{"id":"6b931bfa-e38c-4804-84b0-893c1d7acf4b","arxiv_id":"2501.00738","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"WSINDy discovers interpretable PDE models from simulated and ERA5 atmospheric data, with strong results on simulations and partial results on assimilated data.","lead":"This paper applies the WSINDy equation-discovery method to atmospheric data, recovering known fluid-dynamics equations from simulations and approximate conservation laws from real weather reanalysis. It matters because it shows a path to interpretable, symbolic weather models as an alternative to uninterpretable neural networks.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The assimilated-data demonstration rests on a single-level PV model that omits the vertical-gradient term; if that term is dynamically essential, the ERA5 result is an artifact of missing vertical structure rather than a learned conservation law.","rationale":"I read the paper as a method demonstration, not a claim that WSINDy outperforms NWP. The simulated results are credible: true equations are known for two cases, coefficient errors and true positive ratios are reported, and the forecasting rollouts provide independent validation. The public code is a further point in favor of reproducibility. The load-bearing fragility is exactly where the reader placed it: the only assimilated-data success with substantial explanatory power is the potential vorticity model, and that model substitutes a correlated ω term for an unobserved vertical gradient ω_η. Section 3.3 and the Discussion explicitly acknowledge this substitution and the poor momentum-equation fits; this transparency is a strength of the paper, not a defect. Nevertheless, the central abstract claim about learning effective atmospheric models from assimilated data is only as strong as this PV result, and the paper does not currently show that the ω term is a robust surrogate rather than an artifact of regressing on a single pressure level. My proposed check, using multi-level ERA5 data to admit ω_η into the library, would settle whether the discovered equation changes form when the missing dimension is supplied. Since the reader already judged the paper CONDITIONAL and my analysis supports that same judgment, no verdict adjustment is needed; the appropriate recommendation remains conditional acceptance with the assimilated-data claim qualified. I found no additional concern that would warrant rejection, and the simulated-data portion is not threatened by the vertical-structure issue.","tokens_in":22844,"tokens_out":3618,"duration_ms":37440,"concrete_test":"Obtain ERA5 fields at three pressure levels bracketing 200 hPa (e.g., 150, 200, and 250 hPa) over the same July 2024 period, compute the vertical derivative ω_η and the vertical velocity ηdot at 200 hPa, and add ω_η to the candidate library used for the ω_t model. Then re-run WSINDy at the same query points and support radii ℓ = (25, 25, 9), and also at several neighboring ℓ values. If the selected model becomes ω_t = −α(u·∇)ω + β ηdot ω_η and the standalone ω coefficient goes to zero, the published single-level result is a missing-dimension artifact. If the ω-term coefficient remains significant alongside ω_η across the ℓ range, the effective model is robust. In either case, report bootstrap or split-sample coefficient uncertainties for the ERA5 fits.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The simulated-data demonstrations (Spherical, Barotropic, Stratified in Table 1) are strong: correct terms are recovered, E∞ and TPR are reported, and forecasts provide independent validation. The assimilated-data half of the central claim therefore carries the risk. In Table 2 and Section 3.3, the ERA5 potential vorticity model is reported as ω_t = −α(u·∇)ω + β ηdot ω, with α = 0.70, β = 0.29Ω, and R2 = 74.4%. As the authors state, the true conservation law is ω_t + α(u·∇)ω − β ηdot ω_η = 0; the vertical-gradient term is omitted because training data are on a single pressure level (p = 200 hPa, Appendix A4), and a correlated ω term is included in its place. The paper gives no coefficient uncertainties, no test-function support sensitivities, and no multi-level validation for the ERA5 case. Because the ut and vt momentum equations explain only 23.2% and 30.7% of variance, this PV equation is the main evidence for the assimilated-data claim. If ω_η is dynamically essential at 200 hPa, the discovered equation is an omitted-variable regression model rather than a physically interpretable conservation law. The authors are transparent about this limitation, but the abstract's claim about assimilated data remains broader than what the single-level result establishes.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23130,"tokens_out":7157,"duration_ms":68258,"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":[{"comment":"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.","section":"Section 3.3, Table 2, Appendix A4"},{"comment":"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.","section":"Section 3.3 and Section 3.1"},{"comment":"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.","section":"Appendix A1 and Table 1"}],"minor_comments":[{"comment":"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.","section":"Table 2"},{"comment":"The first sentence reads \"We use assimilated meteorological data from the from Copernicus Climate Change Service\" and contains a duplicated \"from\".","section":"Section 3.3"},{"comment":"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.","section":"Table 3"},{"comment":"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].","section":"Section 2.4.1"},{"comment":"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.","section":"Figure 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper is a competent application of an established method, and the simulated-data benchmarks are convincing. The main gap between the paper's central claim and its evidence lies in the assimilated-data section, where the single-level PV model omits a vertical-gradient term and no uncertainty quantification is given. A revision that either adds a multi-level ERA5 analysis or tempers the abstract's claim would resolve my main concern. I do not see the omission of the vertical-gradient term as a deliberate overreach; the authors are explicitly transparent about it, which is appreciated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou can send this one to a referee. The paper's real contribution is a careful demonstration that WSINDy, a weak-form sparse regression method, recovers known governing equations from three simulated atmospheric datasets and produces a plausible potential vorticity equation from ERA5 reanalysis data. The simulated half is solid: correct terms recovered with TPR=1, coefficients close to truth, high R2, noise robustness up to ~80% noise, and held-out forecast rollouts that track the true solution. The public code is a plus. The spherical-coordinate adaptation and the augmented-library construction for advection terms that don't integrate by parts are genuine technical additions, even if modest.\n\nThe soft spots are exactly where the stress-test lands. The ERA5 PV result (Table 2) comes from a single pressure level at 200 hPa. The true conservation law contains a vertical-gradient term eta_dot omega_eta; the paper omits it because only one level is available and substitutes a correlated omega term instead. That makes the discovered equation an omitted-variable regression, not conclusively a learned conservation law. The momentum equations from ERA5 explain only ~23-31% of variance, so the PV equation is the main evidence for the assimilated-data claim. The authors are transparent about this limitation, which I credit, but the abstract's phrasing—\"learn effective atmospheric models from ... assimilated data\"—is broader than what a single-level PV equation establishes. There are also no coefficient uncertainties and no sensitivity analysis for test-function support radii or the PV rescaling, so the reader cannot tell how knife-edge the ERA5 result is.\n\nOn balance this is a useful, honest application paper. It does not claim a new physical law, and it does not oversell forecasting—the authors explicitly say GraphCast will outperform them. The simulated benchmarks make it a credible proof of concept, and the ERA5 result is a genuine first step, not a finished result. It deserves serious peer review; I would expect a conditional accept with revision, mainly to qualify the assimilated-data claim and add uncertainty or sensitivity analysis.","headline":"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.","tokens_in":23655,"tokens_out":1407,"would_cite":false,"duration_ms":14157,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["86A10","35Q35","65M99"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["WSINDy","sparse identification of nonlinear dynamics","weak form PDE discovery","atmospheric model discovery","potential vorticity","shallow water equations","barotropic turbulence","reanalysis data"],"falsifier":"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.","tokens_in":22604,"feed_emoji":"🌀","tokens_out":11414,"duration_ms":99330,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sparse symbolic regression rediscovers atmospheric physics","feed_subtitle":"It recovers known PDEs from simulated flows and a conservation law from real reanalysis data.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the WSINDy algorithm for PDEs, including weak-form integration, test-function scale selection, and the MSTLS thresholding routine used in every experiment.","marker":"Messenger & Bortz (2021)"},{"why":"Introduces the SINDy sparse-regression paradigm that WSINDy generalizes, establishing the library-and-sparsity formulation.","marker":"Brunton et al. (2016)"},{"why":"Extends SINDy to PDE discovery and supplies the library-ansatz framework the paper adapts to atmospheric evolution equations.","marker":"Rudy et al. (2017)"},{"why":"Provides the spherical-coordinate advection and divergence operators and the primitive-equation forms that define the candidate libraries.","marker":"White (2003)"},{"why":"Supplies the assimilated reanalysis data from which the potential-vorticity and momentum equations are discovered.","marker":"Copernicus Climate Change Service (2023)"},{"why":"Documents the IFS primitive equations and vertical coordinate whose structure motivates the candidate terms and the interpretation of the reanalysis results.","marker":"ECMWF (2021)"},{"why":"Describes the 4D-Var data assimilation algorithm that produced the reanalysis fields used in the assimilated-data experiments.","marker":"Andersson and Thépaut (2008)"},{"why":"Establishes the meteorological significance of potential-vorticity conservation, framing why the recovered equation matters.","marker":"Kooloth et al. (2022)"}],"fun_headline_variants":["WSINDy rediscovers atmospheric equations from data","Sparse regression extracts physical laws from atmospheric data","Discovering interpretable atmospheric models with WSINDy","WSINDy learns atmospheric PDEs from raw data","Data-driven discovery of atmospheric physics laws"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["WSINDy rediscovers atmospheric equations from data","Sparse regression extracts physical laws from atmospheric data","Discovering interpretable atmospheric models with WSINDy","WSINDy learns atmospheric PDEs from raw data","Data-driven discovery of atmospheric physics laws"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000874,"raw_usage":{"total_tokens":3759,"prompt_tokens":897,"completion_tokens":2862,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":2790}},"tokens_in":513,"tokens_out":2862,"duration_ms":18815,"temperature":1.0,"reasoning_tokens":2790,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:44:06.840744+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"APACrefauthors \\ 2003","cited_arxiv_id":null,"evidence_quote":"Provides the spherical-coordinate advection and divergence operators and the primitive-equation forms that define the candidate libraries."},{"cited_title":"APACrefauthors \\ 2021 10","cited_arxiv_id":null,"evidence_quote":"Documents the IFS primitive equations and vertical coordinate whose structure motivates the candidate terms and the interpretation of the reanalysis results."},{"cited_title":"\\ Th \\'e paut, J N","cited_arxiv_id":null,"evidence_quote":"Describes the 4D-Var data assimilation algorithm that produced the reanalysis fields used in the assimilated-data experiments."},{"cited_title":", Smith, L M","cited_arxiv_id":null,"evidence_quote":"Establishes the meteorological significance of potential-vorticity conservation, framing why the recovered equation matters."}],"review_version":1}