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REVIEW 5 major objections 6 minor 37 references

Discovering Governing Equations in the Presence of Uncertainty

T0 review · 5 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Treating unknown coefficients as random variables, not fixed numbers, recovers governing equations from noisy variable data with quantified uncertainty.

desk verdict 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. read the letter →

arxiv 2507.09740 v1 pith:TCBGXO5P submitted 2025-07-13 stat.ML cs.LGmath.DSphysics.data-an

Ridwan Olabiyi , Han Hu , Ashif Iquebal This is my paper · ORCID
classification stat.MLcs.LGmath.DSphysics.data-an MSC 62F1534A5562G0737M10
keywords physicsdiscoverydynamicalsystemsstochasticinversemodelinguncertaintyquantificationSINDypush-forwardmeasurematched-blockbootstrapgoverningequations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 6 minor

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.

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 (5)
  1. [Section 3, Table 1 and Figures 1–4] 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.
  2. [Section 3 and Eq. (2)] 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.
  3. [Section 3.2] 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.
  4. [Section 2.4, Eqs. (11)–(12)] 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.
  5. [Section 2.1, likelihood expression and bandwidth] 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.
minor comments (6)
  1. [Algorithm 1] The word 'Algortihm' is misspelled in the text preceding Algorithm 1.
  2. [Eq. (6)] The symbol \Q(λ) used in Eq. (6) and elsewhere is not defined; it should be the candidate solution Q(λ).
  3. [Section 3.2 and Figure 3] 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.
  4. [References] References [10] and [12] are the same paper (Brunton et al., 2016); one should be removed or the citations consolidated.
  5. [Conclusion] 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.
  6. [Reproducibility] No code or data availability statement is included; providing the candidate libraries, code, and processed data would substantially improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found; the methodological core is externally cited and the empirical benchmarks are independent.

full rationale

The paper's inference machinery is explicitly adopted from external prior work: the posterior formula (Eq. 6) is attributed to Butler et al. [26], and the consistency Theorem 2 is credited to [34] with proof outsourced. Neither citation is self-referential, and the paper does not claim to derive these results from scratch; it uses them as building blocks. The empirical comparisons to SINDy and UQ-SINDy on simulated, historical, and experimental data are external benchmarks, not fitted redefinitions. The abstract's '82% RMSE reduction' is a comparison against standard baselines, not a quantity that is equivalent to an input by construction. The absence of explicit candidate libraries is a reporting limitation that affects interpretability but does not constitute circularity: the library is a standard input to SINDy-type methods, and the paper does not define its recovered equations in terms of the library itself. The two self-citations ([24], [36]) are peripheral data/context references and are not load-bearing for the central argument. No step reduces, by definition or by self-citation, to its own inputs.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The inference apparatus is largely imported from prior stochastic inversion work (refs [26], [34]), and the empirical setup depends on unspecified hyperparameters and candidate libraries. The only real addition is the combination of sparsity priors and matched block bootstrap with physics discovery.

free parameters (4)
  • KDE bandwidth h = Scott's rule or manually set
    Used in likelihood and push-forward density estimation (Section 2.1); not justified per case and can alter the posterior.
  • Matched block bootstrap block length l
    Never specified in Section 2.2, yet determines the resampled trajectories and hence the likelihood.
  • Number of prior samples N
    Sampling budget in Algorithm 1 is not given; it affects the approximation quality of the posterior.
  • Prior hyperparameters
    Spike-and-slab parameters (p, c) and horseshoe parameters (nu, s, tau0) are not specified, though they control sparsity.
assumptions (4)
  • domain assumption Measurement noise is additive, zero-mean Gaussian (Eq. 4)
    Stated in Section 2 without justification; standard but restrictive.
  • standard math The stochastic inverse posterior formula (Eq. 6) is valid
    Adopted from Butler et al. [26]; no derivation in this paper.
  • standard math The matched-block bootstrap is consistent (Theorem 1)
    Proof referenced to Carlstein et al. [29]; required for the single-trajectory cases.
  • ad hoc to paper The candidate library Theta(X) contains the true governing terms
    Central to all case studies but never explicitly stated; without this, the method cannot recover the true equations.

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Cite this review

Pith. "Pith review of Discovering Governing Equations in the Presence of Uncertainty." pith.science (2026). https://pith.science/paper/TCBGXO5P

@misc{pith2026250709740,
  author       = {Pith},
  title        = {Pith review of: Discovering Governing Equations in the Presence of Uncertainty},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TCBGXO5P}},
  note         = {Machine review of arXiv:2507.09740}
}
read the original abstract

In the study of complex dynamical systems, understanding and accurately modeling the underlying physical processes is crucial for predicting system behavior and designing effective interventions. Yet real-world systems exhibit pronounced input (or system) variability and are observed through noisy, limited data conditions that confound traditional discovery methods that assume fixed-coefficient deterministic models. In this work, we theorize that accounting for system variability together with measurement noise is the key to consistently discover the governing equations underlying dynamical systems. As such, we introduce a stochastic inverse physics-discovery (SIP) framework that treats the unknown coefficients as random variables and infers their posterior distribution by minimizing the Kullback-Leibler divergence between the push-forward of the posterior samples and the empirical data distribution. Benchmarks on four canonical problems -- the Lotka-Volterra predator-prey system (multi- and single-trajectory), the historical Hudson Bay lynx-hare data, the chaotic Lorenz attractor, and fluid infiltration in porous media using low- and high-viscosity liquids -- show that SIP consistently identifies the correct equations and lowers coefficient root-mean-square error by an average of 82\% relative to the Sparse Identification of Nonlinear Dynamics (SINDy) approach and its Bayesian variant. The resulting posterior distributions yield 95\% credible intervals that closely track the observed trajectories, providing interpretable models with quantified uncertainty. SIP thus provides a robust, data-efficient approach for consistent physics discovery in noisy, variable, and data-limited settings.

Figures

Figures reproduced from arXiv: 2507.09740 by the authors.

Figure 1
Figure 1. Comparing the true and recovered distributions of (a) [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Posterior marginal distributions of the Lotka–Volterra coefficients inferred from [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Posterior predictive hare–lynx trajectories for the Hudson Bay data using (a) SIP [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Comparisons of predicted dynamics for (a) low viscous (b) high viscous infiltration [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

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Reference graph

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