Physics-Assisted and Topology-Informed Deep Learning for Weather Prediction

Jiaqi Zheng; Qing Ling; Yerong Feng

REVIEW 3 major objections 4 minor 7 references

Physics-Assisted and Topology-Informed Deep Learning for Weather Prediction

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Spherical physics-AI weather model beats operational IFS T42

desk verdict PASSAT is a genuinely new hybrid architecture with honest, extensive experiments, but the paper's headline claim of outperforming all deep learning baselines is contradicted by its own tables. read the letter →

arxiv 2505.04918 v1 pith:TQGXN5OO submitted 2025-05-08 cs.LG cs.AI

Jiaqi Zheng , Qing Ling , Yerong Feng This is my paper · ORCID

classification cs.LGcs.AI

keywords weatherpredictionphysics-informeddeeplearningsphericalgraphneuralnetworkadvectionequationNavier-Stokesequationstopology-awareforecastingERA5medium-range

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

PASSAT is a deep learning weather model that builds the two recognized drivers of atmospheric evolution, advection and the Earth-atmosphere interaction, directly into its architecture, and solves the governing equations on the sphere rather than on a flat latitude-longitude grid. The paper claims that by numerically integrating the advection equation for each weather variable and the Navier-Stokes equations for the associated velocity fields, while a spherical graph neural network supplies the interaction tendency and initial velocities, the model outperforms both state-of-the-art deep learning baselines and the operational numerical model IFS T42 at coarse 5.625-degree resolution. The central claim matters because it suggests physics constraints and spherical geometry can make data-driven forecasts more accurate and robust than either black-box deep learning or coarse operational NWP.

What carries the argument

The central machinery is the coupling of a spherical advection-Navier-Stokes solver with a two-branch spherical graph neural network. For each weather variable $u$, the advection equation gives the advective tendency $- (v_\theta \partial u/\partial\theta + \frac{v_\varphi}{\cos\theta}\partial u/\partial\varphi)$, and the spherical Navier-Stokes equations (7)-(8) update the velocity components $v_\theta$ and $v_\varphi$ under advection, curvature, Coriolis, pressure-gradient (replaced by the z500 geopotential gradient) and viscous terms. The spherical graph neural network supplies the initial velocity fields and the hourly interaction tendency. Solving on the sphere rather than the planar latitude-longitude grid eliminates the $\sec\theta$ amplification that distorts velocities near the poles, and the 0.2-hour Euler time stepping with an hourly-constant interaction term keeps the autoregressive rollout stable without backpropagating through every sub-step.

What would settle it

Ablate the Navier-Stokes velocity update by replacing it with a neural-network velocity updater of matched capacity, or simple persistence of velocities, while keeping the advection solver and spherical graph fixed; if RMSE does not degrade, the Navier-Stokes equations are not load-bearing. A second check would correlate PASSAT's learned initial velocity fields for t2m and z500 with ERA5 wind components; near-zero correlation would indicate the physics is a learned latent, not atmospheric advection.

Watch

Extended reading notes

Core claim

PASSAT decomposes the total tendency of any weather variable into an advective part and an interaction part. The advective part is computed by solving the advection equation on the unit sphere, with future velocity fields obtained from the Navier-Stokes equations on the sphere, closed by replacing the pressure-gradient force with the gradient of the 500 hPa geopotential. The interaction part, radiation, clouds, and subgrid turbulence, is estimated by a spherical graph neural network whose interaction branch runs once per hour, while its velocity branch generates the initial velocity fields. All equations are integrated with a 0.2-hour Euler step, and the graph is built with nodes at the grid points on the sphere using a Haversine/Gaussian kernel. The paper's central discovery is that this physics-assisted, topology-informed architecture achieves the lowest root-mean-square error and highest anomaly correlation among the five compared deep learning models (GraphCast, ClimODE, Pangu, FourCastNet, SFNO) for all five variables, at all lead times up to 144 hours, and also beats IFS T42 at the two reported NWP lead times of 72 and 120 hours. Ablation studies show that removing either the physics equations or the spherical topology degrades accuracy, with most variables benefiting more from the topology correction.

Load-bearing premise

The learned velocity fields that drive advection behave enough like real atmospheric winds that the Navier-Stokes equations genuinely govern their evolution, rather than the equations merely acting as a smoothness regularizer on learned latent quantities.

Editorial extensions

If this is right

PASSAT achieves the best RMSE and ACC among the five parameter-matched deep learning baselines for all five variables (t2m, t850, z500, u10, v10) at every lead time from 6 to 144 hours.
It outperforms the operational IFS T42 model at the two reported lead times (72 and 120 hours) for t2m, t850, and z500, despite IFS T42 running at a finer 2.8-degree resolution.
Ablation results show that both the physics equations and the spherical topology contribute to the gain; for t2m the two contributions are roughly equal, while for t850, z500, u10, and v10 the topology information is the larger contributor.
The 0.2-hour integration step with hourly interaction updates is sufficient for stable 144-hour forecasts, suggesting the Euler scheme is not a practical bottleneck at this resolution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: the learned velocity fields for scalar variables such as t2m and z500 are probably not physical winds; if the observed wind fields were used inside the advection equation instead, the results would reveal whether the Navier-Stokes update is genuinely physical or merely a structured smoothing prior.
Inference: the topology benefit likely comes mostly from the $1/\cos\theta$ metric factor in the spherical advective derivative, which prevents the polar amplification of planar velocities; this could be tested by keeping the spherical graph but switching the advection derivative back to the planar form.
Inference: because the machinery is variable-agnostic, the same architecture could be applied to other advection-dominated spherical fields, such as ocean surface tracers or atmospheric chemical concentrations, provided interaction terms can be learned.
Inference: the comparison with IFS T42 is limited to two lead times and one coarse resolution; a natural extension would test PASSAT at higher input resolution and against IFS T63-equivalent skill, where the paper reports the gap closes.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

What you should know: PASSAT combines spherical advection and a Navier-Stokes update on the sphere with a spherical graph network. That combination is new relative to ClimODE and NowcastNet, and the paper presents it cleanly. The experiments are more complete than usual for this area: five variables, six lead times, five retrained baselines, ablation studies for topology and physics, and a working code link. The ablation story is coherent, and the strongest gains are on z500, u10, v10, and longer lead times, which is where the physics prior should help.

The soft spots are real but not fatal. First, the abstract and Section 4 claim PASSAT outperforms all deep learning baselines in all weather variables across lead times. Their own Table 8 shows GraphCast beating PASSAT on t2m at 24h and 48h by small margins (RMSE 1.21 vs 1.25; 1.52 vs 1.54), and Table 9 shows FourCastNet ahead on t2m ACC at 120h. So strict dominance is false; the defensible claim is that PASSAT improves most variables and leads, especially at longer range, and beats IFS T42 at the reported 72h and 120h cells. That is still a useful result, but smaller, and the text should say it.

Second, the baseline comparison is weaker than it looks. The original GraphCast, Pangu, FourCastNet, and SFNO have tens to hundreds of millions of parameters; the authors shrink all of them to around 1.15M and retrain from scratch. That is not an unreasonable way to compare architectures, but calling those compressed versions "state-of-the-art" overstates them. The fair summary is that PASSAT beats comparably sized versions of those architectures, not the operational-scale models.

Third, the Navier-Stokes viscosity mu is never given. It appears as a constant in Eqs. (7)-(8) and in Algorithm 1. A fixed value or fitted value would both be acceptable; an unstated value is missing information for a paper whose hook is physics.

The weakest conceptual point is not the math but the interpretation. Velocity fields are learned separately for each weather variable, and the paper never checks whether these fields resemble physical winds. If they are merely reparameterizations, the Navier-Stokes block is functioning as a smoothness regularizer rather than a physical law. That weakens the "seamlessly integrates physics" claim, though it does not invalidate the empirical result. A simple diagnostic comparing the learned velocities to ERA5 winds would settle it.

Bottom line: the paper deserves a serious referee. The central claim needs rewriting, mu must be reported, and a physical plausibility check of the velocities would meaningfully strengthen the argument. I would send it to review with those conditions.

Referee Report

3 major / 4 minor

Summary. The paper proposes PASSAT, a deep learning weather-prediction model that combines a spherical graph neural network with numerical solves of the advection equation and the Navier-Stokes equations on the sphere. The GNN estimates the Earth-atmosphere interaction tendency and the initial velocity fields, while the PDE solver advances the variables and their velocities in time. Experiments are conducted on the 5.625-degree ERA5 dataset for t2m, t850, z500, u10, and v10, with lead times up to 144 hours. The paper claims that PASSAT outperforms both state-of-the-art deep learning weather models and the operational IFS T42 model, and ablation studies are used to argue that both physics and topology information contribute to the gains. The appendices provide architecture details, training procedures, modifications of the baseline models, and full comparison tables.

Significance. If the central claims were fully supported, the paper would be a useful contribution to physics-assisted, topology-aware data-driven weather forecasting. The most valuable parts are the spherical treatment of the advection equation, the attempt to couple learned interaction terms with PDE-based advection, and the availability of code and checkpoints. The ablation results are internally consistent and show clear gains over the authors' own variants. However, the headline claim of outperforming state-of-the-art deep learning models is not supported as written, because the baseline models were downscaled to roughly 1.15 million parameters before comparison, and because the paper's own tables contain counterexamples to the strict-dominance statement. The comparison with IFS T42 is also limited to three variables and two lead times. With recalibrated claims and additional validation of the learned velocity fields, the core idea remains worth publishing.

major comments (3)

[Section 4 and Appendix F, Tables 8-9] The statement in Section 4 that 'PASSAT outperforms the other deep learning models in all weather variables across different lead times' is directly contradicted by the paper's own results in Tables 8 and 9. For t2m, GraphCast has lower RMSE than PASSAT at 24 h (1.21 vs 1.25), 48 h (1.52 vs 1.54), and 72 h (1.84 vs 1.85), while FourCastNet has lower RMSE at 144 h (2.68 vs 2.69) and ties at 96 h and 120 h. On ACC, GraphCast leads t2m at 24 h (0.972 vs 0.970), 48 h (0.956 vs 0.955), and 72 h (0.937 vs 0.935), and FourCastNet leads at 120 h (0.888 vs 0.886) and 144 h (0.865 vs 0.862). The t850 result at 24 h is tied on both RMSE and ACC. The abstract and the conclusion should be revised to state that PASSAT improves on most variables and lead times, especially z500, u10, v10, and longer leads, rather than claiming superiority in every cell.
[Section 4, 'Baseline deep learning models'; Appendix E, Tables 3-7] The comparison with 'state-of-the-art' deep learning models is not a comparison with the published models themselves. GraphCast, Pangu, FourCastNet, and SFNO were reduced from 36.7M, 64M, 59.1M, and 107M parameters, respectively, to about 1.15M parameters and retrained from scratch at 5.625 degrees. A 1.15M-parameter GraphCast is not the GraphCast from the Science paper, and the same holds for the other baselines. This makes the controlled comparison legitimate but does not support the abstract's claim that PASSAT 'outperforms the state-of-the-art deep learning-based weather prediction models.' The claim should be explicitly limited to parameter-matched reimplementations trained on this dataset, or the authors should add comparisons with the original pretrained models at their native parameter counts.
[Section 3.5, Section 3.7, and Algorithm 1] The paper does not provide evidence that the learned per-weather-variable velocity fields v(t,s) correspond to physical atmospheric winds. Each variable t2m, t850, z500, u10, and v10 is assigned its own velocity field, and the Navier-Stokes equations (7)-(8) are then solved for these fields. For variables such as t2m and z500, there is no separately measured physical velocity in the dataset, so these fields are learned latents whose physical interpretation is unclear. If the learned velocities do not resemble real winds, then equations (7)-(8) act as a learned regularizer rather than as a physical law, and the claim that PASSAT 'seamlessly integrates the physics of weather evolution' is overstated. A concrete test would be to compare the velocity-branch output for u10 and v10 against ERA5 wind fields, or to couple all advected variables to a single shared wind field. The authors should either add such validation or substantially soften the physics-language claims.

minor comments (4)

[Section 3.5] The viscosity coefficient mu appears in equations (7) and (8) and in Algorithm 1 but its value is never reported in the paper. Since this coefficient controls the strength of the friction term, the authors should state its value and, ideally, report sensitivity to it.
[Appendix D.2] There is a typo in 'GeForece RTX 2080'; it should read 'GeForce RTX 2080.' Additional typos include 'Naiver-Stokes' in Section 5 and 'planner' in Figure 2's caption, and 'Pacth Size' in Table 4.
[Tables 8 and 9] The ClimODE rows show ACC values that collapse to 0.510 at 72 h and 0.020 at 96 h before becoming NAN at 120 h and 144 h. If this reflects numerical instability or a training failure, it should be stated explicitly so that readers do not interpret the missing entries as an unqualified failure of the baseline.
[Section 4, comparison with IFS T42] The comparison with IFS T42 is limited to t2m, t850, and z500 at 72 h and 120 h, as the paper notes. The abstract's claim that PASSAT outperforms the operational NWP model IFS T42 should be qualified accordingly, since no claim is made for u10, v10, or other lead times.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the physics-assisted solver is a coupled PDE system with learned source and velocity terms, not a derivation from its own outputs.

full rationale

The paper's derivation chain is self-contained. The advection equation (Eq. 4) and the Navier-Stokes equations (Eqs. 7-8) are standard external PDEs, cited to [Cha22] and [LTW92]; no fitted parameter is renamed as a prediction. The velocity fields are produced by the same spherical GNN that estimates the Earth-atmosphere interaction, and the pressure-gradient term in Section 3.7 is replaced by the gradient of the model's own predicted z500, but this is a closed dynamical coupling rather than a definitional equivalence: u and v co-evolve through the numerical solver, and the predicted weather is not the fitted quantity itself. The reference list contains no self-citations by the present authors, so there is no load-bearing self-citation chain and no imported uniqueness theorem. The learned per-variable velocities for scalars such as t2m and z500 are not validated against observed winds (Section 3.5 and Algorithm 1), which weakens the physical-interpretation claim, and the Section 4 statement that PASSAT outperforms all deep learning models in all variables is contradicted by the paper's own Tables 8 and 9 (e.g., GraphCast has lower t2m RMSE at 24 h and 48 h). Both are correctness/consistency concerns rather than circularity. The ablation section also declines to isolate the individual effects of the advection and Navier-Stokes equations (Section 5), but that is an acknowledged experimental limitation, not a circular argument. No circular step can be exhibited, so the score is 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central claim rests on a small number of hand-chosen constants (mu, lambda weights, velocity clamp bounds) and on modeling assumptions that are standard for coarse global weather prediction: spherical geometry, two-dimensional dynamics, and a split between a resolved advective part and a learned residual interaction. The most consequential invented entity is the per-variable velocity field, which makes the Navier-Stokes solver a learned latent dynamical system rather than a pure physical law.

free parameters (3)

mu (viscous friction coefficient) = not reported
Appears in Eqs. (7)-(8) and Algorithm 1; value not stated in the paper, and the term mu/cos^2(theta) v is described as 'viscous friction in the Laplacian' though it is not the spherical Laplacian.
lambda1, lambda2, lambda3 (loss penalty weights) = 10, 1, 1
Hand-chosen penalties for initial velocity field magnitude and smoothness in Eqs. (23)-(25); affect training and thus the learned model.
Velocity clamp bound = 0.005 (in units of 6731 km/hour)
Initial velocity fields are projected onto [-0.005, 0.005] to ensure stability (Section 3.7); hand-chosen.

assumptions (6)

domain assumption Earth's surface is modeled as an ideal unit sphere
Section 3.1: radius is set to 1 (6371 km); ignores oblateness and surface details.
domain assumption Vertical actions are neglected
Section 3: the paper disregards vertical motions, focusing on 2D spherical dynamics; it says extension to vertical is straightforward but not implemented.
domain assumption Tendency decomposes into advection plus Earth-atmosphere interaction
Eqs. (1)-(2): total tendency is exactly the sum of -v·grad u and h; no other process (e.g., diffusion, radiation coupling) is modeled explicitly.
standard math The advection and Navier-Stokes equations on the sphere from [LTW92] are the governing equations
Sections 3.3 and 3.5 use classical PDEs from the literature; treated as given.
ad hoc to paper The interaction tendency h is constant within each one-hour interval
Section 3.6: to reduce backprop, h is recomputed once per hour instead of every 0.2h step; this is an approximation specific to PASSAT.
ad hoc to paper Pressure gradient force is replaced by z500 geopotential gradients
Section 3.7: uses gradient of geopotential at 500hPa, with unit conversion, to close the Navier-Stokes equations; this is a crude barotropic approximation used for all variables.

invented entities (1)

Per-weather-variable velocity fields v(t,s)
purpose: Each weather variable (t2m, t850, z500, u10, v10) is assigned its own 2D velocity field, initialized by the velocity branch and evolved by Navier-Stokes; used to compute advection tendencies.
For scalar variables like temperature, there is no physically defined v; these are learned latent fields. The paper does not verify they correspond to actual winds, so the 'physics' is partially learned.

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Cite this review

Pith. "Pith review of Physics-Assisted and Topology-Informed Deep Learning for Weather Prediction." pith.science (2026). https://pith.science/paper/TQGXN5OO

@misc{pith2026250504918,
  author       = {Pith},
  title        = {Pith review of: Physics-Assisted and Topology-Informed Deep Learning for Weather Prediction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TQGXN5OO}},
  note         = {Machine review of arXiv:2505.04918}
}

abstract

Although deep learning models have demonstrated remarkable potential in weather prediction, most of them overlook either the \textbf{physics} of the underlying weather evolution or the \textbf{topology} of the Earth's surface. In light of these disadvantages, we develop PASSAT, a novel Physics-ASSisted And Topology-informed deep learning model for weather prediction. PASSAT attributes the weather evolution to two key factors: (i) the advection process that can be characterized by the advection equation and the Navier-Stokes equation; (ii) the Earth-atmosphere interaction that is difficult to both model and calculate. PASSAT also takes the topology of the Earth's surface into consideration, other than simply treating it as a plane. With these considerations, PASSAT numerically solves the advection equation and the Navier-Stokes equation on the spherical manifold, utilizes a spherical graph neural network to capture the Earth-atmosphere interaction, and generates the initial velocity fields that are critical to solving the advection equation from the same spherical graph neural network. In the $5.625^\circ$-resolution ERA5 data set, PASSAT outperforms both the state-of-the-art deep learning-based weather prediction models and the operational numerical weather prediction model IFS T42. Code and checkpoint are available at https://github.com/Yumenomae/PASSAT_5p625.

Figures

Figures reproduced from arXiv: 2505.04918 by the authors.

**Figure 1.** Attributions of the weather evolution. 1.1. Enhancing Deep Learning with Physics Combining with the differential equations that characterize the weather evolution can enhance the precisions, efficiency and robustness of deep learning models, because the differential equations provide valuable prior knowledge [Xia+22]. Some works incorporate differential equations into losses during training deep learning models [Daw… view at source ↗

**Figure 2.** Distortions due to planar projection. (a) The spherical and planner representations of the global weather. (b) The same weather patterns on the sphere are distorted on the plane. (c) The convolutions on the sphere are distorted on the plane. 1.3. Contributions In this paper, we propose PASSAT, a novel Physics-ASSisted And Topology-informed deep learning model for weather prediction. PASSAT attributes the weather evo… view at source ↗

**Figure 3.** Overview of PASSAT. unit length (6371km). We denote the unit sphere S = {s ∈ R 3 | ||s||2 = 1} as the Earth’s surface. Any spatial coordinate s on the unit sphere corresponds to a point (𝜑, 𝜃) within the planar latitude-longi- tude grid, where 𝜃 is the latitude and 𝜑 is the longitude. Thus, we use s and s(𝜑, 𝜃) interchangeably. Given any spatial coordinate s, e𝜑(s) ∈ R 3 and e𝜃(s) ∈ R 3 are two orthogonal unit vecto… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Overview of PASSAT’s graph neural network. TOP: The backbone model and two branches. BOTTOM: The basic block. ClimODE and NowcastNet both calculate the the advective derivative according to (6), through estimating 𝑣 ′ 𝜃 (𝑡, s) and 𝑣 ′ 𝜑 (𝑡, s) with neural networks. How…

**Figure 5.** Figure 5: Comparison between PASSAT and other models. The x-axis represents the lead time in hours. Smaller RMSE and larger ACC values indicate better performance. Note that some results of IFS T42 exceed the bounds. Results. As demonstrated in [PITH_FULL_IMAGE:figures/full_fig…

**Figure 6.** Figure 6: Comparison between PASSAT and the three variants. The x-axis represents the lead time in hours. Smaller RMSE and larger ACC values indicate better performance. The RMSEs and ACCs of IFS T42 and IFS T63 are from [Ras+20], only including t2m, t850 and z500 for the lead t…

**Figure 7.** Figure 7: Prediction visualization of t2m. 10 [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]

**Figure 8.** Figure 8: Prediction visualization of t. 11 [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]

**Figure 9.** Figure 9: Prediction visualization of z. 12 [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]

**Figure 10.** Figure 10: Prediction visualization of u10. 13 [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]

**Figure 11.** Figure 11: Prediction visualization of v10. 14 [PITH_FULL_IMAGE:figures/full_fig_p028_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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