temperatures are typically more constrained by prescribed SST.

Lastly, we evaluated NeuralGCM’s capability to generalize to unseen warmer climates by conducting AMIP simulations with increased SST (Supplementary Information section  I.4.2 ). We find that NeuralGCM shows some of the robust features of climate warming response to modest SST increases (+1 K and +2 K); however, for more substantial SST increases (+4 K), NeuralGCM’s response diverges from expectations (Supplementary Fig. 37 ). In addition, AMIP simulations with increased SST show climate drift, underscoring NeuralGCM’s limitations in this context (Supplementary Fig. 38 ).

NeuralGCM is a differentiable hybrid atmospheric model that combines the strengths of traditional GCMs with machine learning for weather forecasting and climate simulation. To our knowledge, NeuralGCM is the first machine-learning-based model to make accurate ensemble weather forecasts, with better CRPS than state-of-the-art physics-based models. It is also, to our knowledge, the first hybrid model that achieves comparable spatial bias to global cloud-resolving models, can simulate realistic tropical cyclone tracks and can run AMIP-like simulations with realistic historical temperature trends. Overall, NeuralGCM demonstrates that incorporating machine learning is a viable alternative to building increasingly detailed physical models 32 for improving GCMs.

Compared with traditional GCMs with similar skill, NeuralGCM is computationally efficient and low complexity. NeuralGCM runs at 8- to 40-times-coarser horizontal resolution than ECMWF’s Integrated Forecasting System and global cloud-resolving models, which enables 3 to 5 orders of magnitude savings in computational resources. For example, NeuralGCM-1.4° simulates 70,000 simulation days in 24 hours using a single tensor-processing-unit versus 19 simulated days on 13,824 central-processing-unit cores with X-SHiELD (Extended Data Table 1 ). This can be leveraged for previously impractical tasks such as large ensemble forecasting. NeuralGCM’s dynamical core uses global spectral methods 36 , and learned physics is parameterized with fully connected neural networks acting on single vertical columns. Substantial headroom exists to pursue higher accuracy using advanced numerical methods and machine-learning architectures.

Our results provide strong evidence for the disputed hypothesis 37 , 38 , 39 that learning to predict short-term weather is an effective way to tune parameterizations for climate. NeuralGCM models trained on 72-hour forecasts are capable of realistic multi-year simulation. When provided with historical SSTs, they capture essential atmospheric dynamics such as seasonal circulation, monsoons and tropical cyclones. However, we will probably need alternative training strategies 38 , 39 to learn important processes for climate with subtle impacts on weather timescales, such as a cloud feedback.

The NeuralGCM approach is compatible with incorporating either more physics or more machine learning, as required for operational weather forecasts and climate simulations. For weather forecasting, we expect that end-to-end learning 40 with observational data will allow for better and more relevant predictions, including key variables such as precipitation. Such models could include neural networks acting as corrections to traditional data assimilation and model diagnostics. For climate projection, NeuralGCM will need to be reformulated to enable coupling with other Earth-system components (for example, ocean and land), and integrating data on the atmospheric chemical composition (for example, greenhouse gases and aerosols). There are also research challenges common to current machine-learning-based climate models 19 , including the capability to simulate unprecedented climates (that is, generalization), adhering to physical constraints, and resolving numerical instabilities and climate drift. NeuralGCM’s flexibility to incorporate physics-based models (for example, radiation) offers a promising avenue to address these challenges.

Models based on physical laws and empirical relationships are ubiquitous in science. We believe the differentiable hybrid modelling approach of NeuralGCM has the potential to transform simulation for a wide range of applications, such as materials discovery, protein folding and multiphysics engineering design.

Differentiable atmospheric model

NeuralGCM combines components of the numerical solver and flexible neural network parameterizations. Simulation in time is carried out in a coordinate system suitable for solving the dynamical equations of the atmosphere, describing large-scale fluid motion and thermodynamics under the influence of gravity and the Coriolis force.

Our differentiable dynamical core is implemented in JAX, a library for high-performance code in Python that supports automatic differentiation 42 . The dynamical core solves the hydrostatic primitive equations with moisture, using a horizontal pseudo-spectral discretization and vertical sigma coordinates 36 , 43 . We evolve seven prognostic variables: vorticity and divergence of horizontal wind, temperature, surface pressure, and three water species (specific humidity, and specific ice and liquid cloud water content).

Our learned physics module uses the single-column approach of GCMs 2 , whereby information from only a single atmospheric column is used to predict the impact of unresolved processes occurring within that column. These effects are predicted using a fully connected neural network with residual connections, with weights shared across all atmospheric columns (Supplementary Information section  C.4 ).

The inputs to the neural network include the prognostic variables in the atmospheric column, total incident solar radiation, sea-ice concentration and SST (Supplementary Information section  C.1 ). We also provide horizontal gradients of the prognostic variables, which we found improves performance 44 . All inputs are standardized to have zero mean and unit variance using statistics precomputed during model initialization. The outputs are the prognostic variable tendencies scaled by the fixed unconditional standard deviation of the target field (Supplementary Information section  C.5 ).

To interface between ERA5 14 data stored in pressure coordinates and the sigma coordinate system of our dynamical core, we introduce encoder and decoder components (Supplementary Information section  D ). These components perform linear interpolation between pressure levels and sigma coordinate levels. We additionally introduce learned corrections to both encoder and decoder steps (Supplementary Figs. 4–6 ), using the same column-based neural network architecture as the learned physics module. Importantly, the encoder enables us to eliminate the gravity waves from initialization shock 45 , which otherwise contaminate forecasts.

Figure 1a shows the sequence of steps that NeuralGCM takes to make a forecast. First, it encodes ERA5 data at t  =  t 0 on pressure levels to initial conditions on sigma coordinates. To perform a time step, the dynamical core and learned physics (Fig. 1b ) then compute tendencies, which are integrated in time using an implicit–explicit ordinary differential equation solver 46 (Supplementary Information section  E and Supplementary Table 2 ). This is repeated to advance the model from t  =  t 0 to t  =  t final . Finally, the decoder converts predictions back to pressure levels.

The time-step size of the ODE solver (Supplementary Table 3 ) is limited by the Courant–Friedrichs–Lewy condition on dynamics, and can be small relative to the timescale of atmospheric change. Evaluating learned physics is approximately 1.5 times as expensive as a time step of the dynamical core. Accordingly, following the typical practice for GCMs, we hold learned physics tendencies constant for multiple ODE time steps to reduce computational expense, typically corresponding to 30 minutes of simulation time.

Deterministic and stochastic models

We train deterministic NeuralGCM models using a combination of three loss functions (Supplementary Information section  G.4 ) to encourage accuracy and sharpness while penalizing bias. During the main training phase, all losses are defined in a spherical harmonics basis. We use a standard mean squared error loss for prompting accuracy, modified to progressively filter out contributions from higher total wavenumbers at longer lead times (Supplementary Fig. 8 ). This filtering approach tackles the ‘double penalty problem’ 47 as it prevents the model from being penalized for predicting high-wavenumber features in incorrect locations at later times, especially beyond the predictability horizon. A second loss term encourages the spectrum to match the training data using squared loss on the total wavenumber spectrum of prognostic variables. These first two losses are evaluated on both sigma and pressure levels. Finally, a third loss term discourages bias by adding mean squared error on the batch-averaged mean amplitude of each spherical harmonic coefficient. For analysis of the impact that various loss functions have, refer to Supplementary Information section  H.6.1 , and Supplementary Figs. 23 and 24 . The combined action of the three training losses allow the resulting models trained on 3-day rollouts to remain stable during years-to-decades-long climate simulations. Before final evaluations, we perform additional fine-tuning of just the decoder component on short rollouts of 24 hours (Supplementary Information section  G.5 ).

Stochastic NeuralGCM models incorporate inherent randomness in the form of additional random fields passed as inputs to neural network components. Our stochastic loss is based on the CRPS 28 , 48 , 49 . CRPS consists of mean absolute error that encourages accuracy, balanced by a similar term that encourages ensemble spread. For each variable we use a sum of CRPS in grid space and CRPS in the spherical harmonic basis below a maximum cut-off wavenumber (Supplementary Information section  G.6 ). We compute CRPS on rollout lengths from 6 hours to 5 days. As illustrated in Fig. 1 , we inject noise to the learned encoder and the learned physics module by sampling from Gaussian random fields with learned spatial and temporal correlation (Supplementary Information section  C.2 and Supplementary Fig. 2 ). For training, we generate two ensemble members per forecast, which suffices for an unbiased estimate of CRPS.

Data availability

For training and evaluating the NeuralGCM models, we used the publicly available ERA5 dataset 14 , originally downloaded from https://cds.climate.copernicus.eu/ and available via Google Cloud Storage in Zarr format at gs://gcp-public-data-arco-era5/ar/full_37-1h-0p25deg-chunk-1.zarr-v3. To compare NeuralGCM with operational and data-driven weather models, we used forecast datasets distributed as part of WeatherBench2 12 at https://weatherbench2.readthedocs.io/en/latest/data-guide.html , to which we have added NeuralGCM forecasts for 2020. To compare NeuralGCM with atmospheric models in climate settings, we used CMIP6 data available at https://catalog.pangeo.io/browse/master/climate/ , as well as X-SHiELD 24 outputs available on Google Cloud storage in a ‘requester pays’ bucket at gs://ai2cm-public-requester-pays/C3072-to-C384-res-diagnostics. The Radiosonde Observation Correction using Reanalyses (RAOBCORE) V1.9 that was used as reference tropical temperature trends was downloaded from https://webdata.wolke.img.univie.ac.at/haimberger/v1.9/ . Base maps use freely available data from https://www.naturalearthdata.com/downloads/ .

Code availability

The NeuralGCM code base is separated into two open source projects: Dinosaur and NeuralGCM, both publicly available on GitHub at https://github.com/google-research/dinosaur (ref. 50 ) and https://github.com/google-research/neuralgcm (ref. 51 ). The Dinosaur package implements a differentiable dynamical core used by NeuralGCM, whereas the NeuralGCM package provides machine-learning models and checkpoints of trained models. Evaluation code for NeuralGCM weather forecasts is included in WeatherBench2 12 , available at https://github.com/google-research/weatherbench2 (ref. 52 ).

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Acknowledgements

We thank A. Kwa, A. Merose and K. Shah for assistance with data acquisition and handling; L. Zepeda-Núñez for feedback on the paper; and J. Anderson, C. Van Arsdale, R. Chemke, G. Dresdner, J. Gilmer, J. Hickey, N. Lutsko, G. Nearing, A. Paszke, J. Platt, S. Ponda, M. Pritchard, D. Rothenberg, F. Sha, T. Schneider and O. Voicu for discussions.

Author information

These authors contributed equally: Dmitrii Kochkov, Janni Yuval, Ian Langmore, Peter Norgaard, Jamie Smith, Stephan Hoyer

Authors and Affiliations

Google Research, Mountain View, CA, USA

Dmitrii Kochkov, Janni Yuval, Ian Langmore, Peter Norgaard, Jamie Smith, Griffin Mooers, James Lottes, Stephan Rasp, Michael P. Brenner & Stephan Hoyer

Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA

Milan Klöwer

European Centre for Medium-Range Weather Forecasts, Reading, UK

Peter Düben & Sam Hatfield

Google DeepMind, London, UK

Peter Battaglia, Alvaro Sanchez-Gonzalez & Matthew Willson

School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

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Contributions

D.K., J.Y., I.L., P.N., J.S. and S. Hoyer contributed equally to this work. D.K., J.Y., I.L., P.N., J.S., G.M., J.L. and S. Hoyer wrote the code. D.K., J.Y., I.L., P.N., G.M. and S. Hoyer trained models and analysed the data. M.P.B. and S. Hoyer managed and oversaw the research project. M.K., S.R., P.D., S. Hatfield, P.B. and M.P.B. contributed technical advice and ideas. M.W. ran experiments with GraphCast for comparison with NeuralGCM. A.S.-G. assisted with data preparation. D.K., J.Y., I.L., P.N. and S. Hoyer wrote the paper. All authors gave feedback and contributed to editing the paper.

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Correspondence to Dmitrii Kochkov , Janni Yuval or Stephan Hoyer .

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Competing interests.

D.K., J.Y., I.L., P.N., J.S., J.L., S.R., P.B., A.S.-G., M.W., M.P.B. and S. Hoyer are employees of Google. S. Hoyer, D.K., I.L., J.Y., G.M., P.N., J.S. and M.B. have filed international patent application PCT/US2023/035420 in the name of Google LLC, currently pending, relating to neural general circulation models.

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Extended data figures and tables

Extended data fig. 1 maps of bias for neuralgcm-ens and ecmwf-ens forecasts..

Bias is averaged over all forecasts initialized in 2020.

Extended Data Fig. 2 Maps of spread-skill ratio for NeuralGCM-ENS and ECMWF-ENS forecasts.

Spread-skill ratio is averaged over all forecasts initialized in 2020.

Extended Data Fig. 3 Geostrophic balance in NeuralGCM, GraphCast 3 and ECMWF-HRES.

Vertical profiles of the extratropical intensity (averaged between latitude 30°–70° in both hemispheres) and over all forecasts initialized in 2020 of (a,d,g) geostrophic wind, (b,e,h) ageostrophic wind and (c,f,i) the ratio of the intensity of ageostrophic wind over geostrophic wind for ERA5 (black continuous line in all panels), (a,b,c) NeuralGCM-0.7°, (d,e,f) GraphCast and (g,h,i) ECMWF-HRES at lead times of 1 day, 5 days and 10 days.

Extended Data Fig. 4 Precipitation minus evaporation calculated from the third day of weather forecasts.

(a) Tropical (latitudes −20° to 20°) precipitation minus evaporation (P minus E) rate distribution, (b) Extratropical (latitudes 30° to 70° in both hemispheres) P minus E, (c) mean P minus E for 2020 ERA5 14 and (d) NeuralGCM-0.7° (calculated from the third day of forecasts and averaged over all forecasts initialized in 2020), (e) the bias between NeuralGCM-0.7° and ERA5, (f-g) Snapshot of daily precipitation minus evaporation for 2020-01-04 for (f) NeuralGCM-0.7° (forecast initialized on 2020-01-02) and (g) ERA5.

Extended Data Fig. 5 Indirect comparison between precipitation bias in X-SHiELD and precipitation minus evaporation bias in NeuralGCM-1.4°.

Mean precipitation calculated between 2020-01-19 and 2021-01-17 for (a) ERA5 14 (c) X-SHiELD 31 and the biases in (e) X-SHiELD and (g) climatology (ERA5 data averaged over 1990-2019). Mean precipitation minus evaporation calculated between 2020-01-19 and 2021-01-17 for (b) ERA5 (d) NeuralGCM-1.4° (initialized in October 18th 2019) and the biases in (f) NeuralGCM-1.4° and (h) climatology (data averaged over 1990–2019).

Extended Data Fig. 6 Yearly temperature bias for NeuralGCM and X-SHiELD 31 .

Mean temperature between 2020-01-19 to 2020-01-17 for (a) ERA5 at 200hPa and (b) 850hPa. (c,d) the bias in the temperature for NeuralGCM-1.4°, (e,f) the bias in X-SHiELD and (g,h) the bias in climatology (calculated from 1990–2019). NeuralGCM-1.4° was initialized in 18th of October (similar to X-SHiELD).

Extended Data Fig. 7 Tropical Cyclone densities and annual regional counts.

(a) Tropical Cyclone (TC) density from ERA5 14 data spanning 1987–2020. (b) TC density from NeuralGCM-1.4° for 2020, generated using 34 different initial conditions all initialized in 2019. (c) Box plot depicting the annual number of TCs across different regions, based on ERA5 data (1987–2020), NeuralGCM-1.4° for 2020 (34 initial conditions), and orange markers show ERA5 for 2020. In the box plots, the red line represents the median; the box delineates the first to third quartiles; the whiskers extend to 1.5 times the interquartile range (Q1 − 1.5IQR and Q3 + 1.5IQR), and outliers are shown as individual dots. Each year is defined from January 19th to January 17th of the following year, aligning with data availability from X-SHiELD. For NeuralGCM simulations, the 3 initial conditions starting in January 2019 exclude data for January 17th, 2021, as these runs spanned only two years.

Extended Data Fig. 8 Tropical Cyclone maximum wind distribution in NeuralGCM vs. ERA5 14 .

Number of Tropical Cyclones (TCs) as a function of maximum wind speed at 850hPa across different regions, based on ERA5 data (1987–2020; in orange), and NeuralGCM-1.4° for 2020 (34 initial conditions; in blue). Each year is defined from January 19th to January 17th of the following year, aligning with data availability from X-SHiELD. For NeuralGCM simulations, the 3 initial conditions starting in January 2019 exclude data for January 17th, 2021, as these runs spanned only two years.

Supplementary information

Supplementary information.

Supplementary Information (38 figures, 6 tables): (A) Lines of code in atmospheric models; (B) Dynamical core of NeuralGCM; (C) Learned physics of NeuralGCM; (D) Encoder and decoder of NeuralGCM; (E) Time integration; (F) Evaluation metrics; (G) Training; (H) Additional weather evaluations; (I) Additional climate evaluations.

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Kochkov, D., Yuval, J., Langmore, I. et al. Neural general circulation models for weather and climate. Nature 632 , 1060–1066 (2024). https://doi.org/10.1038/s41586-024-07744-y

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1. Introduction

2. theoretical background, 2.1. review of the lstm and gru architecture, lstm and gru networks, 2.2. attention mechanism, 2.3. time series forecasting methods, 2.3.1. autoregressive integrated moving averages (arimas), 2.3.2. xgboost (extreme gradient boost), 2.3.3. facebook prophet, 2.4. other recent advancements in the area, 3. preliminary considerations and development of the approach, 3.1. historical context and progression, 3.2. data collection, exploration, and preparation, 3.2.1. train and test split, 3.2.2. data shaping for lstm and gru models, 3.3. preprocessing and normalization, 3.4. model evaluation, 3.5. the architectural diagram, 4. numerical results, 4.1. data preprocessing and exploratory analysis, 4.2. hyperparameter selection process.

• Units: The optimization strategy sets the number of units in each LSTM and GRU model to 128 and 64 in the first and second layers, respectively.
• Batch size: For tuning the model, the batch size is set to 1.
• Learning rate: The learning rate of the Adam optimizer is set at 0.1.
• Dropout layer: During model training, it is common to observe a pattern where the model performs well on the training data but fails to replicate this success on the testing and validation data. This discrepancy, often due to overfitting, is a major concern, especially in deep learning models that require a substantial amount of data for training. Dropout is a simple but effective regularization strategy used in neural networks to mitigate this overfitting problem. The cells of the recurrent neural network are dropped at random. The dropout rate is around 0.2.

4.3. Results of the Models

4.3.1. apple stock prediction, 4.3.2. google stock prediction, 4.3.3. microsoft stock prediction, 4.3.4. amazon stock prediction, 4.4. predicted risk–return tradeoff, 5. additional experiments and validations, 5.1. performance of four selected models on apple stock, 5.2. performance of four selected models on amazon stock, 5.3. performance of four selected models on google stock, 5.4. performance of four selected models on microsoft stock, 5.5. discussion: forecasting accuracy, 5.6. implications of this research, 6. discussions, 6.1. contributions, 6.2. limitations of the study, 6.3. future research, 7. conclusions, author contributions, data availability statement, acknowledgments, conflicts of interest.

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Chang, V.; Xu, Q.A.; Chidozie, A.; Wang, H. Predicting Economic Trends and Stock Market Prices with Deep Learning and Advanced Machine Learning Techniques. Electronics 2024 , 13 , 3396. https://doi.org/10.3390/electronics13173396

Chang V, Xu QA, Chidozie A, Wang H. Predicting Economic Trends and Stock Market Prices with Deep Learning and Advanced Machine Learning Techniques. Electronics . 2024; 13(17):3396. https://doi.org/10.3390/electronics13173396

Chang, Victor, Qianwen Ariel Xu, Anyamele Chidozie, and Hai Wang. 2024. "Predicting Economic Trends and Stock Market Prices with Deep Learning and Advanced Machine Learning Techniques" Electronics 13, no. 17: 3396. https://doi.org/10.3390/electronics13173396

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When A.I.’s Output Is a Threat to A.I. Itself

As A.I.-generated data becomes harder to detect, it’s increasingly likely to be ingested by future A.I., leading to worse results.

By Aatish Bhatia

Aatish Bhatia interviewed A.I. researchers, studied research papers and fed an A.I. system its own output.

The internet is becoming awash in words and images generated by artificial intelligence.

Sam Altman, OpenAI’s chief executive, wrote in February that the company generated about 100 billion words per day — a million novels’ worth of text, every day, an unknown share of which finds its way onto the internet.

A.I.-generated text may show up as a restaurant review, a dating profile or a social media post. And it may show up as a news article, too: NewsGuard, a group that tracks online misinformation, recently identified over a thousand websites that churn out error-prone A.I.-generated news articles .

In reality, with no foolproof methods to detect this kind of content, much will simply remain undetected.

All this A.I.-generated information can make it harder for us to know what’s real. And it also poses a problem for A.I. companies. As they trawl the web for new data to train their next models on — an increasingly challenging task — they’re likely to ingest some of their own A.I.-generated content, creating an unintentional feedback loop in which what was once the output from one A.I. becomes the input for another.

In the long run, this cycle may pose a threat to A.I. itself. Research has shown that when generative A.I. is trained on a lot of its own output, it can get a lot worse.

Here’s a simple illustration of what happens when an A.I. system is trained on its own output, over and over again:

This is part of a data set of 60,000 handwritten digits.

When we trained an A.I. to mimic those digits, its output looked like this.

This new set was made by an A.I. trained on the previous A.I.-generated digits. What happens if this process continues?

After 20 generations of training new A.I.s on their predecessors’ output, the digits blur and start to erode.

After 30 generations, they converge into a single shape.

While this is a simplified example, it illustrates a problem on the horizon.

Imagine a medical-advice chatbot that lists fewer diseases that match your symptoms, because it was trained on a narrower spectrum of medical knowledge generated by previous chatbots. Or an A.I. history tutor that ingests A.I.-generated propaganda and can no longer separate fact from fiction.

Just as a copy of a copy can drift away from the original, when generative A.I. is trained on its own content, its output can also drift away from reality, growing further apart from the original data that it was intended to imitate.

In a paper published last month in the journal Nature, a group of researchers in Britain and Canada showed how this process results in a narrower range of A.I. output over time — an early stage of what they called “model collapse.”

The eroding digits we just saw show this collapse. When untethered from human input, the A.I. output dropped in quality (the digits became blurry) and in diversity (they grew similar).

How an A.I. that draws digits “collapses” after being trained on its own output

If only some of the training data were A.I.-generated, the decline would be slower or more subtle. But it would still occur, researchers say, unless the synthetic data was complemented with a lot of new, real data.

Degenerative A.I.

In one example, the researchers trained a large language model on its own sentences over and over again, asking it to complete the same prompt after each round.

When they asked the A.I. to complete a sentence that started with “To cook a turkey for Thanksgiving, you…,” at first, it responded like this:

Initial A.I. output

Even at the outset, the A.I. “hallucinates.” But when the researchers further trained it on its own sentences, it got a lot worse…

After two generations, it started simply printing long lists.

And after four generations, it began to repeat phrases incoherently.

“The model becomes poisoned with its own projection of reality,” the researchers wrote of this phenomenon.

This problem isn’t just confined to text. Another team of researchers at Rice University studied what would happen when the kinds of A.I. that generate images are repeatedly trained on their own output — a problem that could already be occurring as A.I.-generated images flood the web.

They found that glitches and image artifacts started to build up in the A.I.’s output, eventually producing distorted images with wrinkled patterns and mangled fingers.

When A.I. image models are trained on their own output, they can produce distorted images, mangled fingers or strange patterns.

A.I.-generated images by Sina Alemohammad and others .

“You’re kind of drifting into parts of the space that are like a no-fly zone,” said Richard Baraniuk , a professor who led the research on A.I. image models.

The researchers found that the only way to stave off this problem was to ensure that the A.I. was also trained on a sufficient supply of new, real data.

While selfies are certainly not in short supply on the internet, there could be categories of images where A.I. output outnumbers genuine data, they said.

For example, A.I.-generated images in the style of van Gogh could outnumber actual photographs of van Gogh paintings in A.I.’s training data, and this may lead to errors and distortions down the road. (Early signs of this problem will be hard to detect because the leading A.I. models are closed to outside scrutiny, the researchers said.)

Why collapse happens

All of these problems arise because A.I.-generated data is often a poor substitute for the real thing.

This is sometimes easy to see, like when chatbots state absurd facts or when A.I.-generated hands have too many fingers.

But the differences that lead to model collapse aren’t necessarily obvious — and they can be difficult to detect.

When generative A.I. is “trained” on vast amounts of data, what’s really happening under the hood is that it is assembling a statistical distribution — a set of probabilities that predicts the next word in a sentence, or the pixels in a picture.

For example, when we trained an A.I. to imitate handwritten digits, its output could be arranged into a statistical distribution that looks like this:

Distribution of A.I.-generated data

Examples of initial A.I. output:

The distribution shown here is simplified for clarity.

The peak of this bell-shaped curve represents the most probable A.I. output — in this case, the most typical A.I.-generated digits. The tail ends describe output that is less common.

Notice that when the model was trained on human data, it had a healthy spread of possible outputs, which you can see in the width of the curve above.

But after it was trained on its own output, this is what happened to the curve:

Distribution of A.I.-generated data when trained on its own output

It gets taller and narrower. As a result, the model becomes more and more likely to produce a smaller range of output, and the output can drift away from the original data.

Meanwhile, the tail ends of the curve — which contain the rare, unusual or surprising outcomes — fade away.

This is a telltale sign of model collapse: Rare data becomes even rarer.

If this process went unchecked, the curve would eventually become a spike:

This was when all of the digits became identical, and the model completely collapsed.

Why it matters

This doesn’t mean generative A.I. will grind to a halt anytime soon.

The companies that make these tools are aware of these problems, and they will notice if their A.I. systems start to deteriorate in quality.

But it may slow things down. As existing sources of data dry up or become contaminated with A.I. “ slop ,” researchers say it makes it harder for newcomers to compete.

A.I.-generated words and images are already beginning to flood social media and the wider web . They’re even hiding in some of the data sets used to train A.I., the Rice researchers found .

“The web is becoming increasingly a dangerous place to look for your data,” said Sina Alemohammad , a graduate student at Rice who studied how A.I. contamination affects image models.

Big players will be affected, too. Computer scientists at N.Y.U. found that when there is a lot of A.I.-generated content in the training data, it takes more computing power to train A.I. — which translates into more energy and more money.

“Models won’t scale anymore as they should be scaling,” said ​​ Julia Kempe , the N.Y.U. professor who led this work.

The leading A.I. models already cost tens to hundreds of millions of dollars to train, and they consume staggering amounts of energy , so this can be a sizable problem.

‘A hidden danger’

Finally, there’s another threat posed by even the early stages of collapse: an erosion of diversity.

And it’s an outcome that could become more likely as companies try to avoid the glitches and “ hallucinations ” that often occur with A.I. data.

This is easiest to see when the data matches a form of diversity that we can visually recognize — people’s faces:

A.I. images generated by Sina Alemohammad and others .

This set of A.I. faces was created by the same Rice researchers who produced the distorted faces above. This time, they tweaked the model to avoid visual glitches.

This is the output after they trained a new A.I. on the previous set of faces. At first glance, it may seem like the model changes worked: The glitches are gone.

After two generations …

After three generations …

After four generations, the faces all appeared to converge.

This drop in diversity is “a hidden danger,” Mr. Alemohammad said. “You might just ignore it and then you don’t understand it until it's too late.”

Just as with the digits, the changes are clearest when most of the data is A.I.-generated. With a more realistic mix of real and synthetic data, the decline would be more gradual.

But the problem is relevant to the real world, the researchers said, and will inevitably occur unless A.I. companies go out of their way to avoid their own output.

Related research shows that when A.I. language models are trained on their own words, their vocabulary shrinks and their sentences become less varied in their grammatical structure — a loss of “ linguistic diversity .”

And studies have found that this process can amplify biases in the data and is more likely to erase data pertaining to minorities .

Perhaps the biggest takeaway of this research is that high-quality, diverse data is valuable and hard for computers to emulate.

One solution, then, is for A.I. companies to pay for this data instead of scooping it up from the internet , ensuring both human origin and high quality.

OpenAI and Google have made deals with some publishers or websites to use their data to improve A.I. (The New York Times sued OpenAI and Microsoft last year, alleging copyright infringement. OpenAI and Microsoft say their use of the content is considered fair use under copyright law.)

Better ways to detect A.I. output would also help mitigate these problems.

Google and OpenAI are working on A.I. “ watermarking ” tools, which introduce hidden patterns that can be used to identify A.I.-generated images and text.

But watermarking text is challenging , researchers say, because these watermarks can’t always be reliably detected and can easily be subverted (they may not survive being translated into another language, for example).

A.I. slop is not the only reason that companies may need to be wary of synthetic data. Another problem is that there are only so many words on the internet.

Some experts estimate that the largest A.I. models have been trained on a few percent of the available pool of text on the internet. They project that these models may run out of public data to sustain their current pace of growth within a decade.

“These models are so enormous that the entire internet of images or conversations is somehow close to being not enough,” Professor Baraniuk said.

To meet their growing data needs, some companies are considering using today’s A.I. models to generate data to train tomorrow’s models . But researchers say this can lead to unintended consequences (such as the drop in quality or diversity that we saw above).

There are certain contexts where synthetic data can help A.I.s learn — for example, when output from a larger A.I. model is used to train a smaller one, or when the correct answer can be verified, like the solution to a math problem or the best strategies in games like chess or Go .

And new research suggests that when humans curate synthetic data (for example, by ranking A.I. answers and choosing the best one), it can alleviate some of the problems of collapse.

Companies are already spending a lot on curating data, Professor Kempe said, and she believes this will become even more important as they learn about the problems of synthetic data.

But for now, there’s no replacement for the real thing.

About the data

To produce the images of A.I.-generated digits, we followed a procedure outlined by researchers . We first trained a type of a neural network known as a variational autoencoder using a standard data set of 60,000 handwritten digits .

We then trained a new neural network using only the A.I.-generated digits produced by the previous neural network, and repeated this process in a loop 30 times.

To create the statistical distributions of A.I. output, we used each generation’s neural network to create 10,000 drawings of digits. We then used the first neural network (the one that was trained on the original handwritten digits) to encode these drawings as a set of numbers, known as a “ latent space ” encoding. This allowed us to quantitatively compare the output of different generations of neural networks. For simplicity, we used the average value of this latent space encoding to generate the statistical distributions shown in the article.

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Classical Argument

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A (Very) Brief History of Rhetoric

The study of rhetoric has existed for thousands of years, predating even Socrates, Plato and the other ancient Greek philosophers that we often credit as the founders of Western philosophy. Although ancient rhetoric is most commonly associated with the ancient Greeks and Romans, early examples of rhetoric date all the way back to ancient Akkadian writings in Mesopotamia.

In ancient Greece and Rome, rhetoric was most often considered to be the art of persuasion and was primarily described as a spoken skill. In these societies, discourse occurred almost exclusively in the public sphere, so learning the art of effective, convincing speaking was essential for public orators, legal experts, politicians, philosophers, generals, and educators. To prepare for the speeches they would need to make in these roles, students engaged in written exercises called  progymnasmata . Today, rhetorical scholars still use strategies from the classical era to conceptualize argument. However, whereas oral discourse was the main focus of the classical rhetoricians, modern scholars also study the peculiarities of written argument.

Aristotle provides a crucial point of reference for ancient and modern scholars alike. Over 2000 years ago, Aristotle literally wrote the book on rhetoric. His text  Rhētorikḗ ( On Rhetoric ) explores the techniques and purposes of persuasion in ancient Greece, laying the foundation for the study and implementation of rhetoric in future generations. Though the ways we communicate and conceptualize rhetoric have changed, many of the principles in this book are still used today. And this is for good reason: Aristotle’s strategies can provide a great guide for organizing your thoughts as well as writing effective arguments, essays, and speeches.

Below, you will find a brief guide to some of the most fundamental concepts in classical rhetoric, most of which originate in  On Rhetoric.

The Rhetorical Appeals

To understand how argument works in  On Rhetoric , you must first understand the major appeals associated with rhetoric. Aristotle identifies four major rhetorical appeals: ethos (credibility), logos (logic), pathos (emotion), and Kairos(time). 

• Ethos –  persuasion through the author's character or credibility. This is the way a speaker (or writer) presents herself to the audience. You can build credibility by citing professional sources, using content-specific language, and by showing evidence of your ethical, knowledgeable background.
• Logos –  persuasion through logic. This is the way a speaker appeals to the audience through practicality and hard evidence. You can develop logos by presenting data,  statistics, or facts by  crafting a clear claim with a logically-sequenced argument.  ( See enthymeme and syllogism )
• Pathos –  persuasion through emotion or disposition . This is the way a speaker appeals to the audience through emotion, pity, passions, or dispositions. The idea is usually to evoke and strengthen feelings already present within the audience. This can be achieved through story-telling, vivid imagery, and an impassioned voice.  Academic arguments in particular ​benefit from understanding pathos as appealing to an audience's academic disposition on a given topic, subject, or argument.
• Kairos – an appeal made through the adept use of time. This is the way a speaker appeals to the audience through notions of time. It is also considered to be the appropriate or opportune time for a speaker to insert herself into a conversation or discourse, using the three appeals listed above. A Kairotic appeal can be made through calls to immediate action, presenting an opportunity as temporary, and by describing a specific moment as propitious or ideal.

​*Note:  When using these terms in a Rhetorical Analysis, make sure your syntax is correct. One does not appeal to ethos, logos, or pathos directly. Rather, one appeals to an audience's emotion/disposition, reason/logic, or sense of the author's character/credibility within the text. Ethos, pathos, and logos are themselves the appeals an author uses to persuade an audience. 

An easy way to conceptualize the rhetorical appeals is through advertisements, particularly infomercials or commercials. We are constantly being exposed to the types of rhetoric above, whether it be while watching television or movies, browsing the internet, or watching videos on YouTube.

Imagine a commercial for a new car. The commercial opens with images of a family driving a brand-new car through rugged, forested terrain, over large rocks, past waterfalls, and finally to a serene camping spot near a tranquil lake surrounded by giant redwood trees. The scene cuts to shots of the interior of the car, showing off its technological capacities and its impressive spaciousness. A voiceover announces that not only has this car won numerous awards over its competitors but that it is also priced considerably lower than comparable models, while getting better gas mileage. “But don’t wait,” the voiceover says excitedly, “current lessees pay 0% APR financing for 12 months.”

In just a few moments, this commercial has shown masterful use of all four appeals. The commercial utilizes pathos by appealing to our romantic notions of family, escape, and the great outdoors. The commercial develops ethos by listing its awards, and it appeals to our logical tendencies by pointing out we will save money immediately because the car is priced lower than its competitors, as well as in the long run because of its higher MPG rate. Finally, the commercial provides an opportune and propitious moment for its targeted audience to purchase a car immediately. 

Depending on the nature of the text, argument, or conversation, one appeal will likely become most dominant, but rhetoric is generally most effective when the speaker or writer draws on multiple appeals to work in conjunction with one another. To learn more about Aristotle's rhetorical appeals, click here.

Components and Structure

The classical argument is made up of five components, which are most commonly composed in the following order:

• Exordium –  The introduction, opening, or hook.
• Narratio –  The context or background of the topic.
• Proposito and Partitio –  The claim/stance and the argument.
• Confirmatio and/or Refutatio –  positive proofs and negative proofs of support.
• Peroratio –  The conclusion and call to action.

Think of the exordium as your introduction or “hook.” In your exordium, you have an opportunity to gain the interest of your reader, but you also have the responsibility of situating the argument and setting the tone of your writing. That is, you should find a way to appeal to the audience’s interest while also introducing the topic and its importance in a professional and considerate manner. Something to include in this section is the significance of discussing the topic in this given moment (Kairos). This provides the issue a sense of urgency that can validate your argument.

This is also a good opportunity to consider who your intended audience is and to address their concerns within the context of the argument. For example, if you were writing an argument on the importance of technology in the English classroom and your intended audience was the board of a local high school, you might consider the following:

• New learning possibilities for students (General Audience Concerns)
• The necessity of modern technology in finding new, up-to-date information (Hook/Kairos)
• Detailed narrative of how technology in one school vastly improved student literacy (Hook/Pathos) 
• Statistics showing a link between exposure to technology and rising trends in literacy (Hook/Logos)
• Quotes from education and technology professors expressing an urgency for technology in English classrooms (Hook/Ethos)

Of course, you probably should not include all of these types of appeals in the opening section of your argument—if you do, you may end up with a boring, overlong introduction that doesn’t function well as a hook. Instead, consider using some of these points as evidence later on. Ask yourself:  What will be most important to my audience? What information will most likely result in the action I want to bring about?  Think about which appeal will work best to gain the attention of your intended audience and start there.

The narratio provides relevant foundational information and describes the social context in which your topic exists. This might include information on the historical background, including recent changes or updates to the topic, social perception, important events, and other academic research. This helps to establish the rhetorical situation for the argument: that is, the situation the argument is currently in, as impacted by events, people, opinion, and urgency of some kind. For your argument on technology in the English classroom, you might include:

• Advances in education-related technology over the centuries
• Recent trends in education technology
• A description of the importance of digital literacy
• Statistics documenting the lack of home technology for many students
• A selection of expert opinions on the usefulness of technology in all classrooms

Providing this type of information creates the setting for your argument. In other words, it provides the place and purpose for the argument to take place. By situating your argument within in a viable context, you create an opportunity to assert yourself into the discussion, as well as to give your reader a genuine understanding of your topic’s importance.

Propositio and Partitio

These two concepts function together to help set up your argument. You can think of them functioning together to form a single thesis. The propositio informs your audience of your stance, and the partitio lays out your argument. In other words, the propositio tells your audience what you think about a topic, and the partitio briefly explains why you think that way and how you will prove your point. 

Because this section helps to set up the rest of your argument, you should place it near the beginning of your paper. Keep in mind, however, that you should not give away all of your information or evidence in your partitio. This section should be fairly short: perhaps 3-4 sentences at most for most academic essays. You can think of this section of your argument like the trailer for a new film: it should be concise, should entice the audience, and should give them a good example of what they are going to experience, but it shouldn’t include every detail. Just as a filmgoer must see an entire film to gain an understanding of its significance or quality, so too must your audience read the rest of your argument to truly understand its depth and scope. 

In the case of your argument on implementing technology in the English classroom, it’s important to think not only of your own motivations for pursuing this technology in the classroom, but also of what will motivate or persuade your respective audience(s). Some writing contexts call for an audience of one. Some require consideration of multiple audiences, in which case you must find ways to craft an argument which appeals to each member of your audience. For example, if your audience included a school board as well as parents andteachers, your propositio might look something like this:

“The introduction of newer digital technology in the English classroom would be beneficial for all parties involved. Students are already engaged in all kinds of technological spaces, and it is important to implement teaching practices that invest students’ interests and prior knowledge. Not only would the marriage of English studies and technology extend pedagogical opportunities, it would also create an ease of instruction for teachers, engage students in creative learning environments, and familiarize students with the creation and sharing technologies that they will be expected to use at their future colleges and careers. Plus, recent studies suggest a correlation between exposure to technology and higher literacy rates, a trend many education professionals say isn’t going to change.”

Note how the above paragraph considers the concerns and motivations of all three audience members, takes a stance, and provides support for the stance in a way that allows for the rest of the argument to grow from its ideas. Keep in mind that whatever you promise in your propositio and partitio (in this case the new teaching practices, literacy statistics, and professional opinion) must appear in the body of your argument. Don’t make any claims here that you cannot prove later in your argument.

Confirmatio and Refutatio  

These two represent different types of proofs that you will need to consider when crafting your argument. The confirmatio and refutatio work in opposite ways, but are both very effective in strengthening your claims. Luckily, both words are cognates—words that sound/look in similar in multiple languages—and are therefore are easy to keep straight. Confirmatio is a way to confirm your claims and is considered a positive proof; refutatio is a way to acknowledge and refute a counterclaim and is considered a negative proof.

The confirmatio is your argument’s support: the evidence that helps to support your claims. For your argument on technology in the English classroom, you might include the following:

• Students grades drastically increase when technology is inserted into academics
• Teachers widely agree that students are more engaged in classroom activities that involve technology
• Students who accepted to elite colleges generally possess strong technological skills

The refutatio provides negative proofs. This is an opportunity for you to acknowledge that other opinions exist and have merit, while also showing why those claims do not warrant rejecting your argument. 

If you feel strange including information that seems to undermine or weaken your own claims, ask yourself this: have you ever been in a debate with someone who entirely disregarded every point you tried to make without considering the credibility of what you said? Did this make their argument less convincing? That’s what your paper can look like if you don’t acknowledge that other opinions indeed exist and warrant attention. 

After acknowledging an opposing viewpoint, you have two options. You can either concede the point (that is, admit that the point is valid and you can find no fault with their reasoning), or you can refute their claim by pointing out the flaws in your opponent’s argument. For example, if your opponent were to argue that technology is likely to distract students more than help them (an argument you’d be sure to include in your argument so as not to seem ignorant of opposing views) you’d have two options:

• Concession: You might concede this point by saying “Despite all of the potential for positive learning provided by technology, proponents of more traditional classroom materials point out the distractive possibilities that such technology would introduce into the classroom. They argue that distractions such as computer games, social media, and music-streaming services would only get in the way of learning.” 

In your concession of the argument, you acknowledge the merit of the opposing argument, but you should still try to flip the evidence in a positive way. Note how before conceding we include “despite all of the potential for positive learning.” This reminds your reader that, although you are conceding a single point, there are still many reasons to side with you.

• Refutation: To refute this same point you might say something like, “While proponents of more traditional English classrooms express concerns about student distraction, it’s important to realize that in modern times, students are already distracted by the technology they carry around in their pockets. By redirecting student attention to the technology administered by the school, this distraction is shifted to class content. Plus, with website and app blocking resources available to schools, it is simple for an institution to simply decide which websites and apps to ban and block, thereby ensuring students are on task.”

Note how we acknowledged the opposing argument, but immediately pointed out its flaws using straightforward logic and a counterexample. In so doing, we effectively strengthen our argument and move forward with our proposal.

Your peroratio is your conclusion. This is your final opportunity to make an impact in your essay and leave an impression on your audience. In this section, you are expected to summarize and re-evaluate everything you have proven throughout your argument. However, there are multiple ways of doing this. Depending on the topic of your essay, you might employ one or more of the following in your closing:

• Call to action (encourage your audience to do something that will change the situation or topic you have been discussing).
• Discuss the implications for the future. What might happen if things continue the way they are going? Is this good or bad? Try to be impactful without being overly dramatic.
• Discuss other related topics that warrant further research and discussion.
• Make a historical parallel regarding a similar issue that can help to strengthen your argument.
• Urge a continued conversation of the topic for the future.

Remember that your peroratio is the last impression your audience will have of your argument. Be sure to consider carefully which rhetorical appeals to employ to gain a desirable effect. Make sure also to summarize your findings, including the most effective and emphatic pieces of evidence from your argument, reassert your major claim, and end on a compelling, memorable note. Good luck and happy arguing!

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60 Aristotelian (Classical) Argument Model

Aristotelian argument.

The Aristotelian or classical argument is a style of argument developed by the famous Greek philosopher and rhetorician,  Aristotle . In this style of argument, your goal as a writer is to convince your audience of something. The goal is to use a series of strategies to persuade your audience to adopt your side of the issue. Although  ethos ,  pathos , and  logos  play a role in any argument, this style of argument utilizes them in the most persuasive ways possible.

Of course, your professor may require some variations, but here is the basic format for an Aristotelian, or classical, argumentative essay:

• Introduce your issue.  At the end of your introduction, most professors will ask you to present your thesis. The idea is to present your readers with your main point and then dig into it.
• Present your case  by explaining the issue in detail and why something must be done or a way of thinking is not working. This will take place over several paragraphs.
• Address the opposition.  Use a few paragraphs to explain the other side. Refute the opposition one point at a time.
• Provide your proof.  After you address the other side, you’ll want to provide clear evidence that your side is the best side.
• Present your conclusion.  In your conclusion, you should remind your readers of your main point or thesis and summarize the key points of your argument. If you are arguing for some kind of change, this is a good place to give your audience a call to action. Tell them what they could do to make a change.

For a visual representation of this type of argument, check out the Aristotelian infographic below:

Introduction to Aristotelian Argument

The Aristotelian or classical argument is a style of argument developed by the famous Greek philosopher and rhetorician, Aristotle. In this style of argument, the writer’s goal is to be convincing and to persuade your audience to your side of the issue through a series of strategies.

Start here!

Before you begin, review your assignment and ask yourself questions about what you might want to write about.

Use prewriting activities, such as brainstorming or listing, to help develop ideas for topics and angles.

Do your research! Find credible sources to help you build your argument.

But there’s more! There are some important concepts you need to learn about.

Modes of Persuasion

Ethos=credibility

Pathos=emotions

Logos=logic

Know Your Audience!

When writing a classical or Aristotelian argument, think about how you are going to be convincing to your audience!

Things to remember along the way…

Clear thesis

Support thesis

Opposing views

Cite sources

Sample Essay

For a sample essay written in the Aristotelian model, click here .

Aristotelian (Classical) Argument Model Copyright © 2020 by Liza Long; Amy Minervini; and Joel Gladd is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Chapter 3: classical linear regression models.

In this chapter, we will introduce the classical linear regression theory, including the classical model assumptions, the statistical properties of the Ordinary Least Squares (OLS) estimator, the t -test and the F -test, as well as the Generalized Least Squares (GLS) estimator and related statistical procedures. This chapter will serve as a starting point from which we will develop modern econometric theory.

• Autocorrelation
• Classical linear regression
• Conditional heteroskedasticity
• Conditional homoskedasticity
• Hypothesis testing
• Mean Squared Error (MSE)
• Model selection criterion
• Multicollinearity
• Normal distribution
• Strict exogeneity

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Outline of Classical Model of Argumentation

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• > Journals
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• > Volume 24 Issue 5
• > A CLASSICAL MODEL OF EDUCATION, GROWTH, AND DISTRIBUTION

Article contents

A classical model of education, growth, and distribution.

Published online by Cambridge University Press:  05 November 2018

• Supplementary materials

We develop a classical macroeconomic model to examine the growth and distributional consequences of education. Contrary to the received wisdom, we show that human capital accumulation is not necessarily growth-inducing and inequality-reducing. Expansive education policies may foster growth and reduce earning inequalities between workers, but only by transferring income from workers to capitalists. Further, the overall effect of an increase in education depends on the actual characteristics of the educational system and on the nature of labor market relations. Although the primary aim of the paper is theoretical, we argue that the model identifies some causal mechanisms that can contribute to shed light on recent stylized facts on growth, distribution, and education for the USA.

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Special thanks go to Peter Skott, Luca Zamparelli, and two anonymous referees for detailed and perceptive comments. We are grateful to Richard Arena, Philip Arestis, Deepankar Basu, Riccardo Bellofiore, Laura Carvalho, Meghnad Desai, Gerald Epstein, Giulio Fella, Peter Flaschel, John King, Gilberto Lima, Neri Salvadori, Rajiv Sethi, Engelbert Stockhammer, Daniele Tavani, Alessandro Vercelli, and audiences in London, Paris, Bristol, New York, Berlin, Kingston, and Washington for comments and suggestions. We also thank Kurt von Seekam for excellent research assistance. Roberto Veneziani worked on this project while visiting the University of Massachusetts, Amherst. Their hospitality and support are gratefully acknowledged. The usual disclaimer applies.

Dutt and Veneziani supplementary material

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• Volume 24, Issue 5
• Amitava Krishna Dutt (a1) and Roberto Veneziani (a2)
• DOI: https://doi.org/10.1017/S1365100518000755

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Brain-inspired classical conditioning model

Yuxuan zhao.

1 Research Center for Brain-inspired Intelligence, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China

2 Center for Excellence in Brain Science and Intelligence Technology, Chinese Academy of Sciences, Shanghai 200031, China

3 National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China

4 School of Artificial Intelligence, University of Chinese Academy of Sciences, Beijing 100190, China

Associated Data

The MATLAB scripts can be downloaded from the GitHub repository of the Brain-Inspired Cognitive Engine at Research Center for Brain-inspired Intelligence, Institute of Automation, Chinese Academy of Sciences: https://github.com/Brain-Inspired-Cognitive-Engine/BICC .

Classical conditioning plays a critical role in the learning process of biological brains, and many computational models have been built to reproduce the related classical experiments. However, these models can reproduce and explain only a limited range of typical phenomena in classical conditioning. Based on existing biological findings concerning classical conditioning, we build a brain-inspired classical conditioning (BICC) model. Compared with other computational models, our BICC model can reproduce as many as 15 classical experiments, explaining a broader set of findings than other models have, and offers better computational explainability for both the experimental phenomena and the biological mechanisms of classical conditioning. Finally, we validate our theoretical model on a humanoid robot in three classical conditioning experiments (acquisition, extinction, and reacquisition) and a speed generalization experiment, and the results show that our model is computationally feasible as a foundation for brain-inspired robot classical conditioning.

Graphical abstract

• • Classical conditioning (CC) is crucial in biological and embodied robot learning
• • A spiking neural network incorporates existing biological findings of CC in one model
• • BICC can explain a broader set of findings than other existing computational models
• • BICC ensures a robot gets similar biological behavior and speed generalization capability

Neuroscience; cognitive neuroscience; artifical intelligence; robotics

Introduction

Classical conditioning is regarded as a basic learning method for animals in which an association is built between a conditioned stimulus ( C S ) and a conditioned response ( C R ). The best-known experiment of classical conditioning was performed by Pavlov (1927) . When a dog is presented with food (unconditioned stimulus, U S ), it will start to salivate (unconditioned response, U R ). In Pavlov's research, a dog would hear a tone ( C S ) before being presented with food every time. After a number of trials, the dog started to salivate ( C R ) upon hearing a tone.

Computational model

Classical conditioning has attracted the interest of many researchers, and attempts have been made to build a computational model to reveal its mechanism. Rescorla and Wagner (1972) presented the first computational model of classical conditioning, which is named the Rescorla-Wanger model. This model can predict some important classical phenomena of classical conditioning, and it has led to a great deal of research, including modifications and alternative models. The Sutton-Barto (SB) model ( Sutton and Barto, 1981 ) is a temporally refined extension of the Rescorla-Wanger model. This model learns to increase its response rate in anticipation of increased stimulation, and it can account for more phenomena observed in classical conditioning than the Rescorla-Wanger model can; furthermore, it has served as the precursor of many later models. The temporal difference (TD) model ( Sutton and Barto, 1987 ) is an extension of the SB model. This model takes the form of a temporal difference prediction method and can successfully model the inter-stimulus interval (ISI) effect, which is regarded as a primary real-time effect of classical conditioning. Harry Klopf (1988) proposed a neuronal model that is modified from the Hebbian model to be more consistent with animal learning phenomena; this model can predict a wide range of classical conditioning phenomena. Schmajuk and DiCarlo (1992) presented a multilayer neural network called the S-D model. They mapped the nodes and connections onto regional cerebellar, cortical, and hippocampal circuits to obtain a model that can correctly describe the effects of hippocampal and cortical lesions on conditioning. Balkenius and Moren (1999) described a computational model of classical conditioning that is built on the assumption that the goal of learning is the prediction of a temporally discounted reward or punishment based on the current stimulus situation; notably, this model is well suited for robotic implementation. Johansson and Lansner (2002) presented an associative model of classical conditioning that is composed of a number of interconnected Bayesian confidence propagation neural networks (BCPNNs) implemented on the basis of Hebbian learning, and the output of this model fits the results of classical conditioning experiments. Zuo et al. (2005) introduced a spiking-neuron-based cognitive model. This model can simulate the learning process of classical conditioning with a reflex arc structure and a reinforcement learning method based on the Hebb rule, and an inverted pendulum experiment validated the effectiveness of this model. Liu et al. (2008) built a model with classical conditioning behaviors based on a Bayesian network (CRMBBN). This model constructs cause-effect relationships between classical sensing and nonclassical conditional sensing by means of a Bayesian network, and it can successfully simulate many phenomena, such as acquisition, inter-stimulus effects, and extinction. Liu and Ding (2008) presented a dynamic policy adaptation framework inspired by classical conditioning. This model can successfully realize the self-learning process of classical conditioning and achieve adaptive network policy management. Antonietti et al. (2017) developed a detailed spiking cerebellar microcircuit model that can reproduce eyeblink classical conditioning and successfully fits real experimental datasets from humans.

Here, we build a brain-inspired classical conditioning (BICC) model that integrates and adopts existing biological findings of classical conditioning. With a quaternionic-rate-based synaptic learning rule, which is equivalent to spike-timing-dependent plasticity (STDP) ( Bi and Poo, 2001 ) on a timescale of seconds, the BICC model could predict the majority of classical conditioning phenomena. The computational model of biological classical conditioning enables a robot gets similar learning behavior and the capability of speed generalization. Furthermore, the changes in synaptic weights in this model may hint at the biological mechanism of classical conditioning.

Classical experiments

There are several classical conditioning experiments that can be used to verify the effectiveness of computational models. To enable reasonable comparisons with other well-known computational models, we follow the classical conditioning experiments outlined by Balkenius and Moren (1998) . The stimulus before the + presents first, and the stimulus after the + presents later. The parenthesis indicates that the stimuli are presented and end simultaneously. The ⇒ indicates the result of training. The → indicates the result of the stimulus.

Acquisition

Acquisition is the ability to establish an association between a stimulus and a response, and it is the most fundamental process in classical conditioning. In an acquisition experiment, a C S is presented first and a U S is presented subsequently after a small time interval for several trials; then, the response will be induced when the C S is presented on its own. This acquisition progress can be described as follows. For an acquisition experiment involving an eyelid response in the albino rabbit, the response level forms an S-shaped curve similar to a sigmoid function ( Balkenius and Moren, 1998 ; Schneiderman et al., 1962 ).

Inter-stimulus interval effect

The ISI effect is a primary real-time effect of classical conditioning ( Sutton, 1990 ). The ISI represents the time interval between the presentation of the C S and U S , and it can be divided into three types ( Balkenius and Moren 1999 ): delay conditioning A, delay conditioning B, and trace conditioning ( Figure S1 ). In delay conditioning A, the U S is presented immediately when the C S terminates. In delay conditioning B, the C S is still present when the U S is presented, and the CS and US terminate simultaneously. In trace conditioning, the CS and US have fixed lengths, and the CS terminates before the onset of the US. In empirical studies conducted by Schneiderman and Gormezano (1964) and Smith et al. (1969) , the ISI-CR frequency function revealed a concave-down shape during the acquisition and extinction process.

In an extinction experiment, the acquired response will disappear gradually if only a C S is presented without the subsequent U S . The extinction process can be described as follows:

Reacquisition effect

The reacquisition effect is demonstrated when an animal relearns a previously extinguished association, and the relearning phase is faster than the initial learning phase.

Blocking refers to the following phenomenon: when a first stimulus C S 1 has been associated with a response and a second stimulus C S 2 then is presented and ends simultaneously with C S 1 , the attempt to associate the second stimulus C S 2 with the response will be blocked. Blocking experiments show that the association of a stimulus with a response is not independent of earlier learning. The blocking process can be described as follows, where the parentheses are used to indicate that C S 1 and C S 2 are presented and end simultaneously.

Secondary conditioning

In a secondary conditioning experiment, C S 1 has been associated with the response induced by the U S , and C S 1 is then used as the U S for C S 2 to build an association to the response. The effect of such secondary conditioning is typically weak, and C S 1 will undergo extinction, whereas C S 2 is reinforced. The secondary conditioning progress can be described as follows:

Conditioned inhibition

In a conditioned inhibition experiment, C S 2 and C S 0 have been associated with the response, and then a third stimulus C S 1 is presented and ends simultaneously with one of the previously conditioned stimuli C S 0 without the U S . In the test phase, C S 1 will inhibit the ability of C S 2 to induce the response. The conditioned inhibition process can be described as follows, where the parentheses are used to indicate that the stimuli are presented and end simultaneously.

Facilitation by an intermittent stimulus

Under normal acquisition conditions, the conditioning to C S 1 is weak in the case of trace conditioning due to the long ISI. Under conditions of facilitation, the conditioning to C S 1 is facilitated by an additional C S 2 . The facilitation process can be described as follows:

Overshadowing

The C S 1 and C S 2 are presented and ended simultaneously under the conditions of overshadowing; the associative strength acquired by C S 1 and C S 2 are weaker than the C S 1 or C S 2 conditioned alone in the normal acquisition condition ( Angulo et al., 2020 ). The overshadowing process can be described as follows:

Overexpectation

The C S 1 and C S 2 have been associated with the response, respectively, then the following C S 1 - C S 2 presentations result in a weight decrement ( Rescorla and Wagner, 1972 ). The overexpectation process can be described as follows:

Recovery from overshadowing

In the overshadowing experiment, the extinction of the C S 1 will lead to an increased responding to the C S 2 ( Matzel et al., 1985 ). The recovery from overshadowing process can be described as follows:

Recovery from forward blocking

In the forward blocking experiment, the extinction of the blocker C S 1 will lead to an increased response to the blocked C S 2 ( PinenO et al., 2005 ). The recovery from forward blocking process can be described as follows:

The neural circuit underlying delay eyeblink conditioning has been well described in ( Hansel et al., 2001 ; Ten Brinke et al., 2019 ; Wang et al., 2018 ; Hogri et al., 2015 ; Takehara-Nishiuchi, 2018 ), and we propose our BICC model based on these findings. The architecture of the BICC model is shown in Figure 1 .

The architecture of the BICC model

The arrows and dots represent excitatory and inhibitory synapses, respectively, and the dotted lines represent excitatory or inhibitory synapses depending on the results of synaptic plasticity. The CS pathway and the US pathway are used for recognizing the CS and US through traditional pattern recognition methods and for transferring the information to the PN and IO , respectively. The PN projects the information on the CS to the IPN and GC via an mf . The IO projects the information on the US to the IPN and PU via a cf . The GC transfers the stimulus from the PN to the PU and Int.N via a pf . The PU receives inhibitory stimulation from the Int.N , excitatory stimulation from the IO via a cf , and stimulation from the GC via a pf . The IPN receives inhibitory stimulation from the PU and excitatory stimulation from the IO via a cf and from the PN via an mf . The motor control pathway receives excitatory stimulation from the IPN if the IPN is activated and then induces the CR , or it receives excitatory stimulation from the US pathway and then induces the UR .

Model evaluation

In this section, we use the changes in synaptic weight between the PN (pontine nuclei) and the IPN (interpositus nucleus) to evaluate this model; N P C S i represents the neuron population of the corresponding C S in the PN, and N P R represents the neuron population that controls the response in the IPN. The PN deliver the information from the C S , and the IPN generates the C R via the motor control pathway, as introduced in the Methods section.

Inter-stimulus interval effects

We use both delay and trace conditioning experiments to test our model, and the variation in the synaptic weight between N P C S and N P R is shown in Figure 2 A. The curves initially show a marked increase and then exhibit a concave-down shape, which is consistent with the results of the rabbit experiment presented in Schneiderman and Gormezano (1964) and Smith et al. (1969) . There is an optimal interval time for learning under every condition in this model, being 1.3 s for trace conditioning and 3.1 s for both types of delay conditioning.

The results of inter-stimulus interval effects, learning curves, acquisition and extinction, and reacquisition experiments

(A) The inter-stimulus interval effects in delay and trace conditioning experiments. (1) In the delay conditioning A, the duration of the C S varies from 0 to 6 s, and then the U S is presented immediately after the C S and continues for 2 s. The ISI is equal to the length of the C S . (2) In the delay conditioning B, the duration of the CS varies from 1 to 7 s, whereas the U S continues for 1 s, and the C S and U S end simultaneously. The ISI is equal to the difference between the length of the C S and the length of the U S . (3) In trace conditioning, the C S and U S each continue for 2 s. The ISI is equal to the difference between the start time of the C S and the start time of the U S , and it varies from 0 to 6 s.

(B) The learning curves in the model. (1) In delay conditioning A, the C S and U S each continue for 2 s, so the interval time is 2 s. (2) In delay conditioning B, the C S continues for 3 s and the U S continues for 1 s , so the interval time is 2 s . (3) In trace conditioning, the C S is presented at 0 s and continues for 2 s, and the U S is presented at 2.5 s and continues for 2 s, so the interval time is 2.5 s.

(C) The variations in synaptic weight between N P C S and N P R . W A c q u i s i t i o n 1 and W A c q u i s i t i o n 2 represent the weight variations in the acquisition experiment when the initial weight is 0 and 0.5, respectively. W E x t i n c t i o n represents the weight variations in the extinction experiment, where there is only a C S .

(D) Reacquisition experiment. It is easy to see that fewer trials are needed in the reacquisition condition (fewer than 10 trials) than in the acquisition condition (approximately 25 trials) to achieve the same weight.

Learning curves

Figure 2 B shows the results for the learning curves in the model. Under each condition, the curve is an S-shaped acquisition curve. With fewer than eight trials, the synaptic weight between N P C S and N P R is greater in the case of delay conditioning B than in the case of delay conditioning A, and the smallest weight is observed in the trace conditioning case. As the number of trials increases, the synaptic weight increases to a stable constant; ultimately, the synaptic weight under delay conditioning A is greater than that under trace conditioning, and the smallest final weight is observed in the case of delay conditioning B. We therefore select delay conditioning A for the remainder of the experiments unless otherwise stated, with both the C S and the U S continuing for 2 s .

Acquisition, extinction, and reacquisition experiments

Figure 2 C shows the variations in synaptic weight between N P C S and N P R in the acquisition and extinction experiments. In the acquisition experiment, the C S is presented at 0 s and ends at 2 s and the U S is presented at 2 s and ends at 4 s. In the extinction experiment, only a C S is presented at 0 s and ends at 2 s, without a U S . The results of the reacquisition experiment are shown in Figure 2 D. It is easy to see that in the reacquisition experiment, fewer trials are required to achieve a higher synaptic weight between N P C S and N P R than in the acquisition experiment. This is because during acquisition or extinction, not only the synaptic weight but also the number of synapses involved changes. In the reacquisition experiment, more synapses are involved in the synaptic weight changes, so the learning rate is faster than in the acquisition stage. We combine the acquisition, extinction, and reacquisition experiments into a single overall experiment. The results are shown in Figure S2 .

Blocking experiment

Figure 3 A shows the results of the blocking experiment. It is easy to see that the synaptic weight between N P C S 2 and N P R is too small to induce a response. In the blocking stage, the changes in single synaptic weights between N P C S 1 and N P R and between N P C S 2 and N P R are identical because of the synchronization of C S 1 and C S 2 , but there are many more synapses involved in N P C S 1 than in N P C S 2 , which causes the change in W R , C S 1 to be much greater than that in W R , C S 2 .

The results of blocking, secondary conditioning, conditioned inhibition, and facilitation experiments

(A) Blocking experiment. In the first 10 trials, only C S 1 and U S are presented to build a conditioned response. In the remainder of the trials, C S 1 and C S 2 are presented and end simultaneously, and then the U S is presented subsequently.

(B) Secondary conditioning experiment. In the first 25 trials, C S 1 is presented first and U S is presented subsequently, as in the normal acquisition experiment. In the last 25 trials, C S 1 is treated as the unconditioned stimulus and is presented after C S 2 .

(C) Conditioned inhibition experiment. In the first 25 and second set of 25 trials, C S 2 and C S 0 , respectively, are combined with U S for conditioning on the response. In the subsequent 25 trials, C S 0 and C S 1 are presented and end simultaneously, without U S . In the last 25 trials, only C S 1 and C S 2 are presented.

(D) Facilitation experiment. C S 1 continues for 2 s , C S 2 continues for 3 s , and the U S continues for 1 s . During normal acquisition, C S 1 is presented first, and 2 s later, the U S is presented. During facilitated acquisition, C S 1 is presented first, C S 2 is presented immediately when C S 1 ends, and 2 s after C S 1 ends, the U S is presented. C S 2 and the U S end simultaneously.

Secondary conditioning experiment

Figure 3 B shows the results of the secondary conditioning experiment. Here, C S 1 is treated as the U S to build an association between C S 2 and the response, and the corresponding synaptic weight is typically weak because the synaptic weight between N P C S 1 and N P R exhibits an extinction effect at the same time.

Conditioned inhibition experiment

Figure 3 C shows the results of the conditioned inhibition experiment. In the first and second sets of 25 trials, excitatory synapse connections to N P R are built for N P C S 2 and N P C S 0 , respectively. In the third set of 25 trials, N P C S 0 exhibits an extinction effect, whereas N P C S 1 builds inhibitory synapse connections to N P R because of synchronism. In the last 25 trials, the number of synapses between N P C S 1 and N P R increase because of the inhibitory connections and the negative weight changes. At the beginning of the last set of 25 trials, the inhibition effect from C S 1 is not sufficient to completely inhibit the excitatory input from C S 2 , so C S 2 can still induce the response. With the enhancement of the inhibition effect from C S 1 , at the end of the last set of 25 trials, C S 1 can inhibit the response induced by C S 2 . In the extinction experiment, more than 50 trials are needed for C S 2 to lose the ability to induce the response, whereas in the conditioned inhibition experiment, fewer than 25 trials are needed because of the inhibition effect from C S 1 .

Facilitation experiment

Figure 3 D shows the results of the facilitation experiment. It is easy to see that the synaptic weight between N P C S 1 and N P R is stronger under facilitated acquisition than under normal acquisition. Under normal acquisition, the synaptic weight is weak because of the long ISI. Under facilitated acquisition, C S 2 is conditioned on the response, and the response will be induced twice, by C S 2 and the U S , thus causing C S 1 to build stronger synaptic connections.

Overshadowing experiment

Figure 4 A shows the results of the overshadowing experiment. The synaptic weights between N P C S 1 and N P R and N P C S 2 and N P R are identical because of the synchronization of C S 1 and C S 2 . It is easy to see that the synaptic weight in the overshadowing condition is weaker than that in the normal acquisition condition. In the overshadowing condition, the N P R is stimulated by C S 1 and C S 2 simultaneously, and they contribute equally to build an association to response. So, the synaptic weight in the overshadowing condition is weaker, and it is about half of that in the normal acquisition condition.

The results of overshadowing, overexpectation, recovery from overshadowing, and recovery from forward blocking experiments

(A) Overshadowing experiment. In the normal acquisition condition, the C S 1 and C S 2 build a conditioned response with the U S . In the overshadowing condition, the C S 1 and C S 2 are presented and end simultaneously and then the U S is presented subsequently. The dotted line shows the weight changing in the normal acquisition condition. The solid line shows the weight changing in the overshadowing condition.

(B) Overexpectation experiment. In the first 25 trials, the C S 1 is presented first and the U S is presented subsequently. In the subsequent 25 trials, the C S 2 is presented first and the U S is presented subsequently. In the last 25 trials, the C S 1 and C S 2 are presented and end simultaneously, and then the U S is presented subsequently.

(C) Recovery from overshadowing experiment. The first 25 trials are the overshadowing process, the C S 1 and C S 2 are presented and end simultaneously, and then the U S is presented subsequently. The last 25 trials are the recovery process, the C S 1 is ended, and the C S 2 presented first and the U S is presented subsequently.

(D) Recovery from forward blocking experiment. The first 30 trials are the blocking process. In the first 10 trials, the C S 1 is presented first and the U S is presented subsequently. In the subsequent 20 trials, the C S 1 and C S 2 are presented and end simultaneously, and then the U S is presented subsequently. The last 20 trials are the recovery process, wherein the C S 1 is ended, and the C S 2 presented first and the U S is presented subsequently.

Overexpectation experiment

Figure 4 B shows the results of the overexpectation experiment. The synaptic weight between N P C S 1 and N P R and N P C S 2 and N P R have decreased in the last 25 trials, and it provided that the following C S 1 - C S 2 presentations result in a weight decrement. In the first and subsequent 25 trials, the C S 1 and C S 2 build a C R with US, respectively. In the last 25 trials, the C S 1 and C S 2 stimulate the N P R simultaneously. So, the firing rate of the response neuron increases faster and lasts longer, and it makes an extinction effect until the model is stable again.

Recovery from overshadowing experiment

Figure 4 C shows the results of the recovery from overshadowing experiment. The synaptic weight between N P C S 2 and N P R has increased when the C S 1 ended, and it provided that the extinction of the C S 1 will lead to an increased respondse to the C S 2 . The N P R is only stimulated by C S 2 when the C S 1 ended, and there is an acquisition effect when the U S is presented.

Recovery from forward blocking experiment

Figure 4 D shows the results of the recovery from forward blocking experiment. The synaptic weight between N P C S 2 and N P R has increased when the C S 1 ended, and it provided that the extinction of the blocker C S 1 will lead to an increased response to the blocked C S 2 . Similar to the recovery from overshadowing experiment, the N P R is only stimulated by C S 2 when the C S 1 ended, and there is an acquisition effect when the U S is presented.

The results of the presented model in the various experiments and the comparison results with existing models are summarized in Table 1 .

Comparison results with existing models

This table is adapted from Johansson and Lansner (2002) and Balkenius and Moren (1998) . An ∗ means that the model could reproduce the correlation feature, o means that the model could reproduce partially, and - means that it is not mentioned or was unable to reproduce.

SB ( Sutton and Barto 1981 ), TD ( Sutton and Barto 1987 ), Klopf ( Harry Klopf 1988 ), SD ( Schmajuk and DiCarlo 1992 ), Balkenius ( Balkenius and Moren 1999 ), BCPNN ( Johansson and Lansner 2002 ), CRMBBN ( Liu et al., 2008 ), Liu ( Liu and Ding 2008 ).

Robotic classical conditioning experiments

We use acquisition, extinction, and reacquisition experiments and speed generalization experiment to evaluate this model on a humanoid robot.

We selected three classical conditioning experiments—acquisition, extinction, and reacquisition—to evaluate this model on a humanoid robot. A red robot was used as the participant robot, and the various stimuli used in the experiments are shown in Figure 5 A. We use template matching to identify different stimuli. The sequences of stimuli in these experiments are shown in Figure 5 B. In the acquisition experiment, the participant robot first was shown a red fist toy ( C S ), then was shown a blue robot ( U S ), and subsequently took an avoidance action ( U R ). After learning, the participant robot would take an avoidance action ( C R ) when it saw the red fist toy ( C S ). In the extinction experiment, only the C S was presented; after several trials, the participant robot did not perform the C R upon seeing the red fist toy ( C S ). In the reacquisition experiment, the participant robot could establish the C R in fewer trials than in the acquisition experiment. The experimental results are shown in Figures 5 C and 5D.

The acquisition, extinction, and reacquisition experiments on humanoid robot

(A) The stimuli used in the experiments. (a) The blue robot is an unconditioned stimulus that can be regarded as the participant robot's natural enemy; therefore, the participant robot will take an avoidance action upon seeing the blue robot. (b) The red fist toy is a conditioned stimulus. (c) The yellow duck is an interference stimulus used to prove that only the learned stimulus can induce the response. (d) Nothing detected means that there is no stimulus.

(B) The sequences of stimuli for acquisition (upper) and extinction (below).

(C and D) (C) Dynamic changes of weight in robotic classical conditioning experiments. (D) Dynamic changes of response in robotic classical conditioning experiments. The red line indicates the response threshold. The total number of trials is 16, consisting of 3 acquisition trials, 8 extinction trials, 1 reacquisition trial, and 2 extinction trials in addition to 2 interference stimuli (in trials 4 and 7) in the first extinction experiment.

Speed generalization experiment

The speed generalization experiment is that the robot trained at a slow speed on a navigation task; then it could navigate the track at a higher speed without training ( Herreros et al., 2013 ). The humanoid robot Nao is a biped robot. According to its documentation, the speed parameter is the fraction of the maximum speed, such as 100% means the full speed. According to our verification, may be due to the aging problem, the robot cannot move accurately at a given speed, and often deviates from the direction when walking in a straight line. So, we test the speed generalization capabilities of the model in both real and simulation environments.

The real environment is shown as Figure 6 A. In the real environment, the robot is trained at 50% speed and tested at 100% speed. We use simple image recognition algorithms to recognize the CS (black line, Figure 6 B) and the US (red line, Figure 6 C). For the black line recognition, we retain the central area of the image, then convert the image to a binary image according to the given threshold value, delete the small-area object, and finally perform horizontal line detection to complete the recognition. For the red line recognition, we extract the matching color area by setting the threshold value and then detect the horizontal line to complete the recognition. The result of the real environment is shown as Figure 7 .

Speed generalization experiment in real environment

(A) The real environment. The white runway is the track for the robot to navigate. The red line is the US, which means the robot needs to turn right to avoid leaving the runway, and the black line is the CS.

(B) The CS perceived in robot vision.

(C) The US perceived in robot vision.

The result of speed generalization experiment in real environment

(A–D) The V C R is the firing rate of response neuron when it receives CS. For easy comparison, we show the firing rate of response neuron when it receives only US ( V U R ). The red line indicates the response threshold. Compared with (A and B), with the increase of speed, the density of CS increased, so the V C S is stronger. (C) The firing rate of CR and UR at 50% speed. It is easy to see that after training, the robot could perform CR before UR. (D) The firing rate of CR and UR at 100% speed. It is easy to see that without training, the robot could perform CR before UR.

The simulation environment is shown as Figure 8 A. In the simulation environment, we use two experiments to test the speed generalization capacity of the model. In the first experiment, we test the induced time of CR and UR at a given speed in the range of 100%–200% with 10% speed increments. The result is shown as Figure 8 B. In the second experiment, we test the induced distance of CR at a given speed at 100%, 120%, 150%, 200%, 300%, 350%, and 400%. The result is shown as Figure 8 C. And if the speed is greater than 350%, the robot experiment fails.

Speed generalization experiment in simulation environment

(A) Simulation environment. The black square is the CS, and the red square is the US. The US indicates that the robot should turn right to avoid leaving the runway.

(B) The induced time of CR and UR at different speeds. It easy to see that the robot could perform CR before UR.

(C) The induced distance of CR at different speeds. The robot successfully passed the experiment with a maximum speed of 350%. When the speed reached 400%, the CS could not induce the CR, and the experiment failed.

The results of real and simulation environments show that the proposed model has the capacity of speed generalization.

The BICC model exhibits long-term depression (LTD) at GC (granule cell)-PU (Purkinje cell) synapses and long-term potentiation (LTP) at PN-IPN synapses, which is consistent with the findings from electrophysiological experiments on eyeblink conditioning presented in Koekkoek et al. (2003) and Steuber et al. (2007) and Pugh and Raman, (2006 , 2008 ), respectively. LTD is also exhibited at PN-IPN synapses in this model, representing another mechanism of synaptic plasticity that is consistent with the electrophysiological experiments reported in Zhang and Linden (2006) , but there have been fewer reports of this mechanism than of the former.

LTD at GC-PU synapses

If only the C S is presented before learning, the GC (granule cell) receives the C S from the PN and then projects it to an Int.N (inhibitory interneuron). The inhibitory input from the Int.N will inhibit the excitatory input from the GC and the spontaneous firing of the PU, so the firing rate of the PU will immediately drop to zero; in other words, the firing of the PU will be paused. The synaptic weight between the GC and PU will not change because the postsynaptic neurons in the PU is not fired.

If only the U S is presented before learning, the U S is projected to the motor control pathway and induces the U R . In parallel, the IO (inferior olive) receives the U S from the US pathway and then projects it to the IPN and PU via the cf (climbing fiber). The excitatory input from the IO to the PU will enhance the inhibition from the PU to the IPN. The synaptic weight between the GC and PU remains unchanged because the presynaptic neuron in the GC is not fired.

In the acquisition experiment, the firing rate of the PU will gradually decay to zero because of the additional excitatory input from the U S . With the increased firing rate of the presynaptic neurons in the GC and the decreasing firing rate of the postsynaptic neurons in the PU, the synaptic weight between the GC and PU will decrease and exhibit LTD. An intuitive explanation of how spike-based STDP can influence synaptic efficiency through a rate-based mechanism can be found in Bengio et al. (2017) . In short, the synaptic weight is updated in proportion to the product of the presynaptic firing rate and the temporal rate of change in activity on the postsynaptic side. The synaptic weight is updated in our model based on more factors, as well, as detailed in Qiao et al. (2017) . The synaptic weight between the GC and PU will decrease with repeated pairings of the C S and U S , whereas the inhibitory input from the Int.N and the excitatory input from the IO remain unchanged, so the Purkinje cell will pause spontaneous firing. This phenomenon is consistent with the electrophysiological experiments on eyeblink conditioning presented in Wetmore et al. (2014) and Hansel et al. (2001) .

LTP and LTD at PN-IPN synapses

The single synaptic weight changes in a single trial in the acquisition and extinction experiments are shown in Figure 9 A, and the firing rates of the neurons in the acquisition and extinction experiments are shown in Figures 9 B and 9C, respectively. In the acquisition experiment, the change in the synaptic weight is negative because the temporal rate of change of the postsynaptic neurons in the IPN, which is represented as V R in Figures 9 B, is smaller than that of the presynaptic neuron in the PN, which is represented as V C S in Figure 9 B, from 0 to 2 s. The C S ends at 2 s; then the U S is presented, continues for 2 s, and finally ends at 4 s. From 2 to 4 s, the firing rate of the presynaptic neuron decreases because the C S has ended, whereas the firing rate of the postsynaptic neuron increases because the U S is being presented. The change in the synaptic weight is positive because of the decreasing firing rate of the presynaptic neuron and the increased firing rate of the postsynaptic neuron. This model exhibits the acquisition effect if the positive term is greater than the negative term and exhibits the extinction effect if the positive term is less than the negative term, and the model reaches a steady state when the positive term is equal to the negative term.

The synaptic weight changing and firing rates of neurons in acquisition and extinction experiments

(A) Single synaptic weight changes in a single trial in the acquisition and extinction experiments.

(B) Firing rates of neurons in the acquisition experiment. The C S is presented at 0 s and ends at 2 s . The U S is presented at 2 s and ends at 4 s .

(C) The C S is presented at 0 s and ends at 2 s and there is no U S .

In the acquisition experiment, the number of excitatory synapses between the PN and IPN increases, which is consistent with the electrophysiological experiments on eyeblink conditioning presented in Kleim et al. (2002) and Weeks et al. (2007) .

Our model suggests that the cerebellar cortex, especially the IPN, plays a critical role in classical conditioning. In our model, the LTD at GC-PU synapses leads to a reduction in the excitatory input from the GC. Although there is an excitatory input from the IO when the U S is presented, the PU will be paused due to the loss of excitatory input from the GC. In the BICC model, classical conditioning can be achieved without the PU but not without the IPN, which is consistent with Lavond and Steinmetz (1989) .

In this article, we propose a BICC model and use 11 classical conditioning experiments to validate this model. The results of experimental validations in a simulation environment and on a humanoid robot indicate that this model can handle almost all classical conditioning experiments and endows the robot with the ability to establish a C R .

Limitations of the study

Our model cannot reproduce the experiment of latent inhibition ( Lubow and Moore, 1959 ), spontaneous recovery ( Pavlov, 1927 ), and unblocking ( Bradfield and Mcnally, 2008 ). The latent inhibition effect is that a familiar stimulus takes longer to build an association to response than a new stimulus. Our model cannot reproduce the latent inhibition experiment because the model doesn't distinguish between familiar stimulus and new stimulus. The spontaneous recovery effect is the reappearance of a response that had been extinguished. Our model cannot reproduce this experiment, and we think that the spontaneous recovery effect may require involvement of more brain regions. The unblocking effect is that the responding of the blocked C S 2 increases by increase in the intensity or duration of the U S , or increase in the number of the U S . Our model failed in the unblocing effect experiment, if the C S 1 has built an association to response, the C S 2 cannot build the association, no matter how the U S changes. In the future, we will improve our model to reproduce more experiments.

Resource availability

Lead contact.

Further information and requests for resources should be directed to and will be fulfilled by the Lead Contact, Yi Zeng ( [email protected] ).

Materials availability

This study did not generate new unique reagents.

Data and code availability

All methods can be found in the accompanying Transparent methods supplemental file .

Acknowledgments

This work is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB32070100), the new generation of artificial intelligence major project of the Ministry of Science and Technology of the People's Republic of China (Grant No. 2020AAA0104305), the Beijing Municipal Commission of Science and Technology (Grant No. Z181100001518006), the Key Research Program of Frontier Sciences, CAS (Grant No. ZDBS-LY-JSC013), the CETC Joint Fund (Grant No. 6141B08010103), and the Beijing Academy of Artificial Intelligence (BAAI).

Author contributions

Conceptualization, Y. Zhao and Y. Zeng; Methodology, Y. Zhao, Y. Zeng, and G.Q.; Software, Y. Zhao and G.Q.; Validation, Y. Zhao and Y. Zeng; Formal Analysis, Y. Zhao and Y. Zeng; Investigation, Y. Zhao and Y. Zeng; Data Curation, Y. Zhao; Writing – Original Draft, Y. Zhao and Y. Zeng; Writing – Review & Editing Draft, Y. Zhao and Y. Zeng; Visualization, Y. Zhao; Supervision, Y. Zeng; Project Administration, Y. Zeng; Funding Acquisition, Y. Zeng.

Declaration of interests

The authors declare no competing interests.

Published: January 22, 2021

Supplemental Information can be found online at https://doi.org/10.1016/j.isci.2020.101980 .

Supplemental information

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Keynesian and classical theories: static and dynamic perspectives

• Published: 15 March 2021
• Volume 19 , pages 343–367, ( 2022 )

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• Hiroki Murakami 1  

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This paper reexamines the fundamental difference between the Keynesian and classical theories from both static and dynamic perspectives. It is shown that the rigidity of wages plays a pivotal role in the distinction between these theories in statics and that they can be differentiated in terms of long-run stability in dynamics.

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Keynesian Business Cycle Theory, Part Deux: Inflexible Prices and Wages

Bastard Keynesianism

Hicks, john richard (1904–1989).

The formal theory developed in Keynes’ General Theory is substantially static. Since the view is widespread that the IS-LM system provides a “distorted” interpretation of Keynes’ General Theory, it may be useful to argue against this view. Keynes did not oppose but endorse Hicks ( 1937 ) as a summary of his General Theory. This fact can be confirmed by the following correspondence between Keynes and Hicks:

At last long I have caught up with my reading and have been through the enclosed [Hicks ( 1937 )]. I found it very interesting and really have next to nothing to say by way of criticism. (the letter from Keynes to Hicks in Keynes ( 1973 , p. 80)).

The IS-LM system is taken as an expression of the static Keynesian system in this paper, but as recognized in Modigliani ( 1944 ), it can be adapted to a dynamic context if time lags are appropriately introduced. For a recent study on the dynamic effects of time lags concerning consumption and investment in the IS-LM system, see Murakami ( 2017 ).

Although it is argued in Keynes’ General Theory that aggregate investment is determined by the marginal efficiency of capital (or the expected profit on capital) and the rate of interest, Eq. ( 2 ) may be regarded as consistent with his theory of investment because expected effective demand, which is a major determinant of the marginal efficiency of capital, is influenced largely by current effective demand or aggregate output. Equation ( 2 ) is also consistent with the profit principle of investment postulated in the Keynesian theories of business cycles of Kalecki ( 1935 , 1939 ) and Kaldor ( 1940 , 1951 ).

It seems that the reason why the investment and saving functions and the equilibrium condition for the goods-services market are separately posited in Modigliani’s ( 1944 , 1963 ) system is that he appreciated the difference between ex ante investment and saving. The following statement is made in Modigliani ( 1944 ):

In our case, the equilibrium of the “money market” is a condition of short-run equilibrium (that determines the rate of interest for each period) because it is the result of decisions that can be carried out into effect immediately. The condition saving \(=\) investment, on the other hand, is a condition of long-run equilibrium because the equality of ex ante saving and investment cannot be brought about instantaneously. This is a different way of stating the familiar proposition that the multiplier takes time to work out in its full effect. (p. 62).

Equation ( 6 ) is equivalent to

It can, thus, be taken as a behavioral equation for firms’ price setting. Also, if the curvature of the production function is gradual, then the price level is almost proportionate to the nominal wage and, hence, almost fixed for the given nominal wage.

According to Keynes’ interpretation, the first postulate just describes the correlation between aggregate employment and the real wage, as the following statement indicates:

It [the first postulate] means that, with a given organisation, equipment and technique, real wages and the volume of output (and hence of employment) are uniquely correlated, so that, in general, an increase in employment can only occur to the accompaniment of a decline in the rate of real wages. Thus I am not disputing this vital fact which the classical economists have (rightly) asserted as indefeasible. In a given state of organisation, equipment and technique, the real wage earned by a unit of labour has a unique (inverse) correlation with the volume of employment. Thus if employment increases, in the short period, the reward per unit of labour in terms of wage-goods must, in general, decline and profits increase. (Keynes 1936 , p. 17)

His interpretation is more explicit in his debate with Dunlop and Tarshis on real wages:

In the passage quoted above [Keynes (Keynes 1936 , pp. 9–10)] I was dealing with the relation of real wages to changes in output , and had in mind situations where changes in real and money wages are reflection of changes in the level of employment caused by changes in effective demand. (Keynes 1939 , p. 35)

It seems that Keynes should have abandoned ( 6 ) and replaced it with a naive but realistic mark-up pricing hypothesis to establish the principle of effective demand as a foundation of his theory. In the Keynesian system based on the principle of effective demand (( 4 ) and ( 5 )), Eq. ( 6 ) may be better taken as a mark-up pricing rule than as a first order condition for optimality. Indeed, it can be written as

where \(\beta\) stands for the employment elasticity of production. If \(\beta\) is approximately constant (and less than unity), this equation is consistent with the mark-up principle. In this sense, Kalecki’s ( 1939 , 1971 ) theory, based on the mark-up principle, is more faithful to the principle of effective demand. Indeed, in examining the Keynesian theory from dynamic perspectives (in the next section), we will adopt a hypothesis on price settings based on Kalecki’s theory rather than Keynes’, because the former is more “Keynesian” than the latter.

This view is asserted in Modigliani ( 1944 ) as follows:

It is usually considered as one of the most important achievements of the Keynesian theory that it explains the consistency with the presence of involuntary unemployment. It is, however, not sufficiently recognized that, except in a limiting case to be considered later, this result it due entirely to the assumption of “rigid wages” and not to the Keynesian liquidity preference. (p. 65) The liquidity-preference theory is not necessary to explain underemployment equilibrium; it is sufficiently only in a limiting case; the “Keynesian case.” In the general case it is neither necessary nor sufficient; it can explain this phenomenon only with the additional assumption or rigid wages. (pp. 75–76).

As noted in the outset of Modigliani’s ( 1944 ) analysis, this view on the Keynesian theory is only valid in a static context. See Sect.  2.3 .

The equivalence of the second postulate and full employment is recognized in Keynes’ General Theory:

At different points in this chapter, we have made the classical theory to depend in succession on the assumptions: (1) that the real wage is equal to the marginal disutility of the existing employment; (2) that there is no such thing as involuntary unemployment in the strict sense; (3) that supply creates its own demand in the sense that the aggregate demand price is equal to the aggregate supply price for all levels of output and employment. These three assumptions, however, all amount to the same thing in the sense that they all stand and fall together, any one of them logically involving the other two. (Keynes 1936 , pp. 21–22)

This view is made explicit in Ball et al. ( 1988 ) as follows:

According to the Keynesian view, fluctuations in output arise largely from fluctuations in nominal aggregate demand. These changes in demand have real effects because nominal wages and prices are rigid. (p. 1)

This position is taken in Patinkin ( 1965 ) as the following statements indicate:

Thus Keynesian economics is the economics of unemployment dis equilibrium. It argues that as a result of interest-inelasticity, on the one hand, and distribution and expectation effects, on the other, the dynamic process of Chapter XIII:3—even when aided by monetary policy—is unlikely to converge either smoothly or rapidly to the full-employment position. Indeed, if these influences are sufficiently strong, they may even render this process unstable. In such a case the return to full employment would have to await the fortunate advent of some exogenous force that would expand aggregate demand sufficiently. (pp. 337–338) While our interpretation takes off the analytical edge of Keynesian economics in one direction, it sharpens it in another, more vital one. It makes unmistakably clear—what should always have been clear—that the involuntary unemployment of the General Theory need not have its origin in wage rigidities. Indeed, in this respect we are more Keynesian than Keynes. For by unequivocally placing the center of emphasis on the inadequacy of aggregate demand in the commodity market, and by recognizing the resulting involuntary unemployment to be phenomenon of economic dynamics, we have freed ourselves from the necessity of static analysis to connect decreases in employment with increases in the real wage rate. We have been able to explain the existence of involuntary unemployment without placing any restrictions on the movement of the real wage rate. (pp. 340–341)

A similar argument is made in Tobin ( 1975 ):

Very likely Keynes chose the wrong battlefield. Equilibrium analysis and comparative statics were the tools to which he naturally turned to express his ideas, but they were probably not the best tools for his purpose. (p. 195) The real issue is not the existence of a long-run static equilibrium with unemployment, but the possibility of protracted unemployment which the natural adjustments of a market economy remedy very slowly if at all. So what if, within the recherché rules of the contest, Keynes failed to establish an “underemployment equilibrium”? The phenomena he described are better regarded as disequilibrium dynamics. Keynes’s comparative statics were an awkward analytical language unequal to the shrewd observations and intuitions he was trying to embody. (pp. 195–196)

Equation ( 9 ) is consistent with the marginal efficiency theory of investment (Keynes 1936 , Chap, 11) and also with the profit principle of investment (cf. Kalecki 1935 , 1939 ; Kaldor 1940 , 1951 ). For an intertemporal microeconomic foundation on the Keynesian theory of investment, see Murakami ( 2016a ).

Throughout this paper, \(\dot{q}\) denotes the time derivative of q , i.e., \(\dot{q}={\text {d}}q/{\text {d}}t.\)

Since aggregate ex ante investment and saving are distinguished, aggregate ex post capital formation may not be equal to aggregate ex ante investment net of capital depreciation. For simplicity, however, the difference between ex ante and ex post investment is ignored in ( 16 ). For a more general formalization of capital formation, see Stein ( 1969 ); for a further generalization, see Murakami ( 2014 ).

If the natural rate of unemployment is constant over time, the natural-rate level of aggregate employment is proportional to aggregate supply of labor.

Provided that this condition holds, the rate of change in aggregate supply of labor, which can be identified with the rate of change in population, can be negative.

As verified by Muth ( 1961 ), the “adaptive expectations” rule is “rational” (or efficient) if the actual rate of inflation is composed of both permanent and transitory disturbances.

This is a common view as observed in Leijonhufvud ( 1968 ):

In general equilibrium flow models, prices are the only endogenous variables which enter as arguments into the demand and supply functions of individual households. Tastes and initial resource endowments are parametric. In “Keynesian” flow models the corresponding arguments are real income and the interest rate. Of these, real income is a measure of quantity, not of prices. On a highly abstract level, the fundamental distinction between general equilibrium and Keynesian models lies in the appearance of this quantity variable in the excess demand relation to the latter. The difference is due to the assumptions made about the adjustment behavior of the two systems. In the short run, the “Classical” system adjusts to changes in money expenditures by means of price-level movements; the Keynesian adjusts primarily by way of real income movements. (p. 51)

Government expenditure or net exports can be included in aggregate effective demand as exogenous factors, provided that they are proportional to the natural-rate level of aggregate output.

The long-run stability of aggregate share of labor has been confirmed by Jones ( 2016 ). According to him, the U.S. share of capital was almost constant (about 34.2 percent) until around 2000, though it has recently been rising (to 38.7 percent by 2012).

System ( K ) has a lot in common with the Kaldor–Tobin models (synthesizing Kaldor ( 1940 ) and Tobin ( 1975 )) of Asada ( 1991 ), Chiarella and Flaschel ( 2000 , Chap, 6), Chiarella et al. ( 2013 , Chap, 13) and Murakami and Asada ( 2018 ) but differs from them in the following points: (i) the rate of utilization is not identified with the output–capital ratio; (ii) the price level is determined based on the natural-rate (long-run) levels (not actual levels) of employment and output (cf. ( 31 )). It may also be viewed as a long-term extension of the short-term Keynesian model of Flaschel et al. ( 1997 , Chap, 7) and of the medium-term Keynesian models of Murakami ( 2014 , 2016b ). The purpose of our analysis is not to present a generalized Keynesian model integrating the related ones but to consider the difference in view on stability between the Keynesian and classical theories making use of fairly standard models.

\(k=0\) or \(m=0\) is ruled out as an equilibrium value of k or m .

A similar assumption is made for the same purpose in Asada ( 1991 ).

The interest elasticity of liquidity preference is given by

The principle of effective demand in statics, represented by ( 4 ), can be taken as the limiting case of \(\alpha \rightarrow \infty\) in the quantity adjustment described by ( 30 ).

This is consistent with the conclusion in Yoshikawa’s ( 1981 ) dynamic Keynesian model abstracting from capital formation.

This conclusion supports the following view on the flexibility of wages, presented in Keynes’ General Theory:

To suppose that a flexible wage policy is a right and proper adjunct of a system which on the whole is one of laissez-faire, is the opposite of truth. (Keynes 1936 , p. 269)

For discussions on the existence (and uniqueness) of persistent business cycles in related Keynesian models, see Murakami ( 2014 , 2018 , 2019 , 2020 ) and Murakami and Asada ( 2018 ).

For different formalizations of Harrod’s ( 1939 ) theory, see Yoshida ( 1999 ) and Sportelli ( 2000 ).

The solution of Eq. ( 49 ) can be given as

Equation ( 50 ) is a modified version of the law of price dynamics in the “Keynes–Wicksell” model (cf. Stein 1969 ; Fischer 1972 ).

System ( C ) may be regarded as a long-term extension of Tobin’s ( 1975 ) M (Marshall) model. In most studies on the dynamic Keynesian theory (cf. Flaschel et al. 1997 , Chap, 7; Chiarella and Flaschel 2000 , Chap, 6; Chiarella et al. 2013 , Chap, 13; Murakami 2014 , 2016b ; Murakami and Asada 2018 ), the effects of price or wage flexibility on the Keynesian system are examined in detail, but the differences in stability properties between the Keynesian and classical systems are not thoroughly studied (especially in a long-run context). Our analysis studies the stabilizing effect of wage flexibility in the classical system to elucidate the fundamental difference between the Keynesian and classical systems.

The exact values of \(g_1\) and \(g_2\) in ( 61 ) are not necessary for our analysis.

For the possibility of persistent cyclical fluctuations in a classical model (with public capital), see Murakami and Sasaki ( 2020 ).

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Acknowledgements

The author would like to thank the anonymous referees for their comments on the earlier version of this paper. This work was supported by Chuo University Personal Research Grant and JSPS KAKENHI Grant number 18K12748.

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Murakami, H. Keynesian and classical theories: static and dynamic perspectives. Evolut Inst Econ Rev 19 , 343–367 (2022). https://doi.org/10.1007/s40844-021-00205-5

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Structured expert judgment using the Classical Method

Expert opinions are frequently sought when complex decisions must be made in situations where appropriate information cannot be acquired from existing data and models. Experts are asked to quantify their uncertainty over quantities of interest that inform the decision-making process. Furthermore, the experts are unlikely to be in complete agreement with one another. In such situations, expert judgement can be employed to quantify the uncertainty that ensues and to aggregate expert opinion.

Roger Cooke, the Chauncey Starr Senior Fellow at Resources for the Future, and Emeritus Professor at Delft University of Technology, created the Classical Model, also known as Cooke’s method, for quantifying uncertainty when using expert opinion. Throughout the three decades since its formulation, the Classical Model has been used to perform structured expert judgment in a diverse range of applications, including climate change, disaster management, epidemiology, public and global health, ecology, aviation, nuclear safety, environment, and ecology. In addition, together with Dr Tina Nane, also from Delft University of Technology, and Dr Anca Hanea, from the University of Melbourne, Professor Cooke presents the only available online module on structured expert judgment.

Expert judgment can range from asking an individual expert for their best guess, to following a formal, structured approach to systematically obtain and combine probabilistic judgments. This synthesis of opinions is called expert elicitation. The validation of expert judgments is challenging since they are only called for when other data are unavailable. Measuring their accuracy is, therefore, an arduous task.

Structured expert judgment Structured expert judgment aggregates experts’ uncertainty distributions. Cooke explains that structured expert judgment methods are intended to ‘quantify uncertainty, not to remove it from the decision process’. If expert data is to be recognised as scientific data, it should be subjected to the same quality controls as any other kind of data. He proposed a class of methods, known as structured expert judgment, that satisfy four principles required for any method described as ‘scientific’. These are scrutability/accountability, neutrality, fairness, and empirical quality control. Cooke’s Classical Model is arguably the most rigorous method for quantifying uncertainty by using expert opinion.

Structured expert judgment methods are intended to ‘quantify uncertainty, not to remove it from the decision process’.

The Classical Model The naming of the ‘Classical Model’ highlights the method’s association with classical statistics. The Classical Model uses objective performance measures to validate expert opinion. The experts assess uncertain target questions together with a set of calibration questions. The calibration questions are from the experts’ field of knowledge, have observed true values and often involve data from official reports that have not yet been made public. The experts are scored on their performance in assessing the calibration questions. Validation is achieved by assessing the statistical accuracy of an expert’s assessments together with how much information they provide. These two quantitative measures of performance are used to calculate performance-based weights.

The Classical Model combines experts’ distributions using these performance-based weights, optimising the performance of the combined expert, or ‘Decision Maker’. Several interesting mathematical issues arise from optimising this performance-based aggregation. In non-technical terms, Cooke describes how the performance measure must reward both the experts’ statistical accuracy and informativeness, while discouraging the experts from stating judgments that differ from their true opinions. The performance of the Decision Maker can be evaluated in the same way as that of the experts, using the same performance measures. Performance-optimised Decision Makers correspond to virtual experts and can be adopted by the real-life decision maker. Cross validation is also applied whereby subsets of calibration variables are used to form weights and predict the excluded calibration variables.

Applications Numerous studies have been conducted using the Classical Model. Highlights among these include its application to nuclear safety in the 1990s in research being carried out by the European Union and the United States Nuclear Regulatory Commission. During the prolonged volcanic eruption on the island of Montserrat in the West Indies, from 1995 to 2018, the Classical Model was a key decision-support procedure. Harvard University and the Kuwait government used Cooke’s method in their 2004–2005 study of fine particulates, pollution in the form of tiny particles or droplets suspended in the air. It was also used in an investigation into foodborne diseases for the World Health Organization in 2011–2013.

Illnesses transmitted by food and water More recently, the Centers for Disease Control and Prevention and the University of Florida, together with Roger Cooke, Tina Nane, and Willy Aspinall, performed a large, structured expert judgment study using Cooke’s Classical Model in their work to control and prevent illnesses transmitted through food and water in the United States. The expert elicitation took place at a two-day workshop in May 2017 and involved 48 experts from various professional and scientific backgrounds. Estimates were obtained for the proportion of 33 pathogens including bacteria, such as Salmonella and Legionella, and viruses, such as norovirus and hepatitis A, attributed to each of five major transmission pathways (foodborne, waterborne, person-to-person, animal contact, and environmental), and six associated sub-pathways.

The researchers commented on how the method made it possible for the estimates to be informed by multiple data sources, such as outbreak surveillance data, studies of sporadic illnesses, case reports, and the experts’ professional knowledge. They also pointed out that using calibration questions to weigh expert responses, a unique feature of the Classical Model, ‘introduces mathematical rigor not found with other elicitation methods’. The findings provide an understanding of the multiple transmission pathways for the identified pathogens and support the targeting of resources and prioritisation of public health interventions, as well as informing policy.

Climate change Climate change is riddled with uncertainty, and Cooke observes that both the scientific community and the general population make errors when reasoning under uncertainty, and fail to convey it accurately. Faced with multiple uncertain quantities, most people will identify what they consider the most likely outcome for each quantity and then reason as if those values were certain. Uncertainty is ‘taken into account’ after the fact by adding qualifiers such as ‘highly confident’, ‘most likely’, and ‘virtually certain’. This may be satisfactory when deciding whether to take an umbrella to work, but not when deciding how society should deal with climate issues impacting life as we know it. The uncritical aggregation of high confidences sets and baits the ‘confidence trap’: thinking that high confidence in each of several statements confers high confidence in all statements jointly. Consider: you may be highly confident that a six will not come up on the first throw of a dice, and on the second, and third and fourth. However, the probability is about one half that a six will come up on one of the four throws. Reasoning under uncertainty must obey the laws of probability, even if the probabilities are subjective. Communicating uncertainty to the lay public is difficult, but if the communicators themselves don’t understand uncertainty, then it is well-nigh impossible. This, in Cooke’s opinion, is a major challenge of dealing with climate change. We must decide before the facts are in. That means deciding under uncertainty – which we do very badly.

In a review of 49 professionally contracted studies, Cooke highlighted the challenges involved in ensuring that the use of expert subjective probabilities is scientific. This evaluation revealed pervasive overconfidence among experts. It gave insight into how the role of domain expertise and experience can affect statistical accuracy and informativeness. Moreover, the review demonstrated the need for cross validation, to gauge how well performance on calibration variables predicts performance on the variables of interest.

Researchers at the University of Bristol, UK, Princeton University and Rutgers University in the US, and Resources for the Future, have completed a significant structured expert judgement study into climate change. They employed the Classical Model to investigate the contribution made by the dynamic effects of ice sheets to the global mean sea‑level rise.

Forecasting the imminent rise in sea level is challenging. Even so, the quantification of future sea-level rise uncertainties, particularly upper-end estimates, are urgently required to inform adaptation strategies. Expert elicitation took place at two separate, two-day workshops held in the US and UK in 2018 and involved 22 experts. The format and questions were identical, so that the findings could be combined using the Classical Model. The research team found that by 2100, sea-level rise could exceed 2m, more than twice the upper value put forward by the United Nations Intergovernmental Panel on Climate Change in the Fifth Assessment Report. Moreover, this would have profound consequences for humanity with a potential land loss of 1.79 million km2, including areas of food production, and up to 187 million people displaced.

Using calibration to weight expert responses, a unique feature of the Classical Model, ‘introduces mathematical rigor not found with other elicitation methods’.

The article detailing this study, published in the Proceedings of the National Academy of Sciences of the United States of America (PNAS), has received a great deal of attention. It has been mentioned in 331 news stories from 263 outlets, and as the 70th most-discussed paper in 2019, it sits in the top 5% of all research outputs scored by altmetric.com.

Online module The Classical Model is also the focus of a Massive Open Online Course (MOOC): ‘Decision Making Under Uncertainty: Introduction to Structured Expert Judgment’, presented by Roger Cooke, Tina Nane, and Anca Hanea. This course is arranged into six parts, combining theory and applications in an interactive and engaging manner.

Learners are introduced to structured expert judgment methods, particularly the Classical Model, and advised as to why and when they should be applied. They learn how to account for uncertainty assessments and biases within a complex decision-making setting. There is also the opportunity to analyse expert data with EXCALIBUR or Anduryl, software packages implementing the Classical Model, and obtain answers to questions of interest. Optional modules include exploring dependence elicitation and eliciting probabilities, applying structured expert-judgment methods to real-world scenarios, and using a different method, the IDEA Protocol module. This is the only available module on expert judgment. In its three runs to date, it has attracted more than 7,000 participants from 121 countries.

A follow-up online course is dedicated to applying structured expert judgment. Learners have the opportunity to perform their own study, under close guidance from Tina Nane and Roger Cooke. The learners will design the study and gather, assess the performance of, and combine, expert opinion. They will also carry out the analysis and report the findings of their study.

Evaluation of the Classical Model Cooke has collected expert data from many studies over the years to evaluate the Classical Model and compare it with other possible weighting schemes, including equal weighting and quantile aggregation. The Classical Model outperformed the other aggregation methods considered in the analysis. The findings demonstrate the superiority of the Classical Model, both in terms of in-sample and out-of-sample validation and in terms of point forecast accuracy. Moreover, when compared with other aggregation methods, the performance-based combination of experts generated by the Classical Model is more statistically accurate and more informative.

Personal Response

What inspired you to include performance-based weights in the Classical Model?

Expert subjective probabilities began appearing in technical risk analyses of nuclear power plants in the 1970s. The rigorous reporting in these early studies exposed very wide differences in experts’ judgments and teed up issues of validating and synthesising expert judgment. Any new measurement device, such as Galileo’s telescope, is first ‘calibrated’ by applying it to things we know before employing it to measure things we don’t know. Expert judgment constitutes a new measuring device. Applying this simple idea led to treating experts as statistical hypotheses and validating these hypotheses against calibration variables from their field to which the true values were known.

This feature article was created with the approval of the research team featured. This is a collaborative production, supported by those featured to aid free of charge, global distribution.

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