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REVIEW 3 major objections 5 minor 2 cited by

Learning Physically Interpretable Atmospheric Models from Data with WSINDy

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper demonstrates that WSINDy, a weak-form sparse regression algorithm, can recover physically interpretable governing PDEs from simulated atmospheric flows and, at a single pressure level, a potential-vorticity conservation law from…

desk verdict A clean demonstration that WSINDy recovers known atmospheric PDEs from simulated data; the ERA5 potential vorticity result is suggestive but single-level and should not be oversold. read the letter →

arxiv 2501.00738 v2 pith:L47YX6EW submitted 2025-01-01 physics.geo-ph cs.LGphysics.comp-ph

Seth Minor , Daniel A. Messenger , Vanja Dukic , David M. Bortz This is my paper · ORCID
classification physics.geo-phcs.LGphysics.comp-ph MSC 86A1035Q3565M99
keywords WSINDysparseidentificationofnonlineardynamicsweakformPDEdiscoveryatmosphericmodelpotentialvorticityshallowwaterequationsbarotropicturbulencereanalysisdata
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

This paper attempts to establish that the Weak form Sparse Identification of Nonlinear Dynamics (WSINDy) algorithm, a sparse-regression method that learns equations from data without computing noisy pointwise derivatives, can discover physically interpretable atmospheric models. It generalizes the algorithm to high-dimensional fluid data of arbitrary spatial dimension. On simulated datasets the method recovers the shallow-water equations on a sphere, the barotropic vorticity equation, and an effective transport equation for potential temperature. On assimilated reanalysis data, which are observations blended with a weather forecast model, it recovers a potential-vorticity evolution equation with a conservation-law-like form, although the recovered momentum equations are much less accurate. If the approach generalizes, weather and climate modeling gains a route from data to symbolic physics instead of black-box models with millions of parameters.

What carries the argument

The mechanism that carries the argument is the weak form: every candidate term is integrated against compactly supported test functions $\psi_k$, differential operators are shifted from noisy data onto smooth test functions by integration by parts, and the resulting linear system $b=Gw$ is solved with sparse thresholding. The support radii $\ell$ of the test functions act as a tunable scale selector, so the recovered equations are effective models at chosen length and time scales. An augmented-library step represents advection terms such as $(u\cdot\nabla)\omega$ without pointwise derivatives, and scale-invariant preconditioning keeps the weak library well-conditioned for high-dimensional data.

What would settle it

Run the same pipeline on assimilated data that includes a second pressure level so $\dot{\eta}\omega_\eta$ can enter the candidate library; if the recovered equation still requires the $\omega$ term instead of the vertical-gradient term, or if the explained variance does not improve, the single-level interpretation is not the true conservation law.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Weak form Sparse Identification of Nonlinear Dynamics (WSINDy) algorithm, a sparse-regression method that learns equations by integrating data against localized test functions rather than computing noisy pointwise derivatives, can discover physically interpretable atmospheric models from data. On simulated datasets it recovers the full governing equations, including the shallow-water equations on a sphere, the barotropic vorticity equation $\zeta_t = -\nabla\cdot(\zeta u)$ for turbulent flow, and an effective transport equation for potential temperature in a stratified boundary layer. On assimilated global reanalysis data at a single pressure level, it discovers a potential-vorticity equation $\omega_t = -\alpha(u\cdot\nabla)\omega + \beta\dot{\eta}\omega$ that is a conservation law except for the vertical-gradient term $\dot{\eta}\omega_\eta$, which cannot be constructed from one pressure level. The momentum equations recovered from the same assimilated data explain far less variance, and the paper identifies that as the main obstacle to competitive forecasting with symbolic models.

Load-bearing premise

The assimilated-data result rests on the assumption that observations at a single pressure level plus a hand-selected candidate library contain enough information to represent the dynamics; if vertical structure is dynamically essential, the recovered potential-vorticity equation is a correlation artifact rather than a conservation law.

Editorial extensions

If this is right

  • From noiseless simulation data, the method recovers the true governing equations with $R^2 \approx 100\%$ and no spurious terms, so the discovered symbolic models are faithful to the known physics in those regimes.
  • For the barotropic vorticity equation, the correct term set is recovered in every trial up to about $82.5\%$ added noise, indicating that the weak-form approach is genuinely robust to observational noise.
  • From assimilated reanalysis data, the method extracts a potential-vorticity evolution equation with the structure of a conservation law, showing that latent physical relationships can be recovered from real observations, not only from simulations.
  • Discovered symbolic models can be integrated forward in time and, on the simulated test cases, produce short-horizon forecast errors of the same rough order as current deep-learning weather models, despite using only a few terms.
  • The scale-selection property of the test functions means the same algorithm can target different spatiotemporal scales, so it provides a path toward scale-dependent or climate-scale effective equations rather than a single global model.

Reading between the lines

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

  • Editorial inference: if a second pressure level were added so that $\dot{\eta}\omega_\eta$ enters the library, the $\omega$ term should either vanish or lose explanatory power; that would test whether the discovered reanalysis law is a projection artifact, as the paper's own appendix discussion implies.
  • Editorial inference: the non-normal residual distributions reported by the paper suggest the underlying regression error model is misspecified, so a likelihood-based or iteratively reweighted weak-form estimator should improve coefficient accuracy on assimilated data.
  • Editorial inference: the weak-form scale parameter $\ell$ gives a concrete lever for learning subgrid closures for a known dynamical core, since coarsening the test-function support effectively filters the data to scales where unresolved physics must be parameterized.
  • Editorial inference: if accurate momentum equations could be discovered from reanalysis, the same pipeline would allow term-by-term comparison between the data's effective equations and the equations used in the assimilation's forecast model, potentially exposing missing or misrepresented physics.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper applies the WSINDy (Weak-form Sparse Identification of Nonlinear Dynamics) algorithm to atmospheric data, with the central claim that it can learn physically interpretable governing PDEs from both simulated and assimilated data. For simulated data, the method is demonstrated on three benchmarks: spherical shallow-water equations, equivalent barotropic turbulence, and a stably-stratified boundary layer. For assimilated data, the paper analyzes ERA5 reanalysis on a single pressure level and reports a potential-vorticity evolution equation, along with two momentum equations that explain a much smaller fraction of variance. The simulated-data results show high R2 values, correct term recovery, and meaningful noise robustness, and the paper includes forecast rollouts as an independent validation. The assimilated-data results are presented with transparent acknowledgment of limitations, including the omission of a vertical-gradient term in the potential-vorticity model.

Significance. If the central claim holds, the paper makes a useful contribution to data-driven geophysics by showing that a symbolic regression method can recover interpretable equations from noisy atmospheric data, a goal that is often at odds with the accuracy of black-box neural models. The simulated-data demonstrations are strong: the authors report correct term recovery for multiple geophysical regimes, noise-robustness experiments up to 100% noise levels, and held-out forecast skill. The code is publicly available, and the performance metrics (R2, TPR, E∞, forecast RMSE) are standard and clearly defined. However, the assimilated-data portion is preliminary: the single potential-vorticity model rests on a single-level approximation, and the momentum equations explain only 23–31% of the variance. The paper's main contribution may therefore be as a proof-of-concept for simulated data, with the ERA5 analysis serving as a candid but limited illustration of the challenges ahead.

major comments (3)
  1. [Section 3.3, Table 2, Appendix A4] The ERA5 potential-vorticity model is the primary evidence for the abstract's assimilated-data claim, but the model omits the vertical-gradient term η̇ω_η and instead includes a correlated η̇ω term, because the training data are confined to a single pressure level (p = 200 hPa). The paper states this limitation transparently, but it does not provide any multi-level validation, a quantitative argument that the omitted term is dynamically negligible at 200 hPa, or a sensitivity check of the identified coefficients to this omission. Without such support, the discovered equation is an omitted-variable regression model rather than a validated conservation law, and the abstract's statement that WSINDy can learn effective atmospheric models from assimilated data is broader than what this single-level result establishes.
  2. [Section 3.3 and Section 3.1] The ERA5 coefficient estimates in Table 2 are reported as point values with no uncertainties and no sensitivity analysis with respect to the test-function support radii ℓ, the MSTLS threshold λ, or the preconditioning scales γ. The simulated-data results are accompanied by a noise-robustness study (Figure 3), but no analogous analysis is provided for the assimilated data. Given that the PV equation achieves R2 = 74.4% and the momentum equations explain only 23.2% and 30.7% of the variance, it is essential to know whether the identified coefficients are stable under reasonable hyperparameter perturbations. The absence of this analysis leaves the physical interpretability claim for assimilated data unquantified.
  3. [Appendix A1 and Table 1] The Spherical benchmark includes direct observations of the transport operators in the state vector (e.g., ∇·(hu) and (u·∇)u for the ht and ut models, respectively). As a result, the library columns for these terms are computed from exact simulated quantities rather than reconstructed from the underlying state variables. This makes the recovery of the shallow-water equations a coefficient-fitting exercise rather than a discovery of unknown operators. The paper should clarify this point explicitly and may wish to highlight the Barotropic example, where the terms are reconstructed from raw fields, as the more decisive test of the algorithm's discovery capability.
minor comments (5)
  1. [Table 2] In the row for the u equation, the coefficient tuple is written as "(0.06, 0.04. 0.41, 1.56, 0.26, 0.07)", which contains a period instead of a comma after 0.04; this appears to be a typographical error.
  2. [Section 3.3] The first sentence reads "We use assimilated meteorological data from the from Copernicus Climate Change Service" and contains a duplicated "from".
  3. [Table 3] The "n/a" entry for the Spherical row's ttol/T0 is not explained; please clarify whether the forecast never exceeded the 10% threshold during the test interval, or whether the metric was not computed for that case.
  4. [Section 2.4.1] The text says the forecast remains "90% accurate" when E(t) is below the 10% threshold, but it may be clearer to state that the relative error remains below 10% throughout the interval [0, ttol].
  5. [Figure 3 caption] The caption describes results for both the Spherical and Barotropic datasets, but it does not indicate which curve corresponds to which dataset; please add explicit labels or a legend.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: simulated-data benchmarks are validated against known ground-truth models and held-out forecasts, and the ERA5 potential vorticity model is an explicitly disclosed single-level fit rather than a disguised prediction.

full rationale

I find no step in the paper where a claimed prediction or discovery reduces by construction to its own inputs. The simulated-data demonstrations (Spherical, Barotropic, Stratified) are benchmarks with known governing equations from external sources (Dedalus, PyQG, Johns Hopkins Turbulence Database); WSINDy estimates coefficients and selects terms from a candidate library, and the resulting models are then validated by held-out forecast rollouts (Table 3) and noise-robustness experiments (Figure 3). Although the Spherical example is given direct measurements of transport quantities such as -div(hu) and (u dot grad)u, the output is a sparse linear combination with fitted coefficients (H0, g1, g2), not an identity, and forecasting provides an independent check; supplying known physical terms is a standard benchmark design, not a circular redefinition. For the ERA5 assimilated data, the paper explicitly states that the discovered omega equation differs from a true conservation law by omission of the vertical-gradient term eta_dot omega_eta and includes a correlated omega term in its place (Section 3.3, Table 2). This is a transparent model-misspecification or omitted-variable limitation, not a fitted parameter renamed as a prediction, and the paper does not present the single-level equation as a derived first-principles conservation law. The self-citations to Messenger and Bortz provide the underlying WSINDy methodology, but the load-bearing evidence here is the paper's own independent numerical experiments against known ground-truth models and external data; no uniqueness theorem or unverified self-citation is used to forbid alternatives. Therefore no circular step meets the evidentiary bar.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the WSINDy weak-form machinery from prior work, on physically informed candidate libraries, on hand-chosen filter scales and sparsity thresholds, and on two modeling shortcuts: excluding numerical stability terms from ground truth and using single-level ERA5 data. The coefficients reported in Tables 1 and 2 are fitted outputs rather than pre-specified inputs. No new physical entities are introduced.

free parameters (4)
  • Test function support radii ℓ = Spherical (30,14,35); Barotropic (20,20,20) and (38,35,20); Stratified (11,10,15); ERA5 (25,25,9)
    Hand-chosen in Section 3.1 and Appendix A. They set the filtered length and time scales, and the paper notes that the discovered models change with scale.
  • MSTLS threshold λ and loss weight μ = λ scanned over log10 from -4 to 0; μ=(IJ)^-1
    Controls sparsity and the selected terms in Section 3.1; the final λ is chosen by minimizing the regularized loss.
  • ERA5 potential vorticity rescaling = ω → ω/Ω
    Manual preprocessing in Appendix A4 to improve the condition number of G; it rescales the fitted coefficients in Table 2.
  • Preconditioning scale factors γ = chosen to reduce κ(G)
    Appendix D introduces rescaled coordinates with scaling factors that affect the recovered coefficients, though not the term selection.
assumptions (5)
  • standard math Weak-form integration by parts with compactly supported test functions is a valid and sufficiently accurate representation of the data derivatives.
    Foundation of the WSINDy system in Section 2.3; relies on the test functions being smooth and compactly supported.
  • standard math The discrete convolution computed with FFT and trapezoidal rule on a uniform grid accurately approximates the weak-form integrals.
    Section 2.3 and Section 3.1; the Bernstein-type test functions give O(Δ^{p_i+1}) truncation error.
  • domain assumption Each candidate library contains the terms needed to represent the true dynamics of the dataset.
    Sections 2.3.1 and Appendices A1-A4. If a required term is absent from the library, it cannot be discovered. For the Spherical case, exact transport quantities are supplied as observed state variables.
  • ad hoc to paper The hyperviscosity and small-scale dissipation terms can be neglected when defining the true model coefficients.
    Appendix A1 and A2, noted by daggers in Table 1. These stability terms are excluded from w_true, which makes the recovery metrics more favorable.
  • ad hoc to paper A single pressure level is enough to model potential vorticity dynamics in the ERA5 test, with the omitted vertical gradient term absorbed by a correlated omega term.
    Section 3.3 and Appendix A4. If vertical structure is essential, the discovered PV equation is an artifact of the missing vertical dimension.

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Cite this review

Pith. "Pith review of Learning Physically Interpretable Atmospheric Models from Data with WSINDy." pith.science (2026). https://pith.science/paper/L47YX6EW

@misc{pith2026250100738,
  author       = {Pith},
  title        = {Pith review of: Learning Physically Interpretable Atmospheric Models from Data with WSINDy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L47YX6EW}},
  note         = {Machine review of arXiv:2501.00738}
}
read the original abstract

The multiscale and turbulent nature of Earth's atmosphere has historically rendered accurate weather modeling a hard problem. Recently, there has been an explosion of interest surrounding data-driven approaches to weather modeling, which in many cases show improved forecasting accuracy and computational efficiency when compared to traditional methods. However, many of the current data-driven approaches employ highly parameterized neural networks, often resulting in uninterpretable models and limited gains in scientific understanding. In this work, we address the interpretability problem by explicitly discovering partial differential equations governing atmospheric phenomena, identifying symbolic mathematical models with direct physical interpretations. The purpose of this paper is to demonstrate that, in particular, the Weak form Sparse Identification of Nonlinear Dynamics (WSINDy) algorithm can learn effective atmospheric models from both simulated and assimilated data. Our approach adapts the standard WSINDy algorithm to work with high-dimensional fluid data of arbitrary spatial dimension.

Figures

Figures reproduced from arXiv: 2501.00738 by the authors.

Figure 1
Figure 1. A schematic illustrating the construction of the WSINDy linear system of eq. (6). Each column of the weak library G represents a strided convolution between a discretized test function derivative D iψ and a particular function of the data, fj (u). The result is then sub￾sampled over a set of query points {(xk, tk)} (black dots) and vectorized. Here, the data are snapshots of scalar vorticity ζ from the numerical sim… view at source ↗
Figure 2
Figure 2. Illustrating the effect of increasing the test function support parameter ℓi in the convolution (ψ ∗ ul) to select features at increasingly coarsened length scales. Here, the potential vorticity ul = ω from the ERA5 dataset (§A4) is plotted at t = 45 hours. This and other material in this paper contain modified Copernicus Atmosphere Monitoring Service information [2024]. A unique feature of integrating the data agai… view at source ↗
Figure 3
Figure 3. Overview of numerical experiments with the ‘Spherical’ and ‘Barotropic’ datasets, where the true coefficients are known, and the ‘Stratified’ (§A3) and ‘ERA5’ (§3.3) datasets, where the true coefficients are not known. The top panels illustrate the mean E∞ error (top-left) within 1σ error bars and mean TPR (top-right), obtained using 8 to 11 realizations of Gaussian i.i.d. noise ϵ at each noise level σNR ∈ {0.025 · … view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Forecasting error as a function of the non-dimensionalized integration time. Dataset Test Data, [τ, T ] RMSE (%) R2 (%) ttol/T0 EF (%) Spherical (h) t ∈ [360, 540] 0.01 100 n/a 0.02 Barotropic (ζ) t ∈ [20, 30] 20.7 95.7 1.39 34.5 Stratified (ϑ) t ∈ [5.1, 7.5] 13.6 46.5…
Figure 5
Figure 5. Figure 5: Illustrating the forecast error from the discovered WSINDy model at the final time for the ‘Spherical’ (top) and ‘Stratified’ (bottom) datasets. See §3.5 for a discussion. –25– [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]
Figure 6
Figure 6. Figure 6: Rolling out a forecast for the ‘Barotropic’ dataset for times t ∈ [τ, τ + 10]. –26– [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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Reviewed August 10, 2026 · model on record in the stance chip above.