Neural General Circulation Models for Weather and Climate

Introduction

• NeuralGCM targets GCM biases, manual parameterization tuning, and uncertain long-term projections while addressing ML models’ missing calibrated uncertainty, blurry forecasts, diagnostic-variable errors, and limited long-term skill.Existing hybrid models were commonly trained offline, risking instability and climate drift, and had performed worse than state-of-the-art models under realistic conditions.Source: Introduction · pp. 2–3
• NeuralGCM is presented as the first fully differentiable hybrid atmospheric GCM, achieving physically consistent and stable forecasts comparable to best-in-class models from 1–15-day weather through decadal climate prediction.Differentiability enables online training that optimizes learned components within their interactions with governing equations for large-scale dynamics.Source: Introduction · p. 3

Neural general circulation models

• NeuralGCM combines a differentiable dynamical core for large-scale fluid motion and thermodynamics with a neural learned-physics module for unresolved processes including clouds, radiation, precipitation, and subgrid dynamics.The model encodes ERA5 pressure-coordinate data into sigma coordinates, advances tendencies with an implicit-explicit ODE solver, and decodes predictions back to pressure levels.Source: Figure 1 · p. 3, Neural general circulation models · p. 4

Training

• End-to-end training advances forecasts over rollouts that grow from 6 hours to 5 days, allowing learned physics and dynamics to account for their interactions during optimization.Deterministic and stochastic models use distinct training protocols.Source: Training · p. 5
• Deterministic models combine accuracy, spectral-sharpness, and bias-penalizing losses, while stochastic models use CRPS with learned correlated random fields and two ensemble members per forecast.The deterministic loss design supports models trained on 3-day rollouts that remain stable in simulations lasting years to decades.Source: Training · p. 5

Accuracy

• NeuralGCM-ENS has lower ensemble-mean RMSE, RMSB, and CRPS than ECMWF-ENS across almost all variables, lead times, and levels, while maintaining a spread-skill ratio of approximately one.At short lead times, NeuralGCM-0.7° and GraphCast achieve the best deterministic RMSE, while NeuralGCM models show notably lower tropical specific-humidity bias.Source: Accuracy · pp. 6–7, Figure 2 · p. 7

Physical consistency

• NeuralGCM preserves realistic weather structures: its 0.7° model is sharper than pure ML forecasts, and NeuralGCM-ENS produces representative scenarios despite 8× coarser resolution.Its spectra remain roughly constant over forecast time and improve with resolution, while water-budget diagnostics match ERA5 in mean P-E structure but underestimate tropical extremes.Source: Physical consistency · pp. 8–9
• NeuralGCM generalizes beyond training data without GraphCast’s increasing-error trend, and its geostrophic-wind structure is more accurate than GraphCast’s but slightly less aligned with ERA5 than ECMWF-HRES.The 2.8° model was tested on more than 21 unseen years, while geostrophic-wind structure shows little degradation after the initial few days.Source: Physical consistency · p. 9

Seasonal cycle and emergent phenomena

• NeuralGCM-1.4° completed two-year simulations for 35 of 37 initial conditions and reproduced seasonal global temperature variability, with ensemble-mean RMSE 0.16K versus climatology’s 0.45K.It also captured seasonal hydrological and dynamical patterns, while precipitable-water RMSE was 1.09 mm versus 1.74 mm for X-SHiELD and 1.36 mm for climatology.Source: Seasonal cycle and emergent phenomena · pp. 10–11, Figure 4 · p. 12
• At 1.4° resolution, NeuralGCM produced 83 tropical cyclones versus 86 in ERA5 and realistic trajectories, whereas regridded X-SHiELD produced 40.This demonstrates emergent tropical-cyclone behavior despite NeuralGCM’s coarse resolution.Source: Seasonal cycle and emergent phenomena · p. 11, Figure 4 · p. 12

Decadal simulations - AMIP-like experiments

• Across 40-year AMIP-like simulations, NeuralGCM captured ERA5 warming trends and had smaller spatial biases than CMIP6 AMIP runs, but its response diverged for +4K SST increases and exhibited climate drift.NeuralGCM also reduced upper-tropospheric warming bias relative to AMIP runs, while 22 of 37 simulations remained stable for the full 40 years.Source: Decadal simulations - AMIP-like… · p. 11, Figure 4 · p. 12

• NeuralGCM is the first ML-based model to produce accurate ensemble weather forecasts with better CRPS than state-of-the-art physics-based models, while achieving realistic climate behavior.It matches global cloud-resolving models in spatial bias, simulates realistic tropical cyclone tracks, and reproduces historical temperature trends in AMIP-like simulations.Source: Discussion · pp. 12–13
• NeuralGCM runs at 8-40× coarser resolution than ECMWF IFS and global cloud-resolving models, enabling 3-5 orders of magnitude lower compute costs.NeuralGCM-1.4° simulates 70,000 days in 24 hours on one TPU versus 19 days on 13,824 CPU cores with X-SHiELD, enabling large ensemble forecasting.Source: Discussion · p. 13
• NeuralGCM supports the hypothesis [47] that short-term weather prediction can tune climate parameterizations, as 72-hour-trained models produce realistic multiyear simulations.With historical sea surface temperatures, the models capture seasonal circulation, monsoons, and tropical cyclones, but alternative training strategies [47] may be needed for subtle cloud feedbacks.Source: Discussion · p. 13
• NeuralGCM can combine additional physics and machine learning, but climate projection requires coupling to ocean, land, and atmospheric chemistry components.Remaining challenges include generalizing to unprecedented climates, satisfying physical constraints, and avoiding numerical instabilities and climate drift; physics-based modules such as radiation may help.Source: Discussion · p. 13
• Differentiable hybrid modeling could extend beyond weather and climate to materials discovery, protein folding, and multiphysics engineering design.Source: Discussion · p. 13

Code availability

• The Dinosaur and NeuralGCM projects are publicly available on GitHub, providing the differentiable dynamical core, ML models, and trained-model checkpoints.WeatherBench2 includes NeuralGCM weather-forecast evaluation code.Source: Code availability · p. 14

Appendix A Lines of code in atmospheric models

• The code-complexity measure counts “core model” lines while excluding tests, examples, input/output, framework coupling, and model-running files.This counting rule defines the scope of the atmospheric-model comparison.Source: Appendix A · p. 21

A.1 fv3atm

• The fv3atm total is based on 376 578 counted lines from the atmospheric component of NOAA’s Unified Forecast System weather model, downloaded on 7 November, 2023.The repository source was NOAA-EMC/fv3atm.Source: §A.1 · p. 21
• fv3atm contains 42 387 lines in the FV3 dynamical core and 334 191 lines in the Common Community Physics Package Physics.The counts cover Fortran files in the two major fv3atm modules, excluding paths containing “test” or “example”.Source: §A.1 · p. 21

Appendices

• 20 136 lines of core model code in NeuralGCM contrasts with 376 578 lines counted for fv3atm.Both counts exclude tests, examples, input/output, framework coupling, and model-running files.Source: §A.3 · p. 21, §A.1 · p. 21

A.3 NeuralGCM

• NeuralGCM’s 20 136 core-model lines are split between 8609 lines in its spectral dynamical core and 11 527 lines in machine-learning code.This decomposition identifies the two major NeuralGCM modules included in the count.Source: §A.3 · p. 21

Appendix B Dynamical core of NeuralGCM

• NeuralGCM’s dynamical core supplies strong physics priors from phenomena described as well understood and easy to simulate.Appendix B directs readers to spatial discretization, governing equations, and references and rationale for numerical implementations.Source: Appendix B · p. 22

B.1 Discretization of the dynamical core

• NeuralGCM uses Gaussian grids with terrain-following sigma coordinates, training at 2.8°, 1.4°, and 0.7° horizontal resolution and 32 equidistant vertical levels.The corresponding spectral grids are TL63, TL127, and TL255 for training, with cubic truncations T62, T125, and T253 used for equation solving to reduce aliasing and accelerator padding costs.Source: §B.1 · p. 22

B.2 Primitive equations

• The dynamical core solves primitive equations in divergence-vorticity form for seven prognostic variables: winds, temperature, surface pressure, and three moisture species.Temperature is decomposed into a uniform sigma-level reference temperature and level-wise deviations.Source: §B.2 · pp. 22–23

B.3 Numerics

• NeuralGCM uses semi-implicit time integration, treating right-hand-side terms explicitly or implicitly to avoid severe time-step limits from fast gravity waves.Moisture is advected by the wind and affects dynamics through virtual temperature; the schemes follow Durran’s numerical prescriptions.Source: §B.3 · pp. 23–24

Appendix C Learned physics of NeuralGCM

• Learned physics adds neural-network tendencies Ψ(˜X) to the primitive-equation tendencies Φ(˜X) in encoded state space, starting from an encoded ERA5 initial state.The decoded NeuralGCM state is evaluated against the ERA5 state after integration.Source: Appendix C · p. 25

C.1 Input features for all models

• All models use atmospheric state features plus spatial, geographic, radiative, surface, pressure-level, and learned location-specific features, with shared-weight vertical and surface embedding modules.Surface embeddings additionally use sea-surface temperature and sea-ice concentration; for weather forecasts these inputs remain fixed at prelaunch values, while seasonal and climate forecasts prescribe ERA5 values at 6- or 12-hour intervals.Source: §C.1 · p. 26

C.2 Additional input features for stochastic models

• Stochastic models inject twenty independent space-time correlated Gaussian random fields, using separate ten-field sets in the encoder and forward step to generate statistically independent ensembles.NeuralGCM-ENS initializes field length scales from 85 km to 10,000 km and time scales from 30 minutes to 60 hours, with parameters later optimized.Source: §C.2 · pp. 26–27

C.3 Normalization of input features

• Input features are shifted and rescaled toward zero mean and unit variance, with specific humidity normalized per level and normally distributed learned, random-field, and embedding features excluded.The normalization statistics are estimated at initialization and applied before neural-network processing.Source: §C.3 · p. 27

C.4 Network architecture

• The core network uses a residual Encode-Process-Decode architecture with five 3-layer MLP process blocks operating on 384-dimensional latent vectors.All input features are concatenated before encoding, and the decoded outputs are split into per-level variable values.Source: §C.4 · p. 27
• Model-specific grid-point embeddings and 8- or 32-length embedding vectors account for parameter differences, while all models predict multilevel wind, temperature, and moisture tendencies.Higher-resolution models require more parameters; NeuralGCM-1.4° and 0.7° use 32-length embeddings, whereas NeuralGCM-ENS and 2.8° use 8-length embeddings.Source: §C.4 · p. 28

C.4.1 Vertical embedding network

• A five-layer 1D convolutional network computes vertical embeddings from eight atmospheric variables aligned by their vertical locations.The network uses 64 hidden channels and 32 output channels.Source: §C.4.1 · p. 28

C.4.2 Surface embedding network

• Surface embeddings aggregate land, sea, and sea-ice representations, using SST for sea and model-dependent learned or atmospheric-input networks for land and ice.Each surface-specific network produces an 8-dimensional embedding, combined according to each grid point’s surface fractions.Source: §C.4.2 · pp. 28–29

C.5 Network output scaling

• Learned-physics outputs are scaled by 0.01 standard deviation of each tendency variable estimated from globally averaged ERA5 finite-difference snapshots.The standard deviations use 10 snapshots and 1-hour differences.Source: §C.5 · p. 29

C.6 Interpretability of learned physics tendencies

• Learned physics tendencies show plausible surface heating, upper-atmospheric warming, and near-surface wind counteraction consistent with drag effects.These signatures were identified in NeuralGCM-0.7° tendencies, although the current model cannot separate tendencies by physical process.Source: §C.6 · p. 29

Appendix D Encoder and Decoder of NeuralGCM

• NeuralGCM interfaces sigma-coordinate simulations with ERA5 through encoder and decoder modules that combine pressure–sigma regridding with learned corrections.The overall encoder and decoder data flow is presented schematically in Fig. D8.Source: Appendix D · p. 31

D.1 Encoder

• The encoder converts ERA5 pressure-level snapshots to sigma-coordinate states through surface-pressure calculation, interpolation, and neural correction that substantially reduces initialization shock.Corrections target divergence, vorticity, temperature, logarithmic surface pressure, and moisture species; encoder-derived states retain the overall atmospheric structure seen with interpolation alone.Source: §D.1 · p. 32

D.2 Decoder

• The decoder diagnoses geopotential, interpolates sigma-coordinate fields to pressure levels, extrapolates below-surface values, and adds neural corrections modeled on learned physics.Geopotential and temperature use linear extrapolation, while other variables use boundary values; the procedure approximates ECMWF pressure extrapolation.Source: §D.2 · p. 32
• Decoder corrections are scaled to 0.02 standard deviation, leaving atmospheric structure largely unchanged while mainly affecting high-pressure levels near or below topography.The comparison includes NeuralGCM-0.7° predictions processed with the decoder versus simple sigma-to-pressure interpolation; minor artifacts occur around mountains.Source: §D.2 · p. 32, Figure D10 · p. 34

Appendix E Time integration

• NeuralGCM advances atmospheric states with an IMEX scheme, updating model states every 4–30 minutes while recomputing learned physics tendencies every 30–60 minutes.Dynamical-core tendencies are computed at every step; less frequent learned-physics updates reduce excessive backpropagation through neural networks.Source: Appendix E · p. 35

E.1 Time integration scheme

• The solver treats fast, low-amplitude gravity-wave terms implicitly and remaining dynamical tendencies explicitly, enabling longer stable time steps for the primitive equations.This semi-implicit partitioning is applied within the atmospheric model’s ODE integration.Source: §E.1 · p. 35
• NeuralGCM uses the semi-implicit Lorenz three-cycle Runge–Kutta scheme, SIL3, instead of traditional semi-implicit leapfrog integration.SIL3 avoids tracking multiple time steps and time filtering and was developed for global spectral weather models.Source: §E.1 · p. 35

E.2 Filtering

• Two exponential hyper-diffusion filters stabilize spectral dynamics, and a hard filter also suppresses high-frequency noise from learned physics outputs.Filtering strength determines the simulation’s effective length scale, generally slightly larger than grid spacing.Source: §E.2 · pp. 35–36

Appendix F Evaluation metrics

• Evaluations regrid forecasts and targets to 1.5° and use WeatherBench2 metrics, with ECMWF forecast analyses as lead-time-zero references to avoid data-driven advantages.Primary metrics are RMSE, RSMB, CRPS, and skill-spread ratio; ERA5 is the ground truth in most cases.Source: Appendix F · p. 37

F.1 Root mean square error (RMSE)

• RMSE compares forecasts with ground truth at each lead time, variable, pressure level, and area-weighted spatial location, averaged over initial times.The nodal norm weights latitude to prevent densely sampled polar regions from dominating the score.Source: §F.1 · p. 37
• For deterministic forecasts, RMSE favors the conditional mean and is mainly used at shorter lead times, while ensemble RMSE uses the ensemble mean but cannot assess probabilistic skill alone.Identical blurry ensemble members can achieve good ensemble-mean RMSE while missing extremes and probabilistic information.Source: §F.1 · p. 37

F.2 Root mean squared bias (RMSB)

• RMSB measures persistent forecast–truth differences by computing the spatially aggregated RMSE of biases for each variable, level, lead time, and coordinate.Biases are averaged over time before the RMSE and spatial aggregation are applied.Source: §F.2 · p. 38

F.3 Continuous Ranked Probability Score (CRPS)

• CRPS evaluates ensemble forecasts by combining deviation from ground truth with ensemble spread and is minimized when forecast and ground-truth distributions match.The multidimensional version sums over components and is estimated using ensemble members; matching marginals can suffice, so CRPS need not require correct joint forecasts.Source: §F.3 · p. 38

F.4 Spread-Skill ratio

• The spread-skill ratio compares ensemble standard deviation with ensemble-mean RMSE, uses a debiased error estimate, and equals 1 when ensemble members match the ground-truth distribution.The ratio is computed across lead times for ECMWF-ENS and NeuralGCM-ENS, with spatial aggregation performed before global ratios.Source: §F.4 · p. 39

Appendix G Training

• NeuralGCM training uses Adam to minimize losses between predictions and coarsened ERA5 trajectories, progressively extending the loss lead-time horizon during training.Trajectory errors are rescaled before loss computation, and deterministic and stochastic models use separate loss functions.Source: Appendix G · p. 40

G.1 Optimizer settings

• All models use Adam with β1 = 0.9, β2 = 0.95, and ϵ = 10−6, while learning rates warm up, remain constant, and then undergo exponential decay; decoder fine-tuning uses a separate schedule.Peak learning rates and total training steps vary by model configuration and are reported in Table G4.Source: §G.1 · p. 40, Table G4 · p. 40

G.2 Training data and unroll schedules

• ERA5 was conservatively regridded to 2.8°, 1.4°, and 0.7° Gaussian grids, while training data ranges varied by model and excluded 2020 before final evaluation.Unroll length increased with training iterations, extending the prediction horizon as models improved.Source: §G.2 · pp. 40–41, Figure G11 · p. 41

G.3 Variable rescaling for losses

• Loss inputs are rescaled by variable-specific temporal-difference scales and by lead time, using (1 + (τ/24))^-1/2 except for spectral losses, which use (1 + (τ/40)^4)^-1/2.Additional factors balance geopotential, specific humidity, logarithmic surface pressure, and cloud moisture contributions.Source: §G.3 · p. 41

G.4 Loss for deterministic models

• Deterministic models use primary training with accuracy, sharpness, and bias losses across data and model representations, followed by decoder fine-tuning.Empirical loss scales are λdata = 20, λspec = 0.1, λmodel = 1 and λbias = 2; spectral and bias terms improved prediction sharpness.Source: §G.4 · p. 42

G.4.1 Accuracy loss: filtered MSE

• Filtered MSE mitigates the double penalty problem by retaining predictable scales, using a relative-error threshold of 0.12 to set lead-time-dependent spectral cutoffs.Model-space terms encourage the encoder/decoder round trip to be the identity and enhance stability.Source: §G.4.1 · p. 43, Figure G12 · p. 43

G.4.2 Sharpness loss: spectrum MSE; G.4.3 Bias loss: spectral bias MSE

• Spectrum MSE uses spectral cutoffs of 42, 80, and 120 for NeuralGCM models at 2.8°, 1.4°, and 0.7°, respectively, while spectral bias MSE discourages bias accumulation.Spectral bias MSE averages over batch and lead-time dimensions before computing the distance norm.Source: §G.4.2 · p. 44, §G.4.3 · p. 44

G.6 Loss for stochastic models

• CRPS training uses exactly two forecasts per observation, allocating compute to more distinct initial times because this reduces variance linearly versus sub-linear gains from larger ensembles.The stochastic loss combines spectral and nodal CRPS; spectral terms preserve long-range correlations and exclude wavenumbers above 80 for dynamical-core stability.Source: §G.6 · pp. 44–45

G.7 Training resources

• Training uses data parallelism with one example per TPU device, and the 0.7° and ensemble models additionally use spatial model parallelism.Resource requirements for the model variants are reported in Table G6.Source: §G.7 · p. 45, Table G6 · p. 46

H.1 Accuracy

• NeuralGCM-0.7° maintains lower bias than ECMWF-HRES for 10 days, while GraphCast’s early low bias degrades quickly and NeuralGCM-ENS has slightly higher bias.The NeuralGCM-ENS comparison with ECMWF-ENS uses RMSE, RMSB and CRPS scorecards over 15-day forecasts.Source: §H.1 · p. 47, Figure H14 · p. 49
• Beyond two days, NeuralGCM-ENS has lower or similar RMSE, RMSB and CRPS to ECMWF-ENS across variables, except specific humidity at 850 hPa at longer lead times.At very early lead times, ECMWF-ENS scores better, but the absolute differences are small; NeuralGCM-ENS also shows similar spatial skill patterns and better, more evenly distributed spatial biases.Source: §H.1 · pp. 47, 49, Figure H14 · p. 49
• NeuralGCM-0.7° generally outperforms ECMWF-HRES, while NeuralGCM-ENS surpasses GraphCast in RMSE at 5–6 days across atmospheric variables.Against GraphCast, short-lead RMSE is similar, with variable-dependent advantages; NeuralGCM-ENS is also more useful as forecast uncertainty dominates at longer lead times.Source: §H.1 · p. 47, Figure H13 · p. 48

H.2 Derived variables and spectra

• Derived diagnostics extend evaluation beyond core fields to lapse rate, wind speed, divergence, vorticity, vertical velocity, kinetic energy, moisture, transport and humidity.The calculations use second-order finite differences and trapezoidal vertical integration, with code added to WeatherBench2 [10].Source: §H.2 · pp. 49–51
• Forecast fields and power spectra are compared after conservative regridding to a 1.5° grid on 37 pressure levels, but NeuralGCM-ENS is omitted because regridding causes aliasing artifacts.Variables requiring unavailable archived pressure levels, derivatives or vertical integrals are also excluded.Source: §H.2 · pp. 53–54

H.3 Visualization of ensemble weather forecasts

• NeuralGCM-ENS forecast maps and vertical profiles look qualitatively similar to ERA5 after conservative regridding to the model’s native resolution.The comparison covers ensemble forecast fields and vertical profiles across the evaluated variables.Source: §H.3 · p. 54
• Specific cloud liquid and ice water content occasionally become slightly negative, which is physically impossible and identified as an opportunity for future improvement.This limitation appears in the qualitative comparison with ERA5.Source: §H.3 · p. 54

H.4 Evaluation of lower resolution models

• Across atmospheric variables, increasing NeuralGCM resolution from 2.8° to 1.4° and 0.7° improves RMSE skill on 2020 evaluations normalized against ECMWF-ENS.NeuralGCM-2.8° skill is evaluated on 5.6° and rescaled using ECMWF-HRES error ratios at 1.5° and 5.6° to estimate performance without upsampling.Source: §H.4 · p. 54

H.5 Diagnosing precipitation minus evaporation

• NeuralGCM-0.7° closely matches ERA5 precipitation-minus-evaporation distributions in the extratropics and spatial means, but underestimates tropical extremes.The calculation assumes the dynamical core supplies all horizontal motions; at 1.4°, resulting climatologies show artifacts whose origin remains uncertain.Source: §H.5 · pp. 54–55, Figure H30 · p. 69

Appendix I Additional climate evaluations

• The 1.4° precipitation-minus-evaporation climatology in Appendix I contains clear artifacts, although whether they arise from the diagnostic approximation or another issue is unresolved.The diagnostic error may be small at 0.7° but could be larger at lower resolutions.Source: §H.5 · p. 55

H.6.1 Different loss functions

• Using only MSE improves RMSE by several percent, but removing spectral, bias, or filtering losses degrades spectra over time and may undermine climate simulations.The spectral degradation suggests loss terms can address failures seen in prior ML models, although they may reduce accuracy.Source: §H.6.1 · pp. 55–56, Figure H31 · p. 70

H.6 Ablation tests

• The ablation tests reveal that forecast accuracy and long-term atmospheric realism favor different training choices, motivating the full loss design and extended-rollout training.Simple MSE improves short-term RMSE, whereas longer rollouts improve stability; these results jointly qualify the ablation conclusions.Source: §H.6.1 · p. 55, §H.6.2 · p. 56

H.6.2 Training on shorter rollouts

• Training only on rollouts up to 12 hours worsens RMSE and stability, with a sharp RMSE increase after 13 days linked to model instabilities.A model trained on 1-hour rollouts became unstable after very few days.Source: §H.6.2 · p. 56, Figure H31 · p. 70

H.6.3 Learning curve - pure ML vs hybrid

• The hybrid NeuralGCM outperforms its ML-only counterpart, but neither benefits from more than the last 21 years of training data for weather forecasting.The hybrid’s similar learning curve does not reduce data requirements, while the proposed lower-data advantage of its single-column structure remains speculative.Source: §H.6.3 · p. 56, Figure H33 · p. 72

H.7 Instability case study

• Instability case studies show realistic fields can rapidly develop localized or high-wavenumber temperature tendencies that intensify and propagate through later simulation days.In the 1.4° case, instability begins near the South Indian Ocean on day 389 and spreads by day 392; in the 0.7° case, it begins on day 139 and spreads through the tropics by day 155.Source: §H.7 · p. 57, Figure H34 · p. 73, Figure H35 · p. 73

I.1 Seasonal cycle

• NeuralGCM-1.4° closely matches ERA5’s 850-hPa global-mean temperature mean and variability across stable two-year simulations initialized from 37 conditions.Thirty-five of 37 initial conditions completed the full two years, and the comparison focused on 2020.Source: §I.1 · p. 74
• NeuralGCM-1.4° captures the Hadley cell’s seasonal cycle and amplitude, including realistic Indian monsoon and non-monsoon wind structure.The Hadley circulation is evaluated using the mass streamfunction ψ(ϕ, p).Source: §I.1 · p. 74, Figure I36 · p. 76, Figure I37 · p. 77
• NeuralGCM-1.4° reproduces ERA5’s extratropical zonal-wind structure and seasonal EKE cycle, with noticeable wind differences above 30 hPa.The model also captures EKE’s seasonal and spatial structure, while its upper-level predictions were not optimized above approximately 30 hPa.Source: §I.1 · p. 74, Figure I39 · p. 79
• NeuralGCM-1.4° shows no visible annual drift in atmospheric water or total kinetic energy, and its ensemble precipitable-water RMSE is below climatology despite a low TKE bias.The ensemble mean precipitable water matches ERA5’s 2020 magnitude, whereas ensemble TKE trends lower than ERA5.Source: §I.1 · pp. 74–75, Figure I38 · p. 78

I.2 Tropical cyclone tracking

• The tracking calibration yields 88 native-resolution ERA5 TCs versus 84 at 1.4° with SLP and 86 at 1.4° with vorticity.The 1.4° NeuralGCM tracking omits the 10-m wind criterion because those data were unavailable; ensemble tracking combines SLP and vorticity criteria when 300-hPa geopotential is unavailable.Source: §I.2 · pp. 75, 77, 78, Table I7 · p. 80
• NeuralGCM-1.4° produces tropical-cyclone tracks that closely resemble ERA5 in number, location, and shape under both SLP-based and vorticity-based tracking.Resolution-specific tracker parameters were tuned because TempestExtremes can yield different cyclone counts across resolutions; the resulting ERA5 comparisons were nearly identical.Source: §I.2 · pp. 75, 77, 78

I.3 CMIP6 models used in AMIP runs

• The AMIP comparison analyzed 22 CMIP6 models with prescribed SSTs, using accessible variant identifiers for 17 models and alternatives for five others.E3SM was excluded because of temperature artifacts near Antarctica.Source: §I.3 · pp. 79, 81

I.4 Generalizing to unseen data

• The warmer-year tests assess whether physically informed NeuralGCM models remain accurate under climates and weather conditions outside their training data.The motivation is to improve out-of-sample performance under climate change or systematic changes in Earth observing systems.Source: §I.4 · pp. 81, 84
• The broader unseen-data evaluation also examines NeuralGCM simulations against CMIP6 AMIP models and ERA5-based tropical-cyclone behavior.The supplied figures cover 850 hPa temperature bias, cyclone tracks, densities, annual regional counts, and maximum-wind distributions, but do not provide comparative values here.Source: Figure I42 · p. 82, Figure I43 · p. 82, Figure I44 · p. 83

I.4.1 Weather forecasting in warmer years

• A NeuralGCM-2.8° model trained only on 1979–2000 ERA5 retains mid-range forecast skill when evaluated across 21 future years.The experiment evaluates 500 hPa geopotential height, 850 hPa temperature, and 700 hPa specific humidity across forecast lead times.Source: §I.4.1 · p. 84, Figure I47 · p. 86
• Geopotential forecast skill remains consistent 20 years beyond training, while temperature and specific-humidity forecasts show only limited degradation at longer ranges.Temperature declines noticeably at 4 days but remains stable thereafter; specific humidity deteriorates only slightly in longer-range predictions.Source: §I.4.1 · p. 84
• The study compares NeuralGCM-0.7° and a 2017-trained developmental variant with GraphCast and ECMWF-HRES on ERA5 forecasts from 2018–2022.The 2017-trained models enable comparison on years after their training period; the final NeuralGCM-0.7° model was trained through 2019.Source: §I.4.1 · pp. 84, 86
• The extrapolation model omits surface embeddings and forcing features, removes the memory feature, and uses a different time-scaled loss.These are the three major architectural or training differences from the previously described model.Source: §I.4.1 · p. 86
• ECMWF-HRES and NeuralGCM show little variation across evaluation years, whereas GraphCast error increases by a few percent on 2022 forecasts beyond its training domain.The largest GraphCast error increases occur for geopotential forecasts and short lead times.Source: §I.4.1 · pp. 86–87, Figure I48 · p. 87

Limitations and future work

• Because NeuralGCM was trained only on ERA5, the authors caution that extrapolation beyond its training regime is uncertain and suggest climate-invariant methods, warmer-climate training data, and physical model components.These strategies are proposed to improve generalization to climates not represented in training.Source: §I.4.2 · pp. 87–88

Experimental setup

• The experiments used NeuralGCM-2.8° 34-year AMIP-like simulations with +1K, +2K, and +4K SST, comparing stable runs against CESM AMIP baselines.Six +1K, six +2K, and four +4K NeuralGCM runs remained stable for the full period.Source: §I.4.2 · pp. 87–88

Response to warmer climates

• NeuralGCM’s +1K and +2K SST responses broadly matched CESM, reproducing upper-tropospheric warming, polar amplification, and Southern Hemisphere polar-jet shifts, though tropical zonal-wind changes differed.The +4K simulation showed a substantially different and likely unrealistic response.Source: §I.4.2 · p. 88, Figure I49 · p. 89

Short-term response

• NeuralGCM matched MIROC6 in the short-term 850hPa global-temperature adjustment to +4K SST warming, indicating a physics-based-model-like initial response.The comparison used the 40-day adjustment after the SST increase.Source: §I.4.2 · p. 88, Figure I50 · p. 90

Long-term stability

• After 34 years, NeuralGCM exhibited climate drift under warmer SST, with an approximately 0.25K AMIP-to-+1K difference and stronger drift at +4K, although drift remained below the imposed warming.Drift was also visible in the +2K scenario and was especially evident for +4K SST.Source: §I.4.2 · p. 88, Figure I50 · p. 90