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Neural General Circulation Models for Weather and Climate

Dmitrii Kochkov, Janni Yuval, Ian Langmore, Peter Norgaard, Jamie Smith, Griffin Mooers, Milan Klöwer, James Lottes, Stephan Rasp, Peter Düben, Sam Hatfield, Peter Battaglia, Alvaro Sanchez-Gonzalez, Matthew Willson, Michael P. Brenner, Stephan Hoyer

arXiv:2311.07222v3physics.ao-phcs.LGphysics.comp-ph

TL;DR

NeuralGCM addresses the limited stability and ensemble skill of machine-learning weather models by combining differentiable atmospheric dynamics with learned physics. It achieves competitive deterministic, ensemble, and climate forecasts while reducing computational costs by 3–5 orders of magnitude.

  • Problem

    Machine-learning weather models had not demonstrated improved ensemble forecasting or sufficient stability for long-term weather and climate simulations.

  • Method

    NeuralGCM combines a differentiable dynamical core for atmospheric equations with a neural-network physics module that parameterizes unresolved processes.

  • Results

    NeuralGCM achieves accurate, stable forecasts across 1–15-day weather and decadal climate timescales, with 3–5 orders of magnitude computational savings over comparable traditional models.

  • Takeaways & Limitations

    The results support machine learning as a viable alternative for improving GCMs and enable computationally intensive tasks such as large-ensemble forecasting.

  • Takeaways & Limitations

    For substantial sea-surface-temperature increases of +4K, NeuralGCM’s climate response diverges from expectations and exhibits climate drift.

Abstract

from arXiv · show

General circulation models (GCMs) are the foundation of weather and climate prediction. GCMs are physics-based simulators which combine a numerical solver for large-scale dynamics with tuned representations for small-scale processes such as cloud formation. Recently, machine learning (ML) models trained on reanalysis data achieved comparable or better skill than GCMs for deterministic weather forecasting. However, these models have not demonstrated improved ensemble forecasts, or shown sufficient stability for long-term weather and climate simulations. Here we present the first GCM that combines a differentiable solver for atmospheric dynamics with ML components, and show that it can generate forecasts of deterministic weather, ensemble weather and climate on par with the best ML and physics-based methods. NeuralGCM is competitive with ML models for 1-10 day forecasts, and with the European Centre for Medium-Range Weather Forecasts ensemble prediction for 1-15 day forecasts. With prescribed sea surface temperature, NeuralGCM can accurately track climate metrics such as global mean temperature for multiple decades, and climate forecasts with 140 km resolution exhibit 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. 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.

Introduction

NeuralGCM addresses limitations of standalone and offline-hybrid machine-learning weather models by coupling learned physics with a differentiable atmospheric dynamical core and training the system end to end. The resulting architecture is designed to support accurate, stable, physically consistent weather and climate simulation.

  • Motivation: GCMs underpin weather and climate prediction by solving atmospheric equations while parameterizing unresolved processes such as clouds, radiation and precipitation.Traditional GCMs combine numerical methods for large-scale dynamics with semi-empirical representations of small-scale physics.
  • Limitations of existing approaches: ML weather models achieved state-of-the-art deterministic 1-10 day forecasts using roughly 40 years of ERA5 reanalysis data for training and initialization.These models rely solely on machine-learning techniques and operate at a fraction of the computational cost described in the passage.
  • Limitations of existing approaches: Existing deterministic ML models do not provide calibrated uncertainty estimates, while offline-trained hybrid components can cause instability and climate drift when coupled to governing dynamics.Hybrid models have also mostly been limited to idealized scenarios such as aquaplanets under realistic-condition constraints.
  • Contribution: NeuralGCM is presented as the first fully-differentiable hybrid atmospheric GCM, enabling online training of ML components within interactions with large-scale governing equations.The model is trained on forecasting up to 5-day weather trajectories sampled from ERA5, with the authors reporting accurate and stable forecasts.
  • Model architecture: NeuralGCM combines a differentiable dynamical core for large-scale fluid motion and thermodynamics with a neural learned physics module for unresolved atmospheric processes.The dynamical core operates under gravity and the Coriolis force, while the learned physics module parameterizes unresolved effects using atmospheric-column information.
  • Training and forecasting: The model advances forecasts by integrating tendencies from its dynamical core and learned physics with an implicit-explicit ODE solver, while end-to-end rollouts grow from 6 hours to 5 days during training.Stochastic models add random fields and use CRPS to balance accuracy with ensemble spread.

Results

NeuralGCM matches leading deterministic and ensemble weather forecasts while preserving realistic weather structures and physical consistency. In climate simulations, it reproduces observed temperature trends, global metrics, and tropical-cyclone behavior, but shows limitations under substantial warming.

  • Weather forecasting: For 1–3-day forecasts, NeuralGCM-0.7° and GraphCast achieve the best RMSE results, with performance varying across atmospheric variables.At longer lead times, RMSE increases rapidly.
  • Ensemble forecasting: Beyond about 7 days, NeuralGCM-ENS and ECMWF-ENS ensemble means achieve considerably lower RMSE than deterministic models.At 1.4° resolution, NeuralGCM-ENS has lower ensemble-mean RMSE, RSMB, and CRPS than ECMWF-ENS across almost all variables, lead times, and vertical levels, with a spread-skill ratio of approximately one.
  • Physical realism: NeuralGCM-ENS produces sharp, statistically representative scenarios for tropical cyclones, atmospheric rivers, and the inter-tropical convergence zone despite its 8× coarser resolution.NeuralGCM-0.7° has power spectra closer to ERA5 than other ML forecasts, while its spectra remain roughly constant over the forecast period.
  • Climate simulation: NeuralGCM reproduces observed climate behavior, including global warming trends, global mean temperature variability, and realistic tropical-cyclone counts and trajectories.The ensemble mean temperature RMSE is 0.16K against ERA5; NeuralGCM produces 83 tropical cyclones versus 86 in ERA5, while X-SHiELD produces 40.
  • Climate simulation limitations: NeuralGCM’s climate response remains limited for substantial warming: its response diverges from expectations at +4K SST increases, and these simulations exhibit climate drift.It reproduces some robust warming-response features at +1K and +2K SST increases.

Discussion

NeuralGCM combines differentiable atmospheric dynamics with machine learning to achieve accurate ensemble weather forecasts and realistic climate simulations. Its computational efficiency enables large ensembles and supports the view that short-term weather prediction can tune climate parameterizations.

  • NeuralGCM is the first ML-based model to make accurate ensemble weather forecasts, with better CRPS than state-of-the-art physics-based models.
  • NeuralGCM achieves comparable spatial bias to global cloud resolving models, realistic tropical cyclone tracks, and realistic historical temperature trends in AMIP-like simulations.
  • 8-40× coarser horizontal resolution than ECMWF IFS and global cloud resolving models enables 3-5 orders of magnitude savings in compute.
  • 70 000 simulation days in 24 hours using a single TPU versus 19 simulated days on 13 824 CPU cores with X-SHiELD demonstrates NeuralGCM’s computational efficiency.
  • NeuralGCM models trained on 72-hour forecasts produce realistic multiyear simulations and capture seasonal circulation, monsoons, and tropical cyclones with historical sea surface temperatures.
  • The approach can incorporate more physics or more ML, while future weather systems may use observational data and neural corrections for data assimilation and diagnostics.

Code availability

NeuralGCM’s code is publicly available through two open-source GitHub projects, with separate packages for the differentiable dynamical core and ML models, plus evaluation code in WeatherBench2.

  • Code availability: Two open-source GitHub projects provide NeuralGCM code: Dinosaur implements the differentiable dynamical core, while NeuralGCM provides ML models and trained-model checkpoints.Evaluation code for NeuralGCM weather forecasts is included in WeatherBench2.

Appendices … Appendix B Dynamical core of NeuralGCM

The appendices define a core-model code-counting method, compare fv3atm and NeuralGCM codebase sizes, and describe the dynamical core’s physics-prior role and organization.

  • Appendix A Lines of code in atmospheric models: Core-model complexity was measured by counting lines of code while excluding tests, examples, input/output, framework coupling, and model-running files.The count targeted code belonging to the core model rather than supporting files.
  • A.1 fv3atm: 376 578 lines of core model code were counted for fv3atm, the atmospheric component of NOAA’s Unified Forecast System.The fv3atm code was downloaded from GitHub on 7 November 2023.
  • A.1 fv3atm: 42 387 lines were in fv3atm’s FV3 dynamical core, while 334 191 lines were in CCPP Physics.These counts cover Fortran files in the two major fv3atm modules, excluding paths containing “test” or “example”.
  • A.3 NeuralGCM: 20 136 lines of core model code were counted for NeuralGCM across its spectral dynamical core and machine learning code.The NeuralGCM total is split between two major modules.
  • A.3 NeuralGCM: 8609 lines comprised NeuralGCM’s spectral dynamical core, and 11 527 lines comprised its machine learning code.The two modules together make up the reported NeuralGCM core-model total.
  • Appendix B Dynamical core of NeuralGCM: The dynamical core supplies NeuralGCM with strong physics priors based on phenomena that are well understood and easy to simulate.Appendix B introduces spatial discretization, governing equations, and references and rationale for numerical implementation choices.

B.1 Discretization of the dynamical core … C.3 Normalization of input features

NeuralGCM combines a Gaussian-grid, sigma-coordinate dynamical core solving primitive equations with neural-network tendencies for unresolved physics and computational errors. Its inputs include atmospheric, surface, stochastic, and normalized features designed for deterministic, ensemble, seasonal, and climate simulations.

  • B.1 Discretization of the dynamical core: NeuralGCM discretizes its computational domain with Gaussian grids and terrain-following sigma coordinates for efficient grid-to-spherical-harmonic transformations and non-stationary vertical spacing.Models use 2.8°, 1.4°, and 0.7° horizontal resolutions, corresponding to TL63, TL127, and TL255 grids.
  • B.2 Primitive equations: The dynamical core solves primitive equations using divergence, vorticity, temperature, logarithmic surface pressure, and three moisture species as prognostic variables.The equations combine momentum, thermodynamic, equation-of-state, continuity, and hydrostatic components.
  • B.3 Numerics: NeuralGCM uses Durran-based numerical schemes and semi-implicit time integration, separating right-hand-side terms to avoid severe restrictions from fast gravity waves.Moisture species are advected by wind and affect dynamics through virtual temperature.
  • Appendix C Learned physics of NeuralGCM: Neural networks approximate unresolved physical processes and dynamical-core errors by adding learned tendencies to the primitive-equation right-hand side in encoded space.The tendency pipeline extracts and normalizes features, applies a neural-network forward pass, and rescales the output.
  • C.1 Input features for all models: Core neural-network inputs describe atmospheric structure through divergence, vorticity, wind, temperature deviation, moisture species, and logarithmic surface pressure, supplemented by geographic and radiative features.Supplementary inputs include derivatives, land-sea mask, solar radiation, orography, latitude functions, pressure levels, and learned embeddings.
  • C.1 Input features for all models: During weather forecasts, sea-surface temperature and sea-ice concentration remain fixed at values from the day before initialization, whereas seasonal and climate forecasts prescribe ERA5 values updated every 6 or 12 hours.The 6-hour interval applies to the 2.8° model and the 12-hour interval to the 1.4° model.
  • C.2 Additional input features for stochastic models: Stochastic models add twenty independent space-time correlated Gaussian random fields, with ten used by the encoder and ten by the forward step, using a different seed for every forecast.NeuralGCM-ENS fields are initialized with length scales from 85km to 10,000km and time scales from 30 minutes to 60 hours.
  • C.3 Normalization of input features: Input features are shifted and rescaled toward zero mean and unit variance, with specific humidity normalized separately at each atmospheric level.Other features are normalized uniformly across atmospheric levels.

C.4 Network architecture … D.1 Encoder

NeuralGCM combines residual fully connected networks with vertical and surface embedding networks to predict atmospheric tendencies. Its learned physics outputs are scaled from ERA5 statistics, while encoder modules transform ERA5 pressure-coordinate states into sigma-coordinate model states with learned corrections.

  • C.4 Network architecture: The fully connected networks use an Encode-Process-Decode architecture with five Process MLP blocks and 384-dimensional latent representations.Each Process block is a three-layer MLP with 384 hidden units.
  • C.4 Network architecture: NeuralGCM-1.4° and NeuralGCM-0.7° use 32-length embeddings, whereas NeuralGCM-ENS and NeuralGCM-2.8° use 8-length embeddings.The models predict wind, temperature, and moisture-species tendencies at all levels.
  • C.4.1 Vertical embedding network: A five-layer 1D convolutional network computes vertical embeddings from eight atmospheric-variable channels, using 64 hidden and 32 output channels.The inputs are aligned by their vertical atmospheric locations.
  • C.4.2 Surface embedding network: Surface embeddings aggregate separate land, sea, and sea-ice embeddings, with SST provided as input to the sea embedding network.Land and sea-ice embedding implementations vary across model resolutions and deterministic versus ensemble configurations.
  • C.5 Network output scaling: 0.01 standard deviation scales learned physics tendencies for each variable before they are added to the model state.The standard deviations are estimated from ERA5 using one-hour finite differences across ten globally averaged snapshots.
  • C.6 Interpretability of learned physics tendencies: The learned physics tendencies show physically plausible signatures, including near-surface temperature tendencies resembling land surface heating and nighttime cooling.The current NeuralGCM implementation does not separate tendencies by physical process.
  • Appendix D Encoder and Decoder of NeuralGCM: Encoder and Decoder modules interface sigma-coordinate NeuralGCM states with pressure-coordinate ERA5 data through regridding and learned corrections.The overall encoder and decoder data flows are illustrated in Fig. D8.
  • D.1 Encoder: The Encoder computes surface pressure, linearly interpolates atmospheric variables to sigma coordinates, and adds a neural-network correction to an ERA5 snapshot.Correction inputs come from pressure levels without embedding features; outputs correct divergence, vorticity, temperature, logarithmic surface pressure, and moisture species, scaled by 0.02 standard deviation.

D.2 Decoder … Appendix F Evaluation metrics

NeuralGCM decodes model states to ERA5 pressure coordinates using diagnosed and interpolated fields plus learned corrections, advances atmospheric dynamics with stable IMEX integration and filtering, and standardizes evaluation against regridded reference data.

  • D.2 Decoder: NeuralGCM’s decoder maps sigma-coordinate model states to ERA5 pressure coordinates by diagnosing geopotential and interpolating to pressure levels.Above-surface pressure levels use linear interpolation; below-surface values require extrapolation from terrain-following sigma coordinates.
  • D.2 Decoder: Decoder network outputs correct horizontal wind, temperature, geopotential, and moisture species after scaling by 0.02 standard deviation before combination with interpolation.Figure D10 compares decoded predictions with predictions produced by sigma-to-pressure interpolation without learned decoder components.
  • Appendix E Time integration: NeuralGCM advances atmospheric states by iteratively integrating dynamical-core effects and learned physics parameterizations with an implicit-explicit scheme.Integration updates occur every 4–30 minutes depending on model resolution.
  • E.1 Time integration scheme: The semi-implicit solver treats fast, low-amplitude gravity-wave terms implicitly and computes remaining dynamical tendencies explicitly, enabling longer stable time steps.This partitions primitive-equation tendencies into implicit linear terms and explicit terms.
  • E.1 Time integration scheme: NeuralGCM uses the semi-implicit Lorenz three-cycle scheme, SIL3, instead of traditional semi-implicit leapfrog integration.The IMEX Runge–Kutta method avoids tracking multiple time steps and the time filtering required by leapfrog.
  • E.2 Filtering: Two exponential filters of different strengths, hard and soft, provide hyper-diffusion to stabilize spectral atmospheric dynamics during integration.Both filters are applied after each integration step, and the hard filter also limits high-frequency noise from learned physics outputs.
  • Appendix F Evaluation metrics: Evaluation compares forecasts X with ground truth Y, typically ERA5, after regridding both to 1.5° and standardizes assessment with WeatherBench2.ECMWF-HRES and ECMWF-ENS forecasts use ECMWF-HRES lead time = 0 analysis to prevent an unfair advantage for data-driven methods.

F.1 Root mean square error (RMSE) … G.1 Optimizer settings

The paper evaluates forecasts with complementary deterministic, probabilistic, bias, and ensemble-calibration metrics, while training NeuralGCM with Adam and a scheduled learning rate. RMSE is area-weighted and informative for deterministic accuracy, but ensemble skill also requires assessing spread and probabilistic behavior.

  • F.1 Root mean square error (RMSE): RMSE compares forecasts with ground truth separately by variable, pressure level, and lead time, averaging over initial times.The spatial norm is area weighted using latitude-dependent nodal weights.
  • F.1 Root mean square error (RMSE): Ensemble RMSE uses the ensemble mean, but cannot alone establish probabilistic skill because identical blurry realizations may still score well.This limitation motivates evaluating ensemble spread and distributional behavior separately.
  • F.2 Root mean squared bias (RMSB): RMSB estimates persistent forecast differences from ground truth by computing RMSE of the bias and aggregating over spatial dimensions.Biases are reported across variables, pressure levels, lead times, and latitude-longitude coordinates.
  • F.3 Continuous Ranked Probability Score (CRPS): CRPS balances forecast deviation from ground truth against ensemble dispersion and is minimized when forecast and ground-truth distributions match.The multidimensional version sums over components and is estimated using M ensemble members.
  • F.3 Continuous Ranked Probability Score (CRPS): CRPS can accept distributions with correct marginals even when forecasts are not jointly correct, so it does not penalize such forecasts.This is an explicit limitation of the multidimensional construction.
  • F.4 Spread-Skill ratio: The spread-skill ratio compares ensemble standard deviation with ensemble-mean RMSE and equals 1 when ensemble members are distributed like the ground truth.The ratio is computed across lead times for ECMWF-ENS and NeuralGCM-ENS, with spatial aggregation differing between global and visual analyses.

G.2 Training data and unroll schedules … Appendix H Additional weather evaluations

NeuralGCM training combines conservatively regridded ERA5 data, scheduled unrolling, rescaled multi-objective losses, and separate deterministic and stochastic optimization strategies. These designs address variable scaling, predictability limits, sharpness, bias, stability, decoder artifacts, and ensemble-learning efficiency across model resolutions.

  • G.2 Training data and unroll schedules: ERA5 data were conservatively regridded to Gaussian grids at 2.8°, 1.4°, and 0.7° resolutions for NeuralGCM training.The scheme linearly aggregates contributions according to relative area overlap.
  • G.2 Training data and unroll schedules: Training data covered 1979–2017 for final 2.8° and 1.4° models, 1979–2019 for 0.7° and stochastic variants, with no 2020 exposure before evaluation.Lead-time unroll length increased as training iterations progressed.
  • G.3 Variable rescaling for losses: Losses rescale atmospheric variables by 24-hour temporal-difference standard deviations and adjust trajectory weighting for increasing error variance with lead time.The trajectory factor is (1 + (τ/24))^-1/2, while spectral losses use (1 + (τ/40)^4)^-1/2.
  • G.4 Loss for deterministic models: Deterministic models train in primary and decoder-fine-tuning stages using rescaled-variable MSE objectives combining accuracy, sharpness, and bias terms.The primary objective applies these terms across data and model representations, producing five loss terms.
  • G.4.1 Accuracy loss: filtered MSE: Filtered MSE mitigates the double-penalty problem by filtering forecast and target fields according to ECMWF-HRES predictability horizons, using a relative-error threshold of 0.12.Model-space terms also encourage an identity encoder/decoder round-trip and were found to enhance stability.
  • G.4.2 Sharpness loss: spectrum MSE: Spectrum MSE uses resolution-dependent spectral cutoffs of 42, 80, and 120 for 2.8°, 1.4°, and 0.7° NeuralGCM models, respectively.Spectral and bias terms had small total-loss contributions but positively affected prediction sharpness.
  • G.5 Decoder fine-tuning: Decoder fine-tuning freezes the Encoder and learned-physics components while optimizing Decoder parameters to remove high-frequency artifacts from spherical-harmonic truncation errors.The same phase is described as a short optimization after main training.
  • G.6 Loss for stochastic models: Stochastic training uses spectral and nodal CRPS with exactly two forecasts per observation, the minimum for an unbiased estimate, favoring more initial-time variety over larger ensembles.Spectral CRPS emphasizes long-range correlations through lower wavenumbers and excludes wavenumbers above 80 for dynamical-core stability.

H.1 Accuracy · H.2 Derived variables and spectra · H.3 Visualization of ensemble weather forecasts

NeuralGCM-0.7° generally outperforms ECMWF-HRES, while NeuralGCM-ENS matches or improves on ECMWF-ENS beyond two days across most evaluated metrics. Derived-variable diagnostics use standardized regridding and numerical calculations, and ensemble forecasts qualitatively resemble ERA5 despite occasional negative cloud values.

  • H.1 Accuracy: NeuralGCM-0.7° generally outperforms ECMWF-HRES across RMSE and RMSB scores for core atmospheric variables and pressure levels, except at the atmosphere’s top level.Comparisons also include GraphCast across geopotential, temperature, specific humidity and u-component of wind.
  • H.1 Accuracy: 10 days: NeuralGCM-0.7° maintains consistently lower bias than ECMWF-HRES, while GraphCast’s initially lowest bias quickly degrades.NeuralGCM-ENS has slightly higher biases than NeuralGCM-0.7°, likely because of its coarser spatial resolution.
  • H.1 Accuracy: Beyond two days, NeuralGCM-ENS has lower or similar error within 1% of ECMWF-ENS across RMSE, RSMB and CRPS, except for specific humidity at 850 hPa.ECMWF-ENS performs better at very early lead times, with differences described as small.
  • H.2 Derived variables and spectra: Derived-variable consistency is assessed from geopotential, temperature, horizontal wind velocity and specific humidity using quantities such as lapse rate, wind speed, divergence and vorticity.The diagnostics also include vertical velocity, eddy kinetic energy, geostrophic and ageostrophic winds, moisture quantities, integrated vapor transport and relative humidity.
  • H.2 Derived variables and spectra: All derivatives use second-order finite differences and vertical integrals use trapezoidal integration, with calculation code added to WeatherBench2.These procedures support the standardized derived-variable diagnostics.
  • H.2 Derived variables and spectra: Forecast fields and power spectra are compared after conservative regridding to a 1.5° equiangular grid on 37 pressure levels.Variables requiring unavailable archived fields are excluded, and NeuralGCM-ENS is omitted because regridding introduces aliasing artifacts, particularly for derived variables.
  • H.3 Visualization of ensemble weather forecasts: NeuralGCM-ENS forecast maps and vertical profiles look qualitatively similar to ERA5 after conservative regridding to NeuralGCM-ENS’s native resolution.Specific cloud liquid water content and specific cloud ice water content occasionally take on small-magnitude negative values.

H.4 Evaluation of lower resolution models … Appendix I Additional climate evaluations

Across its appendices, NeuralGCM improves forecast skill with finer resolution and a dynamical core, while revealing trade-offs among loss design, rollout length, precipitation extremes, and long-term stability. The evaluations also identify realistic regional behavior alongside documented instability cases.

  • H.4 Evaluation of lower resolution models: Increasing resolution from 2.8° toward 0.7° improves RMSE across all atmospheric variables, with results normalized against ECMWF-ENS for comparison.The 2.8° model’s performance is evaluated at 5.6° and rescaled using ECMWF-HRES error ratios at 1.5° and 5.6°.
  • H.5 Diagnosing precipitation minus evaporation: NeuralGCM-0.7°’s precipitation-minus-evaporation distribution aligns closely with ERA5 in the extratropics but underestimates tropical extreme events.Its third-day, initial-condition-averaged spatial distribution also aligns well with ERA5, while snapshots show moderated tropical extremes.
  • H.5 Diagnosing precipitation minus evaporation: Diagnosed precipitation-minus-evaporation is generally consistent with ERA5, but dynamical-core errors and neural corrections may introduce diagnostic errors.The calculation assumes the dynamical core supplies all horizontal motions, although the physics module may correct coarse-grid advection errors.
  • H.6.1 Different loss functions: Using solely MSE improves deterministic RMSE by several percent but causes spectral degradation, whereas added loss terms may address that degradation at some accuracy cost.The ablation compared bias, filtered-MSE, and spectral-loss variants in NeuralGCM-2.8° models.
  • H.6.2 Training on shorter rollouts: Training only on predictions up to 12 hours deteriorates RMSE and substantially reduces stability compared with the default NeuralGCM-2.8° training rollout.All other model parameters were kept identical in this comparison.
  • H.6.3 Learning curve - pure ML vs hybrid: NeuralGCM-2.8° does not benefit from more than the last 21 years of data for weather forecasting, while its dynamical core improves performance over the ML-only variant.The hybrid and ML-only models were trained with varying amounts of data and evaluated using the same learning-curve setup.

I.1 Seasonal cycle · I.2 Tropical cyclone tracking · I.3 CMIP6 models used in AMIP runs

NeuralGCM-1.4° reproduces key seasonal atmospheric cycles and tropical-cyclone track characteristics while maintaining stable, drift-free simulations. The comparison also uses 22 CMIP6 AMIP models, including 17 with specified accessible variant identifiers and five requiring alternatives.

  • I.1 Seasonal cycle: 35 of 37 NeuralGCM-1.4° simulations completed the full two-year experiment without model instability.The simulations used initial conditions spaced 10 days apart during 2019 and were evaluated for 2020 against ERA5.
  • I.1 Seasonal cycle: NeuralGCM-1.4° captures the Hadley cell’s seasonal cycle and amplitude and accurately represents Indian monsoon and non-monsoon wind structure.The Hadley circulation is evaluated using the mass streamfunction, while winds are compared spatially with ERA5.
  • I.1 Seasonal cycle: NeuralGCM-1.4° reproduces ERA5-like extratropical zonal-wind structure and the seasonal and spatial structure of eddy kinetic energy.Noticeable zonal-wind differences occur above 30 hPa because the model is not optimized for those levels.
  • I.1 Seasonal cycle: NeuralGCM-1.4° shows no visible drift in global atmospheric water or total kinetic energy and achieves a realistic annual cycle.Its ensemble-mean precipitable water matches ERA5 magnitude with lower RMSE than climatology, while ensemble-mean total kinetic energy trends slightly low.
  • I.2 Tropical cyclone tracking: 88 tropical cyclones were detected at native ERA5 resolution versus 84 at 1.4° resolution after tracker-parameter tuning.The 1.4° configuration reduced the required sea-level-pressure increase to at least 0.6 hPa within a 5.5° great-circle-distance radius.
  • I.2 Tropical cyclone tracking: NeuralGCM tropical-cyclone tracks closely resemble ERA5 in number, locations, and shapes when using tuned SLP and vorticity tracking parameters.The tuned parameters produce similar tracks under both criteria and account for resolution-dependent tracker behavior.
  • I.3 CMIP6 models used in AMIP runs: 22 CMIP6 models were analyzed in prescribed-sea-surface-temperature AMIP runs, with 17 using the accessible “r1i1p1f1” variant and five using alternatives.Alternative identifiers were used for CNRM-CM6-1, CNRM-ESM2-1, HadGEM3-GC31-LL, HadGEM3-GC31-MM, and UKESM1-0-LL.

I.4 Generalizing to unseen data · I.4.1 Weather forecasting in warmer years

NeuralGCM is evaluated for generalization beyond its training distribution, first by testing weather forecasts in warmer future years and then by studying warmer-climate simulations. A model trained only on 1979–2000 data retains broadly stable geopotential forecast skill across 21 later years, while GraphCast degrades by a few percent on 2022 forecasts.

  • I.4 Generalizing to unseen data: NeuralGCM incorporates strong physics priors to improve performance on out-of-sample conditions caused by climate change or observing-system changes.The motivation is that physically consistent weather models should remain accurate under different climates and unfamiliar weather conditions.
  • I.4 Generalizing to unseen data: The broader unseen-data evaluation includes tropical-cyclone tracks, densities, annual regional counts, and maximum-wind distributions from ERA5 and NeuralGCM simulations.The supplied figure passages specify ERA5 periods, NeuralGCM resolutions, and 34 initial conditions for the relevant comparisons.
  • I.4 Generalizing to unseen data: The generalization study first evaluates weather forecasts in warmer years before investigating simulations of warmer climates.This establishes the order of experiments in the section.
  • I.4.1 Weather forecasting in warmer years: NeuralGCM-2.8° was trained exclusively on 1979–2000 ERA5 data and evaluated across 21 future years for mid-range forecasts of geopotential, temperature, and specific humidity.Figure I47 reports relative RMSE normalized to 2001 for 500 hPa geopotential height, 850 hPa temperature, and 700 hPa specific humidity.
  • I.4.1 Weather forecasting in warmer years: Geopotential forecast skill remains remarkably consistent when NeuralGCM-2.8° extrapolates 20 years beyond its training period.The passage also notes a noticeable decline in 4-day forecast performance, but the supplied text does not identify the affected variable.
  • I.4.1 Weather forecasting in warmer years: GraphCast and NeuralGCM-0.7° variants were compared with ECMWF-HRES using forecasts evaluated on ERA5 data from 2018–2022.GraphCast and NeuralGCM-0.7°-2017 were trained through 2017, while the final NeuralGCM-0.7° version was trained through 2019.
  • I.4.1 Weather forecasting in warmer years: The NeuralGCM comparison model omitted surface embeddings and forcing features, the memory feature, and used a different time-scaled loss.Its loss used (1+(τ/24))−1 instead of (1+(τ/40)4)−1/2.
  • I.4.1 Weather forecasting in warmer years: GraphCast accuracy degrades by a few percent on 2022 forecasts, five years after its training domain, whereas ECMWF-HRES and NeuralGCM show little variation across evaluation years.The largest error increases occur for geopotential forecasts and at short lead times.

I.4.2 Extrapolation to warmer climates

NeuralGCM was tested outside its ERA5 training regime using multi-decade simulations with warmer sea surface temperatures. It reproduced several warming responses and short-term temperature adjustment, but warmer-climate runs—especially +4K—developed climate drift.

  • Motivation: All NeuralGCM models were trained solely on ERA5, so their ability to extrapolate to warmer climates was explicitly treated as uncertain.The study tested whether extrapolation remains reasonable when input/output distributions do not substantially change from training conditions.
  • Experimental setup: 34-year AMIP-like simulations with NeuralGCM-2.8° used +1K, +2K, and +4K SST scenarios, with CESM AMIP runs as baselines.Of eight initializations, six +1K, six +2K, and four +4K simulations remained stable for the full period.
  • Warming response: NeuralGCM’s +1K and +2K SST runs broadly matched CESM’s zonal-mean temperature and zonal-wind responses.Upper-tropospheric warming, polar amplification, and Southern Hemisphere polar-jet shift appeared across the simulations.
  • Warming response: The 850hPa global temperature adjusted to +4K SST warming on the same timescale in NeuralGCM and MIROC6.This indicates a short-term response similar to that of a physics-based model.
  • Limitations: Unlike CESM, NeuralGCM’s warmer-SST simulations did not follow the AMIP trend, indicating climate drift that was particularly evident under +4K warming.The passage identifies +4K as a scenario in which NeuralGCM had already been noted to behave unrealistically.
  • Future directions: Improving extrapolation may require climate-invariant methods, training on simulated warmer climates, and physically based components such as radiation schemes.The proposed strategies combine broader training data with physical-model elements.
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