{"id":"18d569bf-2dc4-44b2-a3fa-d972d037dbbd","arxiv_id":"2607.10546","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"LLM-PDESR pairs LLM structural hypotheses with C4 quintic splines and subdomain weighted residuals to recover PDEs from noisy data more robustly than prior symbolic methods.","lead":"LLM-PDESR discovers governing PDEs from noisy data by having a language model propose equation skeletons and scoring them with quintic-spline derivatives and subdomain weighted residuals. It recovers known and novel equations more reliably than prior tools and extracts a simple 1D wind surrogate from ERA5 reanalysis.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"SWR+spline fitness may still rank effective low-order surrogates above exact nested/high-order operators, undermining the claim that recovered skeletons are the true governing PDEs.","rationale":"The reader correctly isolates the faithfulness of the spline/SWR/Pareto landscape as the weakest link supporting the strongest claim. The manuscript supplies transparent partial-recovery and failure cases (App. E) and ablations (Fig. 4) that already show the components matter, yet never directly verifies that the true operator would have been selected had it been proposed. The concrete test above is a minimal, fully specified check that uses only the paper’s own evaluation code path; a negative outcome would convert the CONDITIONAL verdict into a clearer limitation on what “structural recovery” means, while a positive outcome would substantially strengthen the claim. No internal contradiction appears in the reported numbers, so the verdict remains CONDITIONAL rather than REJECT; the concern simply sharpens the same soft spot the reader already flagged.","tokens_in":34002,"tokens_out":675,"duration_ms":25212,"concrete_test":"For Topography Chemotaxis and Forced Quintic Swift-Hohenberg, insert the exact ground-truth skeletons (with free numeric coefficients) into the identical quintic-spline + K=10 SWR + BFGS pipeline used for LLM candidates; record their optimized SWR-MSE and AST complexity, then re-run the Pareto selection of Alg. 1. If either true skeleton is dominated by the published partial recovery (higher MSE or higher complexity at comparable MSE), the landscape is not faithful to the governing operator and the structural-success claims weaken.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires that the C4-quintic + SWR evaluation (Eqs. 2–4, Alg. 2, App. B) plus the 10\times MSE_min Pareto gate (Alg. 1 lines 12–13) produce a landscape in which the true operator is preferred over residual-minimizing phenomenological approximations. On the five novel systems designed to block memorization this is only partially secured: Topography-Constrained Chemotaxis recovers an effective macroscopic advection + logistic form that misses the nested (u(1−u)cos(x))_x term entirely (App. E.2); Forced Quintic Swift-Hohenberg recovers the 4th-order skeleton but replaces −u^{3}+u^{5} and the multiplicative Gaussian pump by a free cubic + additive Gaussian (App. E.4). Both partial skeletons achieve low SWR-MSE and survive the parsimony gate, yet are not the true operators. The same pattern appears in the 14-parameter 1D-CACE skeleton (Eq. 5), whose OOD R^{2} (58–76 %) is obtained only after unrestricted re-fitting of every coefficient; the paper itself frames the xv/xu terms as first-order closures rather than fundamental physics. Thus the reported “structural recovery” and “invariant dynamical mechanisms” rest on the untested assumption that the lowest-complexity high-score skeleton under SWR coincides with the true PDE rather than a flexible effective surrogate.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript proposes LLM-PDESR, a closed-loop PDE discovery framework that couples LLM-generated symbolic skeletons with a noise-robust numerical evaluator: C^4 quintic B-spline spatial derivatives, subdomain weighted residual (SWR) fitness as a low-pass filter, BFGS coefficient identification, and a Pareto front with a 10\times MSE_min parsimony gate for feedback and final selection. It reports near-perfect structural recovery on 23 canonical 1D PDEs under 0–20% additive noise, exact or partial recovery on five synthetically designed novel systems (including a coupled predator–prey PDE), and extraction of a 14-parameter 1D coupled atmospheric circulation skeleton (1D-CACE) from ERA5 winds with spatiotemporal OOD coefficient re-optimization. Ablations isolate spline, SWR, and Pareto contributions; Appendix B supplies error bounds and variance-reduction arguments for the evaluator.","tokens_in":34529,"tokens_out":1445,"duration_ms":27442,"significance":"If the claims hold, the work is a meaningful advance for scientific machine learning: it pairs free-form LLM hypothesis generation with a mathematically motivated continuous evaluator that addresses the well-known finite-difference noise amplification problem, and it supplies both a memorization-resistant novel-PDE suite and a real reanalysis case study. Strengths include multi-baseline comparisons (Fig. 2, Table 1), component ablations (Fig. 4), transparent failure on the traffic-shock system (App. E.5), algebraic-equivalence discussion for Michaelis–Menten recovery, and theoretical support for spline/SWR error and variance reduction (App. B). These elements make the contribution more than an LLM wrapper and of clear interest to the PDE-discovery and SciML communities.","major_comments":[{"comment":"§4.3, Table 1, and App. E.2/E.4: The central claim of “true discovery” / “structural recovery” on the five novel PDEs is only partially secured by the SWR+Pareto landscape. Topography-Constrained Chemotaxis recovers an effective logistic + linear advection surrogate that entirely misses the nested (u(1−u)cos(x))_x term; Forced Quintic Swift–Hohenberg recovers u_xxxx but replaces −u³+u⁵ and the multiplicative Gaussian pump by a free cubic plus additive Gaussian. Both partial skeletons achieve low SWR-MSE and pass the 10×MSE_min gate, yet are not the true operators. The paper labels these “≈” and discusses them in the appendix, but the abstract and §1 still frame the suite as validating genuine symbolic discovery. Please either (i) tighten claims to distinguish exact skeleton recovery from low-MSE phenomenological surrogates, or (ii) add controlled experiments showing when/whether the true","section":"§4.3, Table 1, App. E.2–E.4"},{"comment":"§4.4, Eq. (5), Table 2: OOD validation of 1D-CACE freezes the 14-term skeleton and re-optimizes all coefficients θ1…θ14 in each regime (R² ≈ 58–76%). The paper itself interprets xv/xu as first-order spatial closures rather than fundamental physics. Without controls—e.g., random or dictionary-generated multi-term skeletons of comparable complexity re-fit the same way, or coefficient transfer with only a subset free—it is hard to separate “invariant dynamical mechanisms” from flexibility of a rich ansatz. A short control table or null-model comparison is needed to support the OOD generalization claim as stated.","section":"§4.4, Eq. (5), Table 2"},{"comment":"Algorithm 1 (lines 12–13) and §3.3: Final selection uses the fixed heuristic MSE(F) ≤ 10 × MSE_min then min complexity. This gate is load-bearing for “avoidance of equation bloat,” yet no sensitivity to the factor 10 (or alternatives such as knee-point / AIC-style criteria) is reported. Given that partial novel recoveries and the 14-parameter 1D-CACE both survive this gate, a brief sensitivity or alternative-selection analysis would strengthen the claim that the recovered skeletons are preferred for structural reasons rather than gate choice.","section":"Algorithm 1, §3.3"}],"minor_comments":[{"comment":"Fig. 2 axis labels and some appendix figures appear as Unicode glyph dumps in the preprint PDF; ensure vector fonts render correctly in the camera-ready version.","section":"Figure 2"},{"comment":"Appendix D: DeepSeek-V3 is accessed via a closed API with a temperature schedule “every 30,000 tokens.” For reproducibility, state model snapshot/date if available, and report variance over multiple independent runs (seeds) on at least the novel suite.","section":"Appendix D"},{"comment":"K = 10 subdomains and ω = 2 L_domain/K are fixed throughout; a one-paragraph sensitivity note (or reference to App. B scaling) would help readers port the method.","section":"§3.2, Algorithm 2"},{"comment":"Related work: briefly position against concurrent LLM-SR / LLM-guided discovery lines (e.g., LLM-SR, FunSearch-style program search) beyond EqGPT, so the Pareto-feedback + continuous evaluator distinction is sharper.","section":"§2.2"},{"comment":"Eq. (1) and multivariate prompts: notation for D > 1 is introduced early but most of the numerical pipeline is written for scalar u; a short remark that SWR is applied componentwise (or jointly) would clarify the predator–prey and 1D-CACE cases.","section":"§3, Eq. (1)"}],"recommendation":"major_revision","confidential_remarks":"The technical core (spline + SWR evaluator + Pareto LLM loop) is solid and the authors are unusually transparent about failures and partial recoveries. The main risk is overclaiming “true governing operators” when several headline results are effective surrogates; if the authors tighten language and add the OOD/null controls, this is a strong contribution for a methods-oriented ML or SciML venue. Scope is currently 1D; that is acceptable if framed clearly, but editors should not expect 2D/3D without major extra work."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The useful part of this paper is the evaluation stack, not the LLM. Quintic C4 splines plus subdomain weighted residuals give a cleaner fitness signal than finite differences and pointwise MSE, and the Pareto feedback loop keeps the LLM from pure bloat. On the 23 canonical 1D PDEs they report near-perfect structural recovery under 0–20% noise and beat SGA-PDE, DISCOVER, WSINDy_PDE, and EqGPT on the metrics they define. The five novel equations are a real attempt to block memorization, and the transparent traffic-flow failure plus the appendix error bounds for splines and SWR variance reduction are better than most papers in this area.\n\nWhat is actually new is the specific pipeline (LLM skeletons → spline derivatives → SWR loss → BFGS → Pareto top-3 feedback) plus the ERA5 1D-CACE skeleton with frozen-structure OOD coefficient re-fit. Components already exist separately (LLM-SR, EqGPT, WSINDy weak forms); the contribution is the combination and the empirical showing that it works under noise.\n\nThe soft spot the stress-test flags is real but not hidden. On topography chemotaxis they recover an effective advection + logistic form that drops the nested cos(x) flux; on forced Swift–Hohenberg they keep the fourth-order skeleton but replace the quintic nonlinearity and multiplicative pump with a free cubic plus additive Gaussian. Both survive the 10×MSE_min gate with low SWR error. The paper labels these “partial” and discusses macroscopic approximation, so it is not a contradiction—just a reminder that SWR+parsimony can prefer flexible low-order surrogates over exact nested operators. The 14-parameter 1D-CACE is explicitly framed as a reduced-order closure (xv/xu terms as first-order spatial closures), and OOD R² of 58–76% only after re-fitting every coefficient is consistent with that framing, not with discovering invariant fundamental laws.\n\nNo code or multi-seed LLM statistics are shipped, and free parameters (K, window width, temperature schedule, BFGS timeout) matter. Still, the math in App. B is careful, the ablations in Fig. 4 are informative, and the failure analysis is honest. This is for people who do free-form PDE discovery or reduced-order climate surrogates. It deserves a serious referee; I would bring it to a SciML reading group and would cite the evaluation design if I were building a similar pipeline.","headline":"Solid integration of LLM skeletons with spline/SWR evaluation that beats the usual baselines on noise and anti-memorization PDEs; “structural recovery” sometimes means a good effective surrogate, and the paper is mostly honest about that.","tokens_in":35168,"tokens_out":632,"would_cite":true,"duration_ms":16545,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"LLM-guided PDE discovery works when noise-robust derivatives and residual filters give the language model a clean fitness signal.","keywords":["PDE discovery","symbolic regression","large language models","subdomain weighted residuals","quintic B-splines","noise robustness","Pareto multi-objective optimization","scientific machine learning"],"falsifier":"On the Forced Quintic Swift-Hohenberg or Traffic-Flow novel systems, or under 20 percent noise on the high-order benchmark suite, show that the recovered skeleton either systematically omits essential nonlinear or nested terms while still achieving low SWR loss, or fails to transfer when the structural skeleton is frozen and only coefficients are re-optimized on a held-out spatiotemporal domain.","tokens_in":34889,"feed_emoji":"📐","tokens_out":922,"duration_ms":10783,"temperature":0.7,"pith_summary":"Discovering the partial differential equations that govern a physical system from noisy measurements is hard: the combinatorial space of candidate formulas is huge, and standard finite-difference residuals amplify high-frequency noise so badly that search algorithms chase mathematical artifacts instead of true laws. This paper claims that pairing a large language model as a structural hypothesis generator with a continuous evaluation environment—C4 quintic spline derivatives plus subdomain weighted residuals that act as natural low-pass filters—restores a faithful fitness landscape. A Pareto loop then feeds the best accurate-yet-simple candidates back to the model so it can refine them. On 23 canonical one-dimensional PDEs the method recovers the correct skeletons far more reliably under noise than prior symbolic-regression and sparse-regression baselines; it also recovers exact or core terms on five novel equations built to block memorization, and it extracts a reusable structural skeleton for a one-dimensional atmospheric-wind surrogate from real ERA5 reanalysis data that transfers across geography and season once coefficients are re-fit.","feed_headline":"LLM finds PDEs when residuals stop amplifying noise","feed_subtitle":"Spline derivatives and subdomain filters restore a clean fitness signal, beating prior methods on noisy and novel equations","key_machinery":"Subdomain weighted residual (SWR) evaluation on C4 quintic B-splines: the residual of a candidate PDE is integrated against compact smooth test functions over local windows, producing a low-pass-filtered loss whose BFGS optimization supplies the LLM with non-deceptive fitness scores and a Pareto front that prunes bloat.","core_discovery":"The central claim is that LLM-PDESR—an end-to-end loop that lets a language model propose executable equation skeletons while evaluating them with quintic-spline differentiation and subdomain weighted residuals, then selecting via a Pareto accuracy-complexity front—recovers true governing PDE structures from noisy data more reliably than existing methods and yields a transferable 1D atmospheric surrogate directly from reanalysis observations.","pith_inferences":["If the SWR low-pass property is the dominant reason for success, the same evaluation core could be swapped under non-LLM generators such as genetic programming or reinforcement-learning controllers.","The partial recoveries on topography-constrained chemotaxis and forced Swift-Hohenberg suggest the method may systematically prefer effective homogenized operators over exact micro-scale forcings.","Extending the quintic-spline/SWR pipeline to unstructured 2-D/3-D grids would be the natural next stress test for real multi-physics discovery."],"forward_implications":["Noise-robust continuous evaluation lets language-model search scale to free-form PDEs without hand-crafted dictionaries.","Pareto feedback systematically suppresses equation bloat that plagues genetic and neural symbolic regression.","The same skeleton can be re-parameterized for new geographic or seasonal regimes once the structure is frozen.","Novel out-of-distribution equations become a practical test that distinguishes genuine discovery from corpus memorization."],"fun_headline_variants":["LLM-PDESR recovers true PDEs from noisy data via clean residual signals","Quintic splines and subdomain filters let LLMs discover robust PDEs","LLMs generate PDE hypotheses scored by noise-resistant residuals","Pareto loop with LLM skeletons finds PDEs in noisy reanalysis data","Subdomain residuals fix fitness landscape for LLM-driven PDE discovery"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"That the spline-plus-SWR fitness landscape is faithful enough for the language-model proposals to match the true governing operators rather than merely effective residual-minimizing surrogates, especially for nested high-order terms and the multi-parameter atmospheric skeleton.","fun_headline_variants_meta":{"raw":{"variants":["LLM-PDESR recovers true PDEs from noisy data via clean residual signals","Quintic splines and subdomain filters let LLMs discover robust PDEs","LLMs generate PDE hypotheses scored by noise-resistant residuals","Pareto loop with LLM skeletons finds PDEs in noisy reanalysis data","Subdomain residuals fix fitness landscape for LLM-driven PDE discovery"]},"model":"grok-4.5","effort":"low","cost_usd":0.004886,"raw_usage":{"total_tokens":1405,"prompt_tokens":838,"num_sources_used":0,"completion_tokens":95,"cost_in_usd_ticks":48860000,"prompt_tokens_details":{"text_tokens":838,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":472,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":838,"tokens_out":95,"duration_ms":4412,"temperature":1.0,"reasoning_tokens":472,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T10:54:48.191053+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"On the Forced Quintic Swift-Hohenberg or Traffic-Flow novel systems, or under 20 percent noise on the high-order benchmark suite, show that the recovered skeleton either systematically omits essential nonlinear or nested terms while still achieving low SWR loss, or fails to transfer when the structural skeleton is frozen and only coefficients are re-optimized on a held-out spatiotemporal domain.","supporting_citations":[],"review_version":1}