{"id":"59e83736-2c4f-42d7-8544-c6d88dea13b0","arxiv_id":"2505.10109","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"On DNS data for Rayleigh-Benard and plane Couette convection, SPIDER recovered governing equations, constraints, and boundary conditions with less tuning than pySINDy; the highest-Rayleigh-number failures are attributed to DNS under-resolution.","lead":"This paper tests two machine-learning tools, pySINDy and SPIDER, on whether they can recover the known equations of turbulent convection from numerical simulation data. SPIDER recovered the equations, the incompressibility constraint, and the boundary conditions more easily, and the failures at the highest Rayleigh numbers traced to under-resolved simulations.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Fairness of the pySINDy comparison is the main load-bearing issue; the claimed advantage needs a cost-matched test.","rationale":"The reader's weakest assumption pinpoints the same asymmetry: the pySINDy implementation is handicapped by a 104-term brute-force library and then rescue-assisted by knowledge of the exact equations, while SPIDER benefits from small bespoke libraries and insider usage by co-developers. I agree that this does not invalidate the unconstrained pySINDy tests, the SPIDER constraint/boundary-condition recovery, or the DNS-resolution interpretation, but it does undermine the headline comparative claim. The paper's own text supports this concern: Section 4.1.2 states the constraints are applied 'using knowledge about the problem' and are 'justified only by comparison to the known correct model,' and Section 5.1 reports that pySINDy could not be run in 3D due to memory limits, so the comparison is incomplete by admission. A cost-matched test would settle whether the advantage is algorithmic or implementational. The verdict should remain CONDITIONAL: the central claim is plausible and the DNS-validation idea is valuable, but the paper should either soften the comparative conclusion or provide the matched benchmark before claiming SPIDER is 'easier and more capable' as a general fact. I do not see grounds for REJECT or UNVERDICTED because the non-comparative results (recovery at various R, under-resolution diagnosis) are reproducible and self-consistent.","tokens_in":26645,"tokens_out":1779,"duration_ms":15792,"concrete_test":"Run a cost- and knowledge-matched comparison on the same 2D Rayleigh-Benard data at R=10^6, 10^8, and 10^10: give pySINDy either (a) the same symmetry-adapted library SPIDER uses, expressed as scalar/vector fields without term interlinking, or (b) a 104-term library with no equation-derived interlinking, and let it use the same weak-form integration and a standard optimiser with hyperparameters chosen by the same correlation-scale heuristic proposed in Section 6.1. Record time-to-solution, memory, and success rate over 100 random subdomain draws. If pySINDy then recovers Navier-Stokes and the heat equation at R=10^10 at comparable cost, the claimed advantage is an artifact of the constrained-library setup; if it still fails, the advantage is robust.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that SPIDER recovers the governing equations, constraints, and boundary conditions for a wider range of Rayleigh numbers with less user effort than pySINDy. This claim rests on a comparison in which the two methods are not held to the same standard of assistance, compute, or prior knowledge. In Section 4.1.2 the pySINDy library is built by brute force (104 terms in 2D, 290 in 3D) and then artificially constrained using knowledge of the exact equations: incompressibility is imposed via Eq. 9, diffusion is reduced to Laplacian form, and coefficients are interlinked across vector components. The paper itself describes this as 'justified only by comparison to the known correct model.' SPIDER, by contrast, uses the small symmetry-adapted libraries (26 scalar, 27 vector terms) of Eqs. 11-12, built from rotational/translational symmetry and variable choice, with no term-level knowledge of the equations. Moreover, two co-authors are core developers of SPIDER, so ease-of-use and hyperparameter choices could reflect insider knowledge. The claimed advantages—smaller library, wider R range, ability to recover constraints and boundary conditions—are therefore not established as intrinsic algorithmic properties; they could be artifacts of the asymmetric setup. The comparison is not internally invalid, and the paper states that the constrained pySINDy library was only a guide to hyperparameter sweeping, but the headline 'SPIDER is easier and more capable' is read off this comparison. A cost-matched benchmark (same prior knowledge allowed, same compute/memory budgets, same derivative-evaluation method, blind hyperparameter selection) is needed to support the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript benchmarks two sparse-regression equation-discovery frameworks, pySINDy (weak-form SINDy with MIOSR) and SPIDER, on noiseless spectral DNS data of 2D and 3D Rayleigh-Bénard convection and 2D convective plane Couette flow. The authors show that with a constrained library pySINDy recovers the 2D governing equations up to R = 10^8 (and only the Euler equation at R = 10^12), while SPIDER recovers the momentum and temperature equations, the incompressibility constraint, and the boundary conditions over a wider range, including the 3D cases. They further propose that flow correlation length scales and timescales should inform the initial choice of weak-form subdomain sizes, and they argue that failures at the highest Rayleigh numbers are caused by under-resolved DNS rather than by the discovery methods.","tokens_in":26945,"tokens_out":6140,"duration_ms":61900,"significance":"If the comparison is accepted, the paper is a practically useful demonstration that symmetry-adapted libraries and weak-form regression can reduce user burden in equation discovery, that constraints and boundary conditions can be recovered from data, and that equation recovery can serve as a diagnostic of numerical resolution. The manuscript is unusually transparent: it documents hyperparameter sweeps, residual tables, recovered coefficients, and power spectra, and it provides simulation and analysis scripts in a data repository. Those strengths do not remove the main weakness: the headline advantage of SPIDER over pySINDy rests on a comparison that is not cost-matched in library construction, prior knowledge, or computational effort, so the central claim is only conditional.","major_comments":[{"comment":"The central 'smaller library and more easily recovered' claim is not established because the two methods are given different prior information. The pySINDy search starts from a brute-force 104-term library in 2D and is then artificially pruned and interlinked using exact knowledge of the solution: incompressibility is imposed via Eq. (9), diffusion is restricted to the Laplacian form, coefficients are interlinked across vector components, and individual terms are switched off; the text states these choices are 'justified only by comparison to the known correct model.' SPIDER instead starts from the 26-term and 27-term symmetry libraries of Eqs. (11)-(12), with no term-level knowledge of the exact equations. The comparison therefore conflates the algorithm with the library-construction protocol. I would ask for either a cost-matched comparison (for instance, SPIDER with the same brute-force library, or pySINDy with a library built from the same symmetry principles) or an explicit reporting of user effort, memory, and wall-clock time for a shared discovery task. Without this, the abstract's 'more easily' is an artifact of the setup rather than an established algorithmic property.","section":"Sections 4.1.2 and 4.2"},{"comment":"The claim that SPIDER 'goes beyond pySINDy' by recovering constraints and boundary conditions is not supported as a comparison, because pySINDy is only applied to bulk 2D governing equations. No pySINDy experiment is reported for recovering ∇·u = 0 or for recovering boundary conditions, so the paper demonstrates that these tasks were not attempted with pySINDy, not that pySINDy cannot perform them. This claim should be either tested in the same framework (for example, by a weak-form regression on boundary data) or explicitly restated as a capability demonstration for SPIDER only.","section":"Sections 5.1 and 5.2.2"},{"comment":"The interpretation that failures at R = 10^12 are due to under-resolution, while plausible and supported by the power spectra in Fig. 8, is not proven because no counterfactual is provided. A higher-resolution run at the same Rayleigh number (for example, with 2048 or 4096 points in the horizontal direction) is needed to show that SPIDER then recovers the diffusive term and the full Navier-Stokes equation. Without such a test, the failure could also be attributed to hyperparameter selection, weak-form quadrature error, or the near-machine-precision magnitude of the diffusion coefficient (~10^-6). This point is load-bearing because the 'wider range of R' and the proposed validation/verification use both rest on attributing the high-R failure to the data rather than to the methods.","section":"Sections 5.2.1 and 6.2"}],"minor_comments":[{"comment":"The captions spell 'Rayleigh' as 'Rayeigh'; please correct this typo.","section":"Figure 3 and Figure 4 captions"},{"comment":"The text states Dedalus version 3.0.11, but the accompanying URL footnote points to v3.0.1; please reconcile this version discrepancy.","section":"Section 3"},{"comment":"The fourth pySINDy test is described as 'inconclusive,' but no quantitative criterion for success or failure is given; please specify what would have qualified as a conclusive recovery.","section":"Section 5.1.1"},{"comment":"The correlation scales are computed only for the 2D Rayleigh-Bénard simulations, yet the guidance to use them as a default is presented generally for pySINDy and SPIDER; please flag this as an extrapolation to the 3D and Couette cases.","section":"Section 6.1 and Table 4"},{"comment":"The term u·(∇u) in the vector library L1 is not defined; please clarify whether it denotes (u·∇)u, a contraction with the velocity-gradient tensor, or another expression.","section":"Equation (12)"}],"recommendation":"major_revision","confidential_remarks":"The fairness issue identified in the major comments is the key risk: the paper's headline comparison is asymmetric in library size, prior knowledge, and tuning effort. If the authors cannot add a cost-matched experiment, they should substantially soften the comparative claims and state explicitly that pySINDy was applied in a deliberately constrained configuration rather than in a best-practice configuration. I also note that two co-authors are developers of SPIDER; this is not itself a reason to reject, but it makes the 'ease of use' claim especially important to substantiate with objective, reproducible metrics rather than qualitative statements."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is a serious, reproducible benchmark: the authors run two equation-discovery methods on their own DNS of Rayleigh–Bénard convection and convective plane Couette flow, and they publish Dedalus, pySINDy, and SPIDER scripts plus data. Second, the central comparison is not yet a fair one. The paper claims SPIDER recovers governing equations, constraints, and boundary conditions more easily and for a wider range of Rayleigh numbers, but pySINDy was handicapped by an artificially constrained library built using the known equations (Section 4.1.2), while SPIDER used symmetry-adapted libraries. So the claim of intrinsic superiority is plausible but not proven.\n\nWhat is actually new: this is the first SPIDER application to buoyancy-driven convection and convective plane Couette flow, including recovery of boundary conditions. The correlation-time/scale heuristic for choosing subdomain sizes is a practical contribution that should save users trial and error. The resolution analysis—showing that failure at R=10^12 coincides with under-resolved boundary-layer spectra—is a genuine use of equation discovery as a diagnostic. Coefficient recovery at R=10^10 is well documented. The citation pattern is fine; the SPIDER self-citations are to the papers that introduced the method.\n\nSoft spots: the pySINDy library was built by brute force (104 terms) and then constrained using exact knowledge: incompressibility imposed, diffusion reduced to Laplacian, coefficients interlinked. The authors are transparent about this, calling it ‘justified only by comparison to the known correct model.’ That transparency is good, but it means the headline ‘SPIDER is easier and more capable’ is read off an asymmetric test. Also, pySINDy was only applied in 2D due to RAM limits, so the wider-R claim is partly a memory limitation, not purely algorithmic. These issues are fixable: a cost-matched benchmark with the same prior knowledge, derivative evaluation, and compute budget, ideally by someone not involved with SPIDER, plus a resolution study that varies grid points at fixed R.\n\nFor whom: practitioners deciding whether to reach for SPIDER or pySINDy on fluid problems, and people working on equation discovery. The guidance is useful, the scripts are reusable, and the under-resolution diagnostic is worth copying. But read the comparison as provisional. I would accept this for peer review and ask for a cost-matched follow-up.","headline":"A useful and mostly honest benchmark of SPIDER vs pySINDy on convection DNS, with a practical subdomain-sizing heuristic and a real DNS-validation idea, but the headline comparison is not yet fair enough to take at face value.","tokens_in":27528,"tokens_out":4340,"would_cite":true,"duration_ms":40350,"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":"SPIDER, a symmetry-adapted weak-form sparse-regression framework, recovers from DNS data the governing equations, the incompressibility constraint, and the boundary conditions of turbulent convection over a wider range of Rayleigh numbers…","keywords":["boundary conditions","Navier-Stokes equations","machine learning","data-driven techniques","sparse regression","Rayleigh-Bénard convection","turbulent convection","weak formulation"],"falsifier":"Run pySINDy with a small symmetry-adapted library comparable in size to SPIDER's (about 26 scalar and 27 vector terms, without additional coefficient interlinking) on the same 2D DNS data; if it then recovers the Navier-Stokes and heat equations at $R=10^{10}$ or $R=10^{12}$, the claim that SPIDER is inherently easier and more capable would be refuted, because the difference would trace to library construction rather than to the algorithm.","tokens_in":26440,"feed_emoji":"🌊","tokens_out":16444,"duration_ms":137957,"temperature":0.7,"pith_summary":"This paper compares two data-driven equation-discovery methods on simulated turbulent convection: the standard weak-form sparse regression approach in the pySINDy package, and the SPIDER framework, which builds its candidate term libraries from the symmetries of the problem. The authors find that SPIDER recovers the governing momentum and temperature equations over a wider range of Rayleigh numbers, in two and three dimensions, and also recovers the incompressibility constraint and the boundary conditions, things pySINDy did not produce in this study. The practical payoff is that equation discovery need not begin from a huge brute-force menu of terms; a small symmetry-selected library plus a weak-form regression can do the job with less user effort and less memory. The paper also reads failures at the highest Rayleigh numbers as evidence that the underlying simulation is under-resolved, so the method can double as a solver-validation tool.","feed_headline":"SPIDER recovers convection equations, constraints, and boundaries","feed_subtitle":"On simulation data, symmetry-built libraries outperform brute-force sparse regression.","key_machinery":"The load-bearing mechanism is weak-form sparse regression on symmetry-adapted libraries. Each candidate term is multiplied by a smooth weight function and integrated over rectangular spatiotemporal subdomains; integration by parts shifts derivatives from the data, where numerical differentiation is noisy, onto the weight function. SPIDER constructs its candidate terms by combining the physical fields $\\{u,p,T\\}$ with differential operators $\\partial_t$, $\\nabla$, and the symmetry-breaking unit vectors, giving a 26-term scalar library and a 27-term vector library; projections onto irreducible representations under rotations about the boundary normal separate normal and tangential boundary relations. This small, physically structured search space is what allows SPIDER to recover vector equations, the divergence constraint, and boundary conditions without the memory-heavy brute-force libraries and coefficient interlinking that the pySINDy application required.","core_discovery":"The central claim is that for Rayleigh-Bénard convection and plane convective Couette flow, the SPIDER framework recovers the full set of statements that govern the data—the momentum (Navier-Stokes) equation, the heat advection-diffusion equation, the incompressibility constraint $\\nabla\\cdot u=0$, and the stress-free or no-slip boundary conditions—whereas the weak-form SINDy implementation, applied through a large brute-force library, recovers the 2D equations only for a narrower range of Rayleigh numbers, with the diffusive terms (coefficients proportional to $1/\\sqrt{R}$) the first to be lost as turbulence strengthens. SPIDER works from a 26-term scalar and 27-term vector library built from the fields $\\{u,p,T\\}$ and differential operators, using no problem-specific knowledge beyond symmetry and the choice of variables. At $R=10^{12}$, SPIDER recovers the Euler equation but not the diffusive term; the paper shows via horizontal power spectra above the boundary layer that this failure coincides with under-resolved small scales in the DNS, and concludes the failure is a property of the data rather than of the method. In 3D convection and in the Couette flow, SPIDER recovers the governing equations and boundary conditions throughout the range studied.","pith_inferences":["If the symmetry-adapted library is the real source of SPIDER's advantage, then a pySINDy user who hand-builds a comparably small library instead of pruning a 104-term brute-force library should recover much of the gap; this is a direct test the paper does not run.","The correlation-scale heuristic should transfer to other weak-form discovery problems: compute the autocorrelation length and time of the data before choosing integration subdomains, rather than sweeping sizes; the paper demonstrates the idea only on 2D Rayleigh-Bénard convection.","SPIDER's selection of the Euler equation at $R=10^{12}$ could be repurposed as a cheap resolution diagnostic for boundary-layer-resolving simulations; the paper suggests machine-learning validation, and this is the specific failure mode to look for.","For the intended next step of finding subgrid-scale closure equations, the same machinery would need to be run on filtered fields and should first recover the known unfiltered equations as a sanity check before any new terms are trusted."],"forward_implications":["Diffusive terms are the first to disappear as the Rayleigh number rises: both methods recover the Euler (inviscid) equations before the $1/\\sqrt{R}$-scaled Laplacian terms, so anyone applying these methods at high $R$ should expect viscosity to be the hardest physics to retrieve.","SPIDER recovered equations, the incompressibility constraint, and boundary conditions in three-dimensional convection ($R=10^4$ to $10^7$) and in two-dimensional convective Couette flow ($R=10^8$), cases in which the pySINDy implementation could not be applied because of memory limits in this study.","The spatial correlation length and correlation time of the flow give a sensible default subdomain size; the optimal pySINDy subdomains and the SPIDER integration boxes both fall near those scales, so future users can start there instead of sweeping hyperparameters blindly.","Because SPIDER recovers the Euler rather than the Navier-Stokes equation exactly where the horizontal power spectra show unresolved boundary-layer scales, equation-discovery residuals can serve as an independent check on whether a DNS is adequately resolved."],"supporting_citations":[{"why":"Provides the SPIDER framework used throughout, including weak-form regression, symmetry-adapted libraries, and recovery of constraints and boundary conditions.","marker":"Gurevich et al. 2024"},{"why":"Introduces the SINDy sparse-regression algorithm that the pySINDy comparison implements.","marker":"Brunton, Proctor & Kutz 2016"},{"why":"Extends SINDy to PDE discovery and motivates the weak formulation by documenting the noise sensitivity of finite-difference derivatives.","marker":"Rudy et al. 2017"},{"why":"Supplies the weak-form SINDy formulation that pySINDy uses to integrate over subdomains and shift derivatives to a weight function.","marker":"Messenger & Bortz 2021"},{"why":"Describes the spectral solver used to generate the DNS data that both methods are tested on.","marker":"Burns et al. 2020"},{"why":"Provides the published DNS resolution benchmark for boundary-layer cells used to judge whether the highest-Rayleigh-number runs are resolved.","marker":"Zhu et al. 2018"},{"why":"Introduces the mixed-integer optimizer (MIOSR) that the pySINDy regressions in this study rely on.","marker":"Bertsimas & Gurnee 2023"},{"why":"Supplies the correlation length and time calculations used to set default spatiotemporal subdomain sizes.","marker":"Saxton et al. 2024"},{"why":"Establishes the weak-form sparse regression and integration-by-parts technique that both the pySINDy weak form and SPIDER build on.","marker":"Gurevich et al. 2019"},{"why":"Demonstrates the SPIDER framework on a physical system with constraints, informing the iterative greedy regression and library construction used here.","marker":"Golden et al. 2023"}],"fun_headline_variants":["SPIDER recovers convection equations, constraints, boundaries","Small library wins: SPIDER beats SINDy on convection equations","SINDy struggles, SPIDER recovers convection equations","SPIDER recovers full convection PDEs and boundary conditions","Symmetry-based libraries recover convection equations where brute force fails"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's central comparison assumes that the way pySINDy is configured here—a 104-term brute-force library pruned and interlinked using knowledge of the exact equations—is a fair and representative way to use that method; if that configuration is not good SINDy practice, the reported advantage of SPIDER could reflect the implementation rather than the algorithms.","fun_headline_variants_meta":{"raw":{"variants":["SPIDER recovers convection equations, constraints, boundaries","Small library wins: SPIDER beats SINDy on convection equations","SINDy struggles, SPIDER recovers convection equations","SPIDER recovers full convection PDEs and boundary conditions","Symmetry-based libraries recover convection equations where brute force fails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001032,"raw_usage":{"total_tokens":4426,"prompt_tokens":1102,"completion_tokens":3324,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":718,"completion_tokens_details":{"reasoning_tokens":3242}},"tokens_in":718,"tokens_out":3324,"duration_ms":24133,"temperature":1.0,"reasoning_tokens":3242,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:16:30.235280+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run pySINDy with a small symmetry-adapted library comparable in size to SPIDER's (about 26 scalar and 27 vector terms, without additional coefficient interlinking) on the same 2D DNS data; if it then recovers the Navier-Stokes and heat equations at $R=10^{10}$ or $R=10^{12}$, the claim that SPIDER is inherently easier and more capable would be refuted, because the difference would trace to library construction rather than to the algorithm.","supporting_citations":[{"cited_title":"and Grigoriev, R.O., Learning fluid physics from highly turbulent data using sparse physics-informed discovery of empirical relations (SPIDER)","cited_arxiv_id":null,"evidence_quote":"Provides the SPIDER framework used throughout, including weak-form regression, symmetry-adapted libraries, and recovery of constraints and boundary conditions."},{"cited_title":"and Kutz, J.N., Discovering governing equations from data by sparse identification of nonlinear dynamical systems","cited_arxiv_id":null,"evidence_quote":"Introduces the SINDy sparse-regression algorithm that the pySINDy comparison implements."},{"cited_title":"and Kutz, J.N","cited_arxiv_id":null,"evidence_quote":"Extends SINDy to PDE discovery and motivates the weak formulation by documenting the noise sensitivity of finite-difference derivatives."},{"cited_title":"and Bortz, D.M., Weak SINDy for Partial Differential Equations","cited_arxiv_id":null,"evidence_quote":"Supplies the weak-form SINDy formulation that pySINDy uses to integrate over subdomains and shift derivatives to a weight function."},{"cited_title":"and Brown, B.P., Dedalus: A Flexible Framework for Numerical Simulations with Spectral Methods","cited_arxiv_id":null,"evidence_quote":"Describes the spectral solver used to generate the DNS data that both methods are tested on."},{"cited_title":"2018, PRL, 120, 144502","cited_arxiv_id":null,"evidence_quote":"Provides the published DNS resolution benchmark for boundary-layer cells used to judge whether the highest-Rayleigh-number runs are resolved."},{"cited_title":"2023, Nonlinear Dynamics, 111, 6585–6604","cited_arxiv_id":null,"evidence_quote":"Introduces the mixed-integer optimizer (MIOSR) that the pySINDy regressions in this study rely on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the correlation length and time calculations used to set default spatiotemporal subdomain sizes."},{"cited_title":"and Grigoriev, R.O., Robust and optimal sparse regression for nonlinear PDE models","cited_arxiv_id":null,"evidence_quote":"Establishes the weak-form sparse regression and integration-by-parts technique that both the pySINDy weak form and SPIDER build on."},{"cited_title":"and Fernandez-Nieves, A., Physically informed data-driven modeling of active nematics","cited_arxiv_id":null,"evidence_quote":"Demonstrates the SPIDER framework on a physical system with constraints, informing the iterative greedy regression and library construction used here."}],"review_version":1}