{"id":"d94c5720-1b1f-4c31-b033-83e191d69794","arxiv_id":"2507.09740","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A push-forward stochastic inversion framework with sparsity priors is applied to discover governing equations and coefficient distributions from noisy, variable dynamical system data.","lead":"The paper proposes a stochastic inverse physics-discovery (SIP) framework that treats equation coefficients as random variables and learns their distribution from noisy, variable dynamical system data. It claims to recover the correct governing equations and to lower coefficient error by about 82 percent relative to SINDy and its Bayesian variant across four benchmark systems.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The candidate libraries are never specified, so the claimed recovery of correct equations may be selection among pre-known terms; the 82% RMSE comparison is not interpretable without them.","rationale":"The paper's abstract promises 'consistent identification of the correct equations' and an 'average 82%' RMSE reduction. Both components depend on the candidate library, but Section 3 never defines it. The absence of Table 1 and Figures 1–4 also makes the numerical claims unverifiable, and the reader's REJECT verdict is justified on those grounds alone. However, the most load-bearing substantive concern is the unstated library: even if the missing numbers were supplied, the central claim would remain ambiguous unless the search space is reported and shown to be non-trivial. The reader's weakest_assumption identifies this same issue. A concrete ablation varying library composition would settle whether the method discovers equations or merely selects among supplied candidates. No change to the reader's REJECT verdict is needed.","tokens_in":12714,"tokens_out":10773,"duration_ms":122824,"concrete_test":"Obtain from the authors the exact Θ(X) used in each case study (Section 3). Then rerun SIP and the SINDy/UQ-SINDy baselines on identical data with (a) the original library expanded to include ten additional high-order polynomial, trigonometric, and cross-term distractors, and (b) a library with one true term removed (e.g., uv for Lotka–Volterra). Report the selected equations and per-coefficient RMSE for each variant. If the selected structure or the RMSE improvements change materially, the discovery claim is an artifact of an unstated favorable library.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that SIP 'consistently identifies the correct equations' is a claim about structure discovery, but the search space—the candidate library Θ(X)—is never listed for any of the four case studies in Section 3. In the SINDy-style formulation of Eq. (2), the recovered equations are subsets of Θ(X); if the true terms (e.g., uv in Lotka–Volterra Eq. (13), the xz and xy terms in Lorenz, or the inertial-capillarity terms in infiltration) are not present, no sampling scheme can recover them. The paper does not state whether each library was constructed with the answer already in it or how many distractor terms were used. Consequently, the claimed structural recovery may be a selection among pre-known correct terms rather than a discovery, and the 82% coefficient-RMSE reduction is not interpretable because it depends on the unstated library composition and on the error metric over coefficients of terms included by each method. This is a load-bearing assumption, not a reporting quibble: it determines whether the empirical comparisons test the method's discovery capability or simply confirm the library.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a stochastic inverse physics-discovery (SIP) framework for identifying governing differential equations from noisy, variable, and data-limited observations. The unknown coefficients are treated as random variables with sparsity-promoting priors, and their posterior is inferred by minimizing the KL divergence between the push-forward of simulated trajectories and the empirical data distribution; a matched-block bootstrap is used when only a single trajectory is available. The method is benchmarked on simulated Lotka–Volterra, the Hudson Bay lynx–hare data, the Lorenz system, and fluid-infiltration experiments, with the reported headline result being an average 82% reduction in coefficient RMSE relative to SINDy and UQ-SINDy.","tokens_in":12975,"tokens_out":8369,"duration_ms":98827,"significance":"If the reported results are substantiated, the paper would offer a useful and timely extension of equation-discovery methods to settings with input variability and limited data, and the use of matched-block bootstrap for single-trajectory likelihood estimation is a sensible component. However, the central empirical claims are not verifiable from the submitted manuscript because the results table and all figures are missing, and several methodological details that are load-bearing for the claims remain unspecified. The potential significance is therefore real but cannot be assessed at this stage.","major_comments":[{"comment":"Table 1 is present only as a header with no content, and Figures 1–4, which are referenced throughout Sections 3.1–3.4, are absent from the manuscript. Every quantitative claim in the paper—the average 82% RMSE reduction, the per-case improvement percentages, the recovered coefficient distributions, and the posterior-predictive coverage—is tied to these missing exhibits. The abstract's headline result cannot be checked, and the comparison with SINDy and UQ-SINDy is not reproducible as submitted.","section":"Section 3, Table 1 and Figures 1–4"},{"comment":"The candidate library Theta(X) is never specified for any of the four case studies, even though structural recovery in Eq. (2) is exactly a selection of nonzero coefficients from this library. Without knowing whether the true terms (e.g., uv in Lotka–Volterra, xz and xy in Lorenz, or the inertial-capillarity terms in infiltration) were included and how many distractor terms were present, the claim that SIP 'consistently identifies the correct equations' may amount to selection among pre-specified correct terms rather than discovery. The lynx–hare figure caption lists a library {1,u,v,uv}, but the same information is not given for the other studies.","section":"Section 3 and Eq. (2)"},{"comment":"For the Hudson Bay lynx–hare data, the manuscript reports RMSE 'relative to the ground truth model coefficients' and compares the recovered distributions to 'ground truth' in Figure 2, but this is observational data with no known coefficient values. The ground truth used for these calculations is never defined. Without a clearly stated reference solution (for example, coefficients calibrated by an independent method or taken from a specific ecological study), the reported RMSE reductions of 45–80% for this case are not interpretable.","section":"Section 3.2"},{"comment":"The claimed consistency guarantee is not established in the manuscript. Theorem 1 is stated with a proof outsourced to [29], and Theorem 2 is a restatement of a result from [34] with 'Refer to [34] for details and proof.' Eq. (11) contains undefined objects, and Eq. (12) uses the notation μ_{\\hat Q(μ_Λ)}^Y and \\hat Q^{-1} without definition. The text says 'We subsequently provide the theoretical analysis,' but no derivation, precise assumptions, or connection to the finite-sample rejection sampler in Algorithm 1 is given. If a consistency guarantee is part of the paper's contribution, it needs to be stated in a self-contained and correct form.","section":"Section 2.4, Eqs. (11)–(12)"},{"comment":"The kernel-density likelihood is not well defined as written. In the expression for \\hatπ_Y(Q(λ)|Y), the quantity ([Y_i]_j - [\\hat Y]_j)^2 is a vector of length n, so the exponential is not scalar unless a norm is intended; the authors should specify the norm and the bandwidth units. In addition, the bandwidth rule h ∝ r^{-1/(n+4)} follows Scott's rule for a sample of size r in dimension n, treating the time-series length as the KDE dimension, which is an unusual and potentially degenerate choice for long trajectories. Since this likelihood is used in all posterior computations, the sensitivity of the results to this choice should be discussed.","section":"Section 2.1, likelihood expression and bandwidth"}],"minor_comments":[{"comment":"The word 'Algortihm' is misspelled in the text preceding Algorithm 1.","section":"Algorithm 1"},{"comment":"The symbol \\Q(λ) used in Eq. (6) and elsewhere is not defined; it should be the candidate solution Q(λ).","section":"Eq. (6)"},{"comment":"The text states the Hudson Bay dataset spans 20 years, while the caption for Figure 3 places year 0 at 1900 and the standard dataset is commonly cited as a much longer record; please clarify the exact time interval used.","section":"Section 3.2 and Figure 3"},{"comment":"References [10] and [12] are the same paper (Brunton et al., 2016); one should be removed or the citations consolidated.","section":"References"},{"comment":"The conclusion states that improvements 'often exceed 90%' while the abstract reports an average of 82%; please reconcile these statements and report the range of per-case improvements.","section":"Conclusion"},{"comment":"No code or data availability statement is included; providing the candidate libraries, code, and processed data would substantially improve reproducibility.","section":"Reproducibility"}],"recommendation":"major_revision","confidential_remarks":"The manuscript in its current form is not publishable because the central evidence is missing: Table 1 is empty and all figures are absent. I recommend major revision only if the authors can supply the full results table, all figures, the candidate libraries for every case study, and a clear definition of the ground truth used for the lynx–hare comparison. If these materials are not provided in a revised version, the paper should be rejected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper borrows Butler et al.'s stochastic inverse framework and applies it to sparse physics discovery with sparsity-promoting priors and matched-block bootstrap. That is a legitimate extension, and the choice of case studies — Lotka-Volterra, lynx-hare, Lorenz, infiltration — is sensible. The framing that system/input variability is distinct from measurement noise is a real gap in the SINDy literature, and modeling coefficients as random variables rather than point estimates is appropriate for that setting. The idea is worth taking seriously.\n\nWhat is genuinely new here is the combination, not the machinery: the consistency theorems are imported and cited, not derived, and the rejection sampler is standard. The paper does give credit appropriately, and the sparsity priors and bootstrap are reasonable tools. But the submission is missing Table 1 and every figure; they are referenced in the text but absent from the manuscript. That alone would normally be grounds for return to the authors.\n\nThe more substantive problem is that the candidate function libraries are never specified. The stress-test note is on the mark: without knowing whether the true terms (e.g., uv in Lotka-Volterra, xz in Lorenz, the inertial-capillarity terms in infiltration) are in the library, \"SIP consistently identifies the correct equations\" may just be selection among pre-known terms. The 82% RMSE reduction is not interpretable because the comparison depends on how many distractor terms each method had to choose from. Also, the lynx-hare case reports RMSE against a ground truth that is never defined — real data have no true coefficients unless the authors force them.\n\nThe theory is not the weak point; the missing evidence and unspecified libraries are. I would not send this to peer review in its current state. A serious editor should ask the authors to resubmit with the tables and figures included, the candidate libraries explicitly listed for each case study, and the lynx-hare ground truth defined. With those, the paper could become a useful application note for the scientific machine learning community. Without them, the central claim cannot be evaluated.","headline":"Plausible extension of stochastic inverse modeling to physics discovery, but the manuscript omits all its empirical evidence and never specifies the candidate libraries, so the headline 82% RMSE claim cannot be verified.","tokens_in":13435,"tokens_out":3846,"would_cite":false,"duration_ms":45839,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F15","34A55","62G07","37M10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Treating unknown coefficients as random variables, not fixed numbers, recovers governing equations from noisy variable data with quantified uncertainty.","keywords":["physics discovery","dynamical systems","stochastic inverse modeling","uncertainty quantification","SINDy","push-forward measure","matched-block bootstrap","governing equations"],"falsifier":"Run SIP on a synthetic dynamical system whose true equation includes a term deliberately omitted from the candidate library (e.g., a cubic $x^3$ term not offered in $\\Theta(X)$); the central claim is refuted if the recovered equations claim to match the data while missing that term, or if the 95% credible intervals fail to cover trajectories generated from the true model.","tokens_in":12548,"feed_emoji":"📈","tokens_out":10539,"duration_ms":109706,"temperature":0.7,"pith_summary":"The paper tries to establish that the failure of existing equation-discovery methods on real data is mostly a misspecification: they model measurement noise but ignore variability in the system's own parameters. The proposed stochastic inverse physics-discovery framework instead treats the unknown coefficients as random variables and learns their posterior by minimizing the Kullback-Leibler divergence between the push-forward of posterior samples and the empirical data distribution. Tested on Lotka-Volterra, the Hudson Bay lynx-hare record, the chaotic Lorenz system, and two fluid-infiltration experiments, the method reports recovering the correct equations in every case and cutting coefficient root-mean-square error by about 82% on average relative to SINDy and its Bayesian variant. If this holds, researchers working with noisy, limited, or unrepeatable measurements could obtain both the governing equation and a defensible uncertainty band around its coefficients.","feed_headline":"Random coefficients cut equation discovery error by 82%","feed_subtitle":"A stochastic inversion method recovers the true governing terms where SINDy and its Bayesian variant add spurious ones.","key_machinery":"The load-bearing object is the push-forward acceptance ratio $\\varphi(\\hat Q(\\lambda))=\\pi_Y(\\hat Q(\\lambda)|Y)/\\pi_Q(\\hat Q(\\lambda))$: the likelihood of a simulated trajectory under a kernel-density estimate of the observed data, divided by the push-forward density of prior samples through the candidate equation. Rejection sampling against this ratio converts prior coefficient draws into posterior samples whose simulated trajectories match the empirical data distribution. Around this ratio, the machinery includes sparsity-promoting priors (spike-and-slab and regularized horseshoe) to zero out inactive library terms, a matched-block bootstrap that manufactures dependent sample paths when only one trajectory exists, and a rule-of-thumb bandwidth for the kernel-density likelihood.","core_discovery":"The paper's central claim is that the data-generating process is $Y=\\int \\lambda\\Theta(X)\\,dt+\\epsilon$ with $\\epsilon\\sim N(0,\\sigma^2)$, where the coefficient matrix $\\lambda$ is not a fixed parameter but a random quantity encoding input or system variability. The stochastic inverse problem is solved by selecting a measure $\\mu_\\Lambda^*=\\arg\\min_{\\mu_\\Lambda} D_{\\mathrm{KL}}(\\hat\\mu_Y\\|\\mu_Y)$ so that simulating prior samples through the candidate equations pushes the coefficient distribution forward onto the observed data distribution. The posterior over coefficients takes the push-forward Bayes-like form $\\pi_\\Lambda(\\lambda|Y)=\\pi_\\Lambda(\\lambda)\\,\\pi_Y(\\hat Q(\\lambda)|Y)/\\pi_Q(\\hat Q(\\lambda))$, with sparsity-promoting priors shrinking irrelevant library terms to zero, matched-block bootstrap generating multiple dependent sample paths from a single trajectory, and rejection sampling drawing posterior samples. The paper reports that this consistently identifies the correct equations and recovers coefficient distributions whose 95% credible intervals track observed trajectories across all benchmarks, with an average 82% reduction in coefficient RMSE relative to SINDy and its Bayesian variant.","pith_inferences":["If system variability is real, more data will not shrink coefficient posterior uncertainty to zero; the framework implies an irreducible uncertainty floor set by input variability, a phenomenon standard Bayesian discovery cannot express.","A natural stress test the paper does not run: apply SIP with an intentionally incomplete or misspecified library and ask whether the KL-minimizing push-forward degrades gracefully (wide credible intervals, flagged missing terms) or silently substitutes spurious terms.","A testable extension separating measurement-noise variance $\\sigma^2$ from coefficient variability could let practitioners quantify what fraction of observed dispersion is intrinsic system variability versus observation error; the paper lists disentangling these as future work.","Because the acceptance ratio relies on kernel density estimates, performance may degrade sharply as the state or parameter dimension grows; comparing acceptance rates and accuracy against dimension would probe scalability."],"forward_implications":["On noisy and variable data, SIP recovers the true active physics terms where SINDy and its Bayesian variant add spurious terms or drop real ones.","The recovered coefficient posteriors give 95% credible intervals that track observed trajectories, so predictions can be reported with calibrated uncertainty.","A single observed trajectory suffices: the matched-block bootstrap creates synthetic paths that preserve temporal dependence, extending the method to historical or unrepeatable experiments.","The approach works for chaotic systems and real experimental fluids, suggesting broad applicability across biology and engineering.","Coefficient RMSE drops by about 82% on average, with per-case improvements up to roughly 99% over SINDy and 96% over the Bayesian variant, so the accuracy gain is not limited to one example."],"supporting_citations":[{"why":"Supplies the sparse-regression formulation $\\dot X=\\lambda\\Theta(X)$ and the SINDy baseline that SIP is compared against.","marker":"[10]"},{"why":"The Bayesian UQ-SINDy baseline with sparsifying priors that SIP must beat in coefficient accuracy.","marker":"[18]"},{"why":"Provides the push-forward stochastic-inversion posterior construction, Eq. (6), on which SIP is built.","marker":"[26]"},{"why":"Matched-block bootstrap method used to generate multiple dependent sample paths from a single trajectory.","marker":"[29]"},{"why":"Data-consistent inversion theory that underlies the consistency of the posterior push-forward (Theorem 2).","marker":"[34]"},{"why":"Spike-and-slab prior used to distinguish active from inactive library terms.","marker":"[27]"},{"why":"Regularized horseshoe prior used as the continuous shrinkage sparsity prior.","marker":"[28]"},{"why":"Supplies the rule-of-thumb bandwidth scaling for the kernel-density likelihood estimate.","marker":"[31]"},{"why":"Hudson Bay lynx-hare pelt time series used as the real-world predator-prey benchmark.","marker":"[35]"},{"why":"Infiltration experimental data with known physics and fluid parameters for the ether and silicone-oil cases.","marker":"[37]"}],"fun_headline_variants":["Stochastic inversion recovers equations with 82% less error","SIP framework beats SINDy by 82% in equation discovery","Random-coefficient discovery slashes error 82%","Uncertainty-aware method finds true equations, cuts error 82%","New framework reduces equation discovery error by 82%"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the candidate physics library $\\Theta(X)$ for each case study contains the exact true dynamics terms; the paper never lists these libraries in Section 3, so the method could be selecting among pre-known correct terms, and if the true term is absent, no sampling scheme can recover it.","fun_headline_variants_meta":{"raw":{"variants":["Stochastic inversion recovers equations with 82% less error","SIP framework beats SINDy by 82% in equation discovery","Random-coefficient discovery slashes error 82%","Uncertainty-aware method finds true equations, cuts error 82%","New framework reduces equation discovery error by 82%"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000415,"raw_usage":{"total_tokens":2193,"prompt_tokens":1047,"completion_tokens":1146,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":663,"completion_tokens_details":{"reasoning_tokens":1059}},"tokens_in":663,"tokens_out":1146,"duration_ms":8728,"temperature":1.0,"reasoning_tokens":1059,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:48:35.150073+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run SIP on a synthetic dynamical system whose true equation includes a term deliberately omitted from the candidate library (e.g., a cubic $x^3$ term not offered in $\\Theta(X)$); the central claim is refuted if the recovered equations claim to match the data while missing that term, or if the 95% credible intervals fail to cover trajectories generated from the true model.","supporting_citations":[{"cited_title":"Sparsifying priors for bayesian uncertainty quantification in model discovery,","cited_arxiv_id":null,"evidence_quote":"The Bayesian UQ-SINDy baseline with sparsifying priors that SIP must beat in coefficient accuracy."},{"cited_title":"Combining push-forward measures and bayes’ rule to construct consistent solutions to stochastic inverse problems,","cited_arxiv_id":null,"evidence_quote":"Provides the push-forward stochastic-inversion posterior construction, Eq. (6), on which SIP is built."},{"cited_title":"Matched-block bootstrap for dependent data,","cited_arxiv_id":null,"evidence_quote":"Matched-block bootstrap method used to generate multiple dependent sample paths from a single trajectory."},{"cited_title":"Data-consistent inversion for stochastic input-to- output maps,","cited_arxiv_id":null,"evidence_quote":"Data-consistent inversion theory that underlies the consistency of the posterior push-forward (Theorem 2)."},{"cited_title":"Bayesian variable selection in linear regression,","cited_arxiv_id":null,"evidence_quote":"Spike-and-slab prior used to distinguish active from inactive library terms."},{"cited_title":"Handling sparsity via the horseshoe,","cited_arxiv_id":null,"evidence_quote":"Regularized horseshoe prior used as the continuous shrinkage sparsity prior."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Hudson Bay lynx-hare pelt time series used as the real-world predator-prey benchmark."},{"cited_title":"Inertial capillarity,","cited_arxiv_id":null,"evidence_quote":"Infiltration experimental data with known physics and fluid parameters for the ether and silicone-oil cases."}],"review_version":1}