arxiv: 2604.10854 · v1 · submitted 2026-04-12 · 📊 stat.AP · stat.ML

Recognition: unknown

Uncertainty-Aware Sparse Identification of Dynamical Systems via Bayesian Model Averaging

Shuhei Kashiwamura , Yusuke Kato , Hiroshi Kori , Masato Okada

Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:56 UTC · model grok-4.3

classification 📊 stat.AP stat.ML

keywords Bayesian model averagingsparse identificationdynamical systemsuncertainty quantificationoscillator networksmodel selectiondata-driven modelingposterior inclusion probabilities

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The pith

Bayesian model averaging yields posterior probabilities for each candidate interaction when identifying sparse dynamical systems from data.

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

The paper proposes a framework that merges sparse regression with Bayesian model averaging to infer both the structure and the functional form of interactions in coupled dynamical systems while assigning a credibility score to every term. This matters because limited data or weak identifiability often allows several different models to explain observations equally well, so point estimates alone can be misleading. Experiments on networks of oscillators show that the method recovers known sparse couplings, including higher-order harmonics, phase lags, and multi-body terms, together with quantified uncertainty. The same procedure still identifies useful effective components even when the true equations lie outside the assumed library of basis functions.

Core claim

The authors combine sparse modeling with Bayesian model averaging to produce posterior inclusion probabilities that quantify the credibility of each candidate interaction and basis component; numerical tests on oscillator networks confirm that this recovers sparse interaction structures with uncertainty estimates for higher-order, phase-lag, and multi-body effects, and continues to locate effective terms even when the true governing equations are absent from the model class.

What carries the argument

Bayesian model averaging over sparse regression models for dynamical systems, which supplies posterior inclusion probabilities for every candidate interaction and basis function.

If this is right

Each candidate interaction receives an explicit probability that it belongs in the model, allowing direct comparison of competing explanations.
Higher-order harmonics, phase lags, and multi-body terms can be retained or discarded according to their posterior support rather than ad-hoc thresholds.
When the exact equations are unknown, the procedure still returns an effective functional form accompanied by uncertainty measures.
Model selection becomes probabilistic, so downstream tasks such as prediction or control can incorporate the full posterior over structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The posterior probabilities could serve as a prior for sequential model updates when new observations arrive.
The same averaging machinery might be applied to other data-driven settings such as reaction networks or gene regulatory models where interaction libraries are large.
Uncertainty over model structure could be propagated into forecasts to produce calibrated prediction intervals rather than point predictions alone.

Load-bearing premise

Success on simulated oscillator networks is taken as evidence that the method accurately recovers structures and supplies useful uncertainty estimates in general settings with limited or poorly identifiable data.

What would settle it

A controlled simulation in which the true sparse interaction structure is known but the method assigns high posterior probability to incorrect terms or fails to recover the known couplings.

Figures

Figures reproduced from arXiv: 2604.10854 by Hiroshi Kori, Masato Okada, Shuhei Kashiwamura, Yusuke Kato.

**Figure 2.** Figure 2: FIG. 2: Synthetic phase time-series data [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Posterior inclusion probabilities of two-body interaction indicators [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Posterior inclusion probabilities of the asymmetric three-body interaction indicators [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Posterior inclusion probabilities of the symmetric three-body indicators [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: Posterior inclusion probabilities of the indicator vector [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: Histograms of the marginal posterior distribution [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8: Histograms of the marginal posterior distribution [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9: Dependence of the Hamming distance [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10: Dependence of the MSE [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11: Dependence of the Hamming distance [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12: Dependence of the MSE [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13: (a) Dependence of the Hamming distance [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗

**Figure 14.** Figure 14: FIG. 14: Phase difference dynamics extracted from simulations of two coupled metronomes. The trajectory shows [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗

**Figure 15.** Figure 15: FIG. 15: Posterior inclusion probabilities of the indicator vector for the sine and cosine components. Each panel [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗

**Figure 16.** Figure 16: FIG. 16: Plot of the inferred phase coupling term [PITH_FULL_IMAGE:figures/full_fig_p016_16.png] view at source ↗

read the original abstract

In many problems of data-driven modeling for dynamical systems, the governing equations are not known a priori and must be selected phenomenologically from a large set of candidate interactions and basis functions. In such situations, point estimates alone can be misleading, because multiple model components may explain the observed data comparably well, especially when the data are limited or the dynamics exhibit poor identifiability. Quantifying the uncertainty associated with model selection is therefore essential for constructing reliable dynamical models from data. In this work, we develop a Bayesian sparse identification framework for dynamical systems with coupled components, aimed at inferring both interaction structure and functional form together with principled uncertainty quantification. The proposed method combines sparse modeling with Bayesian model averaging, yielding posterior inclusion probabilities that quantify the credibility of each candidate interaction and basis component. Through numerical experiments on oscillator networks, we show that the framework accurately recovers sparse interaction structures with quantified uncertainty, including higher-order harmonic components, phase-lag effects, and multi-body interactions. We also demonstrate that, even in a phenomenological setting where the true governing equations are not contained in the assumed model class, the method can identify effective functional components with quantified uncertainty. These results highlight the importance of Bayesian uncertainty quantification in data-driven discovery of dynamical models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adds Bayesian model averaging to sparse identification so you get inclusion probabilities on terms for coupled dynamics, but the evidence is only simulated oscillator runs without metrics or calibration.

read the letter

The new piece is the integration of sparse regression for dynamical systems with Bayesian averaging over models. This produces posterior inclusion probabilities for each candidate interaction and basis function, which directly addresses the problem of multiple plausible explanations when data are limited or identifiability is weak. The authors apply it to oscillator networks and show it can surface higher-order harmonics, phase lags, and multi-body terms while also identifying effective components even when the true equations sit outside the library. That last point is practically useful for phenomenological modeling. The approach is straightforward and the math follows standard Bayesian model averaging once the sparse library is set up. What the paper does well is keep the focus on joint inference of structure and functional form rather than treating them separately. The simulations illustrate the idea on networks with some complexity. The soft spots are exactly where the stress-test note points. All claims about accurate recovery and useful uncertainty rest on numerical experiments with simulated oscillators. There are no reported recovery rates, no direct comparisons to plain SINDy or other baselines, no posterior predictive checks, and no coverage diagnostics on held-out trajectories. It is not shown how the method behaves under genuinely scarce data or when identifiability is poor, which are the regimes where the uncertainty quantification would matter most. No real data examples appear either. This work is aimed at people who build data-driven models of coupled physical or biological systems and want a principled way to report which terms are supported. It deserves peer review because the framework is coherent and the target problem is real, but the authors will need to supply quantitative validation and calibration results before the uncertainty claims can be taken as reliable.

Referee Report

3 major / 2 minor

Summary. The paper develops a Bayesian sparse identification framework for dynamical systems that integrates sparse regression with Bayesian model averaging. This yields posterior inclusion probabilities for candidate interaction terms and basis functions, providing uncertainty quantification on both structure and functional form. Numerical experiments on oscillator networks are used to demonstrate recovery of sparse structures (including higher-order harmonics, phase lags, and multi-body interactions) and identification of effective components even under model mismatch.

Significance. If the validation concerns are addressed, the approach could meaningfully extend point-estimate methods such as SINDy by supplying calibrated uncertainty on model selection in poorly identifiable regimes. This is relevant for data-driven modeling in physics and engineering where multiple explanations fit limited data equally well. The numerical demonstrations on coupled oscillators provide a concrete test bed, but the work's broader utility hinges on showing that the uncertainty measures are reliable and actionable beyond the simulated setting.

major comments (3)

[Numerical Experiments] Numerical Experiments section: the claim of 'accurate recovery' of sparse interaction structures is not supported by quantitative metrics (e.g., precision-recall for term inclusion, RMSE on held-out trajectories, or comparison against standard SINDy and other Bayesian baselines). Without these, it is impossible to judge whether the Bayesian averaging improves upon point estimates.
[Results / Numerical Experiments] Results on oscillator networks (and the mismatch experiment): no posterior predictive checks, calibration plots, or coverage diagnostics are reported for the posterior inclusion probabilities or credible intervals. This leaves open whether the quantified uncertainty reflects true epistemic uncertainty or is an artifact of library size and prior specification, which is central to the UQ contribution.
[Model Mismatch Experiment] Model-mismatch demonstration: while effective functional components are identified, the manuscript does not show that these components yield better predictive performance or interpretability than the corresponding point-estimate SINDy solution on the same data. This comparison is required to substantiate the advantage of the Bayesian procedure.

minor comments (2)

[Methods] Notation for the candidate library and the spike-and-slab or horseshoe prior should be defined explicitly in the Methods section with a clear mapping to the posterior inclusion probabilities.
[Figures] Figures displaying posterior inclusion probabilities would benefit from showing results across multiple independent runs or data realizations to illustrate variability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We agree that additional quantitative metrics, validation diagnostics, and direct comparisons will strengthen the manuscript's claims regarding recovery accuracy and the reliability of the uncertainty quantification. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Numerical Experiments] Numerical Experiments section: the claim of 'accurate recovery' of sparse interaction structures is not supported by quantitative metrics (e.g., precision-recall for term inclusion, RMSE on held-out trajectories, or comparison against standard SINDy and other Bayesian baselines). Without these, it is impossible to judge whether the Bayesian averaging improves upon point estimates.

Authors: We acknowledge that the current Numerical Experiments section relies primarily on visual inspection of recovered structures rather than quantitative metrics. In the revision we will add precision-recall statistics for term inclusion, RMSE on held-out trajectories, and explicit comparisons against standard SINDy as well as other Bayesian sparse regression baselines. These additions will provide a clearer, quantitative demonstration of the benefits of Bayesian model averaging over point estimates. revision: yes
Referee: [Results / Numerical Experiments] Results on oscillator networks (and the mismatch experiment): no posterior predictive checks, calibration plots, or coverage diagnostics are reported for the posterior inclusion probabilities or credible intervals. This leaves open whether the quantified uncertainty reflects true epistemic uncertainty or is an artifact of library size and prior specification, which is central to the UQ contribution.

Authors: We agree that posterior predictive checks and calibration diagnostics are necessary to substantiate the uncertainty quantification. We will incorporate posterior predictive checks on held-out trajectories, calibration plots for the inclusion probabilities, and coverage diagnostics for the credible intervals in the revised Results section. These diagnostics will help confirm that the reported uncertainties are well-calibrated and not artifacts of library size or prior choice. revision: yes
Referee: [Model Mismatch Experiment] Model-mismatch demonstration: while effective functional components are identified, the manuscript does not show that these components yield better predictive performance or interpretability than the corresponding point-estimate SINDy solution on the same data. This comparison is required to substantiate the advantage of the Bayesian procedure.

Authors: We recognize that a direct comparison of predictive performance between the Bayesian-averaged model and the point-estimate SINDy solution is needed in the model-mismatch experiment. In the revision we will report predictive RMSE (and, where appropriate, interpretability metrics) on test trajectories for both approaches. This will provide concrete evidence of any advantages offered by the Bayesian procedure. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the Bayesian sparse identification framework

full rationale

The paper proposes a framework that integrates sparse regression with Bayesian model averaging to infer interaction structures and quantify uncertainty via posterior inclusion probabilities. This construction follows directly from standard Bayesian principles applied to a library of candidate terms and does not reduce to any self-defined inputs or fitted parameters renamed as predictions. The central claims are supported by numerical experiments on simulated oscillator networks, which provide independent validation rather than tautological confirmation. No load-bearing self-citations, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation are evident. The derivation chain is self-contained and relies on externally established Bayesian and sparse modeling techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard Bayesian inference and sparse regression assumptions whose concrete implementation details (priors, basis libraries, sampling methods) are not specified in the abstract.

axioms (2)

domain assumption Observed time-series data can be explained by a linear combination of candidate basis functions with additive noise.
Implicit foundation of sparse identification methods referenced in the abstract.
standard math Bayesian model averaging yields meaningful posterior inclusion probabilities for model components.
Core statistical assumption underlying the uncertainty quantification.

pith-pipeline@v0.9.0 · 5524 in / 1220 out tokens · 57919 ms · 2026-05-10T14:56:40.675254+00:00 · methodology

discussion (0)

Reference graph

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+ 3L(3)(N−1)(N−2). b. Indicators and vectorization.For each interac- tion class introduced in Sec. III, we assign a binary struc- tural indicator to every basis function. Specifically, (i) c(2) ij denotes whether the pairwise interactionB (2) ij is ac- tive, (ii)c (3,a) ijk denotes whether the asymmetric three– body basisB (3,a) ijk is active, and (iii)c ...

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Robustness analysis To evaluate the robustness of our method with respect to the number of data points and the magnitude of dy- namical noise, we repeated data generation and estima- tion for various settings and computed the mean and standard error of the metricsE c andE Θ. The results are summarized in Figs. 9a–11b. a. Asynchronous regime (Configuration...
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