{"id":"f792b4cf-62e4-4bb4-b407-a363b4710b6a","arxiv_id":"2505.05869","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The authors train a GPT-style model on 221 handbook PDE structures and use it to generate and select PDEs from data, including a proposed previously unreported equation for pre-breaking surface gravity waves.","lead":"A new system called EqGPT learns the structure of known PDEs from a math handbook, then generates candidate equations to fit observed data. It recovers classic PDEs from sparse noisy data and produces a candidate equation for surface gravity waves heading toward breaking.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The wave-breaking discovery depends on surrogate derivatives on gappy camera data; the paper supplies no check that those derivatives are accurate, so the R²-based selection of Eq. (15) is not yet supported.","rationale":"The reader's conditional verdict already identifies the same load-bearing risk: the wave-breaking claim rests on whether the surrogate model gives accurate derivatives across the gappy experimental domain. My own reading of the manuscript confirms that this is the most fragile link. The proof-of-concept results on canonical PDEs, the irregular-domain tests, and the high-dimensional extensions are informative and are supported by reasonably detailed experimental settings, so I am not raising an objection there. The concern is specifically about the real-data wave section, where the manuscript's own statement that 'no explicit interpolation or reconstruction was applied to address the spatial gaps' exposes the surrogate to uncontrolled smooth interpolation across occluded regions. Since the reward function selects and ranks PDE structures from derivatives of that surrogate, any bias in those derivatives directly propagates into the central scientific claim. The proposed synthetic gappy benchmark is a direct, inexpensive check: if it recovers a known equation, the method's handling of gaps is validated; if not, the current wave equation should be treated as conditional at best. Thus the reader's CONDITIONAL verdict stands unchanged.","tokens_in":23262,"tokens_out":3814,"duration_ms":41947,"concrete_test":"Create a synthetic counterpart of the wave experiment: simulate a known nonlinear evolution equation (e.g., KdV or a Boussinesq-type equation) on the same three-subdomain spatial mask with comparable temporal sampling and noise, run the identical ANN-surrogate-and-EqGPT pipeline, and compare the recovered structure with the known equation. Independently, evaluate the surrogate's ∂_tη, ∂_xη, ∂_xxxη, and ∂_xxxxxη against finite-difference or reference derivatives on the full synthetic grid; if the pipeline fails to recover the known form, or if the surrogate derivatives deviate by more than 10% relative L2 error over the masked domain, the wave-breaking discovery is not supported by the current evidence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The pivotal condition for the headline wave claim is that the ANN surrogate trained per experiment on three non-contiguous spatial subdomains (Fig. 6b; Supplementary S1.1) returns accurate pointwise derivatives, because Eq. (15) is selected by the R² reward computed on automatic-differentiation derivatives of that surrogate (Eq. (1) and Materials and Methods). The paper explicitly says no interpolation or reconstruction addressed the spatial gaps; the fully connected surrogate therefore fills those gaps with a smooth learned function, and third- and fifth-order x-derivatives plus ∂_tη are then evaluated across the whole domain. Near wave-breaking onset, such high-order derivatives of an interpolant are strongly influenced by network initialization, architecture, and regularization rather than by measured data. If the surrogate's derivative field is biased, the least-squares regression and reward ranking select structures that fit surrogate artifacts, not the physics; the 'previously unreported PDE' claim then loses its empirical basis. The absence of out-of-sample validation, error bars, and comparison with established wave-evolution models makes this derivative-reliability question the decisive one.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes EqGPT, a generative approach to PDE discovery that pretrains an autoregressive transformer on 221 PDE structures extracted from a mathematical handbook, encoded as token sequences under the proposed generative representation of equations (GRE). Candidate equations are generated by sampling from this model, evaluated through a reward combining least-squares R2 on surrogate-reconstructed fields with a sparsity penalty, and used to fine-tune the generator in a generation-evaluation-optimization loop. The authors report rediscovery of eight canonical PDEs from sparse and noisy data, comparisons against eight existing PDE-discovery methods, extensions to irregular computational domains and high-dimensional systems, and a real-world case study in which the method yields a reported 'previously unreported' PDE for strongly nonlinear surface gravity waves propagating toward breaking, obtained by fitting a linear propagation term, a quadratic nonlinear dispersive term, and a linear dispersive term to wave-tank data.","tokens_in":23477,"tokens_out":5458,"duration_ms":58001,"significance":"If all claims are substantiated, this is a valuable contribution: the knowledge-guided generative prior is a simple and potentially effective way to expand the search space of PDE discovery beyond fixed candidate libraries, and the public code and data make the benchmark portion reproducible. The benchmark results on canonical PDEs, irregular domains, and high-dimensional systems are largely plausible and would be useful to the community. However, the central scientific-discovery claim for the wave-breaking equation depends critically on the accuracy of automatic-differentiation derivatives of ANN surrogates trained on gappy camera data, and on the stability of the fitted coefficients. As the evidence stands, that headline claim is not fully supported and requires additional validation before it can be accepted.","major_comments":[{"comment":"The wave-breaking result rests on automatic-differentiation derivatives of per-experiment ANN surrogates trained on three non-contiguous spatial subdomains, yet the paper provides no validation that these surrogates, and their spatial derivatives up to third order, are accurate in the gaps or near the breaking onset. Since Eq. (1) computes the R2 reward on these surrogate derivatives and Eq. (15) is selected by that reward, any bias in the surrogate derivatives can select a structure that fits surrogate artifacts rather than the physics. Please provide held-out derivative tests, such as comparison with finite differences on observed frames, train/test splits, or ensemble variability across surrogate initializations, and report how the selected equation changes when the surrogate architecture or training seed is varied.","section":"Supplementary Information S1.1; Materials and Methods ('the surrogate model can ..."},{"comment":"Table S3 shows that the coefficients of Eq. (15) vary substantially across the 12 experiments: c2 ranges from 4.819e-4 to 2.376e-3 and c3 changes sign (negative in cases 1-11, positive in case 12). The statement that the coefficients 'may be slightly different' is therefore inaccurate, and the sign change in the linear dispersive term is physically significant. The paper should report per-experiment R2 values or other goodness-of-fit measures, uncertainties for c1-c3, and a sensitivity analysis with respect to random seeds and surrogate training; without these, the claim that this PDE 'performs well across all experiments' and the physical interpretation of the new nonlinear term are not established.","section":"Table S3; Results, 'Discovery of new PDEs from real-world experimental data'"},{"comment":"The evidence that Eq. (15) is a 'previously unreported' governing equation is incomplete. The paper does not report quantitative out-of-sample skill, does not compare Eq. (15) with classical weakly nonlinear models such as KdV, NLS, or Boussinesq-type equations, and does not show that the discovered equation outperforms plausible alternatives on data withheld from structure selection. The time-marching posterior prediction in Fig. 6e needs a quantitative error metric per experiment; otherwise the central claim of scientific discovery is not falsifiable from the reported material.","section":"Results, 'Discovery of new PDEs from real-world experimental data'; Fig. 6e"}],"minor_comments":[{"comment":"Please make the components of Eq. (1) explicit in the typeset version; in the provided text the expression is difficult to read, and the definitions of alpha0, Nterm, and R2 should appear in one place.","section":"Eq. (1)"},{"comment":"Please typeset Eq. (15) with unambiguous subscripts and superscripts; the current display is difficult to parse and could mislead readers about which terms are multiplied.","section":"Eq. (15)"},{"comment":"The main-text benchmark figures appear to show single representative trials; consider reporting success rates and standard errors in the main text, as already computed in Supplementary S1.2, rather than only in the supplement.","section":"Fig. 2; Supplementary S1.2"},{"comment":"The sparsity penalty alpha0 is set to different values in different experiment groups (0.2, 1, and 0.02), but no sensitivity analysis is given; a brief report on how the selected structure depends on alpha0 would strengthen the robustness claims.","section":"Materials and Methods, 'The experimental settings'"},{"comment":"Please cite the specific handbook chapter or equation number for the term (eta*)^2 eta_xxx so that readers can verify that Eq. (15) itself is not simply memorized from the handbook, rather than being a genuinely new recombination.","section":"Discussion, paragraph on compactons"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the benchmark and framework-comparison portions of the paper are solid and likely publishable after revision, but the headline wave-breaking discovery needs substantially more evidence. Specifically, the authors should supply surrogate derivative validation, per-experiment uncertainties, and out-of-sample predictive tests. If those additions are made, the paper could become acceptable; without them, the central claim is not yet supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The framework part of this paper is genuinely new and worth engaging with. Training a small GPT-style model on 221 PDE structures from a handbook, tokenizing equations into operators and terms, and then steering generation with an R²-based reward loop is a clever way to balance search space and optimization efficiency in PDE discovery. The benchmark results are plausible: EqGPT recovers canonical PDEs from noisy and sparse data, handles unusual terms like u_xt and u_x/x, and the target-deletion ablation shows the search is not just memorizing the training set. The comparisons against eight existing methods are fair enough for a scope/efficiency heatmap. Code and data are public, which makes the method section checkable. I would take the framework claim seriously.\n\nThe soft spot is exactly where the stress-test note points. The headline wave-breaking result depends on derivatives of an ANN surrogate trained on three non-contiguous spatial subdomains of camera images, with no interpolation to fill the gaps. The reward that selects Eq. (15) is computed from automatic differentiation of that surrogate, and high-order spatial derivatives (η_xxx and higher) of a smooth interpolant on gappy data are strongly influenced by network regularization, not measured physics. The paper gives no check on derivative accuracy, no error bars, no out-of-sample validation, no comparison with existing wave-evolution models, and Table S3 shows the coefficients scatter across experiments. That is not enough to claim a previously unreported governing equation. The authors themselves note the limitation about no interpolation, but they do not address its consequence for the derivatives.\n\nI do not think this sinks the paper. The method is the substance, and the wave case is an illustration that needs more work. The circularity concern is real but mild: the search space is a recombination of known forms, and the paper frames the new equation as a selection from that space, which is honest. I would recommend sending this to peer review, but with the expectation that reviewers require (a) a surrogate-derivative validation for the gappy data, e.g., comparing AD derivatives on held-out frames or against a numerical wave model, and (b) uncertainty estimates and a baseline comparison for the discovered wave equation. Without those, the strong claim should be downgraded.","headline":"EqGPT framework is a legitimate step for PDE discovery and deserves serious review, but the 'previously unreported wave-breaking PDE' is currently overclaimed.","tokens_in":24010,"tokens_out":2953,"would_cite":true,"duration_ms":31033,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A generative model trained on sentence-like encodings of handbook PDEs recovers governing equations from sparse, noisy data, including a previously unreported equation for surface gravity waves approaching breaking.","keywords":["PDE discovery","generative representation of equations","scientifically augmented training","knowledge-guided discovery","neural surrogate model","wave breaking","nonlinear dynamics","symbolic regression"],"falsifier":"Train the same pipeline on a synthetic PDE with a known solution, delete a subregion from the training data, and compare automatic-differentiation derivatives from the surrogate against exact derivatives in that subregion; if the reward ranks wrong structures above the true PDE whenever the derivative error is substantial, or if the discovered wave equation fails to time-march a new breaking-wave experiment it was not fitted to, the central claim would be refuted.","tokens_in":23072,"feed_emoji":"🌊","tokens_out":17619,"duration_ms":163049,"temperature":0.7,"pith_summary":"Data-driven PDE discovery usually faces a trade-off: sparse regression is fast but limited to a hand-built candidate library, while free-form symbolic search explores more but is slow and fragile. This paper proposes embedding prior knowledge instead of brute-force search: existing PDEs from a mathematical handbook are tokenized into operators and base terms, and an autoregressive generative model learns which term combinations are plausible. In a generation–evaluation–optimization loop, candidate equations are produced by the model, scored by how well they fit derivatives from a neural surrogate of the data, and the best candidates are used to fine-tune the generator. The paper reports that this recovers canonical PDEs with tens of seconds of compute and tolerates over 50 percent noise, handles unusual terms such as $u_{xt}$ and $u_x/x$, works on irregular and high-dimensional domains, and uncovers a previously unreported equation for strongly nonlinear surface gravity waves moving toward breaking from real wave-tank data.","feed_headline":"Trained on handbook PDEs, model finds new wave-breaking law","feed_subtitle":"It recovers unusual PDE forms and a previously unreported wave equation from real experiments.","key_machinery":"The load-bearing object is the generative representation of equations (GRE): a tokenization that writes a PDE as a sentence of operators and basic terms, so that a transformer language model can learn co-occurrence patterns among terms. The trained model, EqGPT, proposes candidate structures; a neural-network surrogate trained on the observations supplies values and derivatives everywhere, and a reward combining a sparsity penalty with the coefficient of determination $R^2$ of a least-squares regression scores each candidate. The top-scoring structures fine-tune the generator, and the loop drives the model toward equations that are both data-consistent and plausible under the handbook's prior.","core_discovery":"The paper's central claim is that the combinatorial difficulty of PDE discovery can be shifted from the optimizer to a learned prior over equation structure. Concretely, 221 handbook PDEs are rewritten as sentences over a 56-token vocabulary of operators and terms, augmented by term shuffling to 7,072 sentences, and used to train EqGPT, an autoregressive transformer that generates free-form PDEs by next-token sampling under syntactic and dimensional constraints. The generated structures are evaluated by least-squares fitting against meta-data produced by a neural-network surrogate, with a reward that combines the fit's $R^2$ with a sparsity penalty; the top ten structures fine-tune the generator, and the loop repeats. Against this machinery the paper reports rediscovery of eight canonical PDEs, including forms with the mixed derivative $u_{xt}$ and the fraction $u_x/x$, recovery of Poisson's equation on disk, smiley-face, letter-glyph, and three-dimensional shuttle geometries, and reconstruction of coupled oil–water flow equations in three dimensions. Its headline experimental result is a previously unreported PDE for nondimensional surface elevation $\\eta^*$ of strongly nonlinear waves propagating toward breaking, with $\\eta_t^*$ on the left and a combination of spatial derivative terms on the right whose three coefficients are re-fit per experiment; the paper argues the leading linear term agrees with classical potential-flow wave theory and that the newly identified nonlinear term can generate the higher-frequency content seen as breaking approaches.","pith_inferences":["The scaling behaviour implied by the paper's ablation is that EqGPT's discovery power is bounded by the coverage of the handbook corpus; a sharper test would hold out entire families of terms and ask whether the model can compose them from parts it has seen.","Because the reward is computed from surrogate derivatives, the wave-equation result should be checked against a second, independent derivative estimator on the same data; a mismatch would indicate the discovered form may be an artifact of the surrogate.","The same loop could be repurposed for closure discovery by freezing the known operators of a system and letting EqGPT generate only the unknown coupling or source terms, which would turn the method into a hypothesis generator for unresolved physics.","A natural next experiment is to test whether the discovered wave equation predicts surface elevation for wave groups with spectral parameters outside the 12 calibration cases, especially closer to or past the onset of breaking."],"forward_implications":["The fixed candidate-library bottleneck disappears: terms like $u_{xt}$ and $u_x/x$ are generated and can be selected when the data call for them.","PDE discovery becomes feasible on irregular, discontinuous, and three-dimensional geometries, because the surrogate supplies predictions and derivatives at arbitrary observation points.","The loop can operate with as few as hundreds of data points under heavy noise in the tested cases, and each optimization cycle costs seconds, so the method is cheap enough to rerun for many candidate hypotheses.","Growing the equation corpus improves the success rate; with the current 221-equation handbook the reported rediscovery success rates exceed 0.7 even when the target equation is held out.","For multi-variable systems, the method recovers expanded forms of nested-operator equations, giving scientists a ranked set of structurally plausible laws rather than a single guess."],"supporting_citations":[{"why":"the mathematical handbook that supplies the 221 PDE structures forming EqGPT's training corpus.","marker":"[26]"},{"why":"the autoregressive transformer architecture on which EqGPT is built.","marker":"[27]"},{"why":"the surrogate-model technique used to generate meta-data and derivatives by automatic differentiation for evaluating PDEs.","marker":"[28]"},{"why":"the sparse-regression baseline whose fixed candidate library motivates the generative approach.","marker":"[6]"},{"why":"weak-form sparse regression baseline used in the comparison, showing the library limitation persists.","marker":"[8]"},{"why":"LLM-based equation discovery baseline used to compare computational efficiency.","marker":"[17]"},{"why":"the image-processing method that reconstructs wave surface elevation from camera images, providing the experimental wave data.","marker":"[42]"},{"why":"physics-informed neural networks used to refine the coefficients of the discovered PDEs.","marker":"[49]"}],"fun_headline_variants":["Handbook-trained AI discovers a new wave-breaking PDE","EqGPT: Trained on handbook PDEs, finds unreported wave law","From math handbooks to new physics: EqGPT's discovery","AI upends PDE discovery: learns from handbooks, finds new law","Generative discovery: Handbook PDEs prime AI to find new wave equation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the neural surrogate trained on the observations yields accurate values and derivatives everywhere in the domain—including the non-contiguous gaps in the wave-camera images—because every generated equation is judged by how well its terms fit those derivatives.","fun_headline_variants_meta":{"raw":{"variants":["Handbook-trained AI discovers a new wave-breaking PDE","EqGPT: Trained on handbook PDEs, finds unreported wave law","From math handbooks to new physics: EqGPT's discovery","AI upends PDE discovery: learns from handbooks, finds new law","Generative discovery: Handbook PDEs prime AI to find new wave equation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000336,"raw_usage":{"total_tokens":1903,"prompt_tokens":1027,"completion_tokens":876,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":643,"completion_tokens_details":{"reasoning_tokens":786}},"tokens_in":643,"tokens_out":876,"duration_ms":8281,"temperature":1.0,"reasoning_tokens":786,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:53:37.262845+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the same pipeline on a synthetic PDE with a known solution, delete a subregion from the training data, and compare automatic-differentiation derivatives from the surrogate against exact derivatives in that subregion; if the reward ranks wrong structures above the true PDE whenever the derivative error is substantial, or if the discovered wave equation fails to time-march a new breaking-wave experiment it was not fitted to, the central claim would be refuted.","supporting_citations":[{"cited_title":"Nonlinear Partial Differential Equations for Scientists and Engineers","cited_arxiv_id":null,"evidence_quote":"the mathematical handbook that supplies the 221 PDE structures forming EqGPT's training corpus."},{"cited_title":"& Zhang, D","cited_arxiv_id":null,"evidence_quote":"the surrogate-model technique used to generate meta-data and derivatives by automatic differentiation for evaluating PDEs."},{"cited_title":"H., Brunton, S","cited_arxiv_id":null,"evidence_quote":"the sparse-regression baseline whose fixed candidate library motivates the generative approach."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"weak-form sparse regression baseline used in the comparison, showing the library limitation persists."},{"cited_title":"& Zhang, D","cited_arxiv_id":null,"evidence_quote":"LLM-based equation discovery baseline used to compare computational efficiency."},{"cited_title":"M., Fang, Y","cited_arxiv_id":null,"evidence_quote":"the image-processing method that reconstructs wave surface elevation from camera images, providing the experimental wave data."},{"cited_title":"& Karniadakis, G","cited_arxiv_id":null,"evidence_quote":"physics-informed neural networks used to refine the coefficients of the discovered PDEs."}],"review_version":1}