arxiv: 2604.04829 · v1 · submitted 2026-04-06 · 📊 stat.ME · cs.LG· stat.ML

Recognition: no theorem link

A Robust SINDy Autoencoder for Noisy Dynamical System Identification

Kairui Ding

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:51 UTC · model grok-4.3

classification 📊 stat.ME cs.LGstat.ML

keywords SINDy autoencodernoise separationdynamical system identificationsparse regressionlatent coordinatesLorenz systemmeasurement noise

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The pith

Adding a noise-separation module to SINDy autoencoders enables recovery of latent dynamics and noise estimates from noisy observations.

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

The paper proposes an extension to SINDy autoencoders that adds a dedicated noise-separation module to handle measurement errors in data from unknown dynamical systems. Standard SINDy relies on clean data and sparse representation in the observed coordinates, while autoencoder versions learn simpler latent coordinates; noise still disrupts both steps. The new module aims to disentangle signal from noise early, so that sparse regression can still identify parsimonious governing equations in the latent space. A reader would care because most real observations of physical or engineering systems contain noise that currently limits equation discovery methods.

Core claim

By incorporating a noise-separation module into the SINDy autoencoder architecture, the method improves robustness to measurement error. This enables the recovery of interpretable latent dynamics and accurate estimation of the measurement noise from noisy observations, as demonstrated by numerical experiments on the Lorenz system.

What carries the argument

The noise-separation module placed before the autoencoder, which separates the underlying signal from additive measurement noise so that latent coordinate learning and subsequent sparse regression operate on cleaner data.

If this is right

Interpretable latent dynamics can be recovered even when observations contain substantial measurement noise.
The measurement noise level itself can be estimated accurately as part of the identification process.
Sparse governing equations become identifiable in the learned latent coordinates despite noisy inputs.
The overall framework extends sparse identification techniques to a wider range of practical, noise-corrupted datasets.

Where Pith is reading between the lines

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

The same noise-separation idea might transfer to other latent-variable models used for equation discovery in fluid dynamics or biological networks.
Accurate per-sample noise estimates could support downstream tasks such as denoising trajectories before long-term prediction.
Applying the method to systems with structured noise, such as sensor-specific biases, would test whether the module generalizes beyond the white-noise case used in the Lorenz tests.

Load-bearing premise

That inserting a noise-separation module will cleanly disentangle signal from noise without distorting the learned latent coordinates or the sparse regression that follows.

What would settle it

If experiments on noisy Lorenz observations produce latent dynamics that do not match the known equations or yield noise estimates far from the true noise level, the robustness claim would be falsified.

read the original abstract

Sparse identification of nonlinear dynamics (SINDy) has been widely used to discover the governing equations of a dynamical system from data. It uses sparse regression techniques to identify parsimonious models of unknown systems from a library of candidate functions. Therefore, it relies on the assumption that the dynamics are sparsely represented in the coordinate system used. To address this limitation, one seeks a coordinate transformation that provides reduced coordinates capable of reconstructing the original system. Recently, SINDy autoencoders have extended this idea by combining sparse model discovery with autoencoder architectures to learn simplified latent coordinates together with parsimonious governing equations. A central challenge in this framework is robustness to measurement error. Inspired by noise-separating neural network structures, we incorporate a noise-separation module into the SINDy autoencoder architecture, thereby improving robustness and enabling more reliable identification of noisy dynamical systems. Numerical experiments on the Lorenz system show that the proposed method recovers interpretable latent dynamics and accurately estimates the measurement noise from noisy observations.

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

Adds a noise-separation module to SINDy autoencoders and gets supported results on noisy Lorenz data with quantitative metrics in the full text.

read the letter

The main takeaway is that this paper adds a noise-separation module to the SINDy autoencoder framework and demonstrates it recovers latent dynamics plus noise estimates on the Lorenz system. The full text supplies the architecture details, combined loss, training procedure, and metrics on trajectory error, coefficient error, and noise variance recovery, which line up without contradictions per the stress-test note. This makes the central claim hold up on the reported benchmark. What is new is the specific integration of that separation module; everything else reuses standard autoencoder and sparse regression components from prior SINDy work. The paper does this cleanly and focuses on the robustness issue without overclaiming. The soft spots are limited in scope. Evaluation stays on one chaotic system, so broader testing on other dynamics or noise types would help, but that is a normal next step rather than a load-bearing flaw. Hyperparameters and loss weights are the usual free parameters and are not disguised. No circularity or unstated assumptions appear to undermine the Lorenz results. This work is for people already using SINDy-style methods on noisy measurements who want a practical architectural tweak. A reader in applied dynamical systems identification would get usable ideas from the setup and numbers. It deserves peer review because the idea is clear, the experiments are consistent and falsifiable on the benchmark, and the contribution is honest incremental engineering rather than hype.

Referee Report

2 major / 3 minor

Summary. The paper introduces a SINDy autoencoder augmented with a noise-separation module to improve robustness to measurement noise in discovering governing equations of dynamical systems. The architecture learns latent coordinates via an autoencoder while performing sparse regression on the latent dynamics and explicitly estimating additive noise; the combined loss balances reconstruction, sparsity, and noise separation. Numerical experiments on the Lorenz system are reported to recover interpretable latent equations and accurately estimate noise variance from noisy observations.

Significance. If the numerical results hold under broader testing, the work strengthens the SINDy-autoencoder line by directly addressing measurement error, a common obstacle in real data. The explicit noise module and reported metrics (trajectory error, coefficient error, noise variance recovery) on a standard benchmark provide a concrete advance over prior SINDy-AE variants that lacked dedicated noise handling.

major comments (2)

§3.2, combined loss (Eq. 8): the weighting between the noise-separation term and the SINDy sparsity term is treated as a tunable hyperparameter; the manuscript should demonstrate that the reported Lorenz recovery is stable across a reasonable range of these weights rather than a single tuned value, as this directly affects the claim of reliable disentanglement.
§4.2, Table 2 (noise variance recovery): the reported error bars are obtained from 10 random seeds, but the table does not include a baseline comparison against a standard SINDy-AE without the noise module on the same noisy trajectories; adding this would strengthen the central claim that the module is responsible for the improvement.

minor comments (3)

Abstract: the summary sentence on numerical experiments should include at least one quantitative result (e.g., coefficient error or noise variance recovery percentage) to give readers an immediate sense of performance.
§2.1, notation for the library matrix: the symbol Θ is reused for both the full-state and latent libraries; a subscript (e.g., Θ_z) would eliminate ambiguity when the two appear in the same equation.
Figure 3 caption: the color scale for the noise estimate plot is not stated; adding the range or normalization used would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive review and recommendation for minor revision. We address each major comment point by point below, agreeing where changes are warranted and providing clarifications.

read point-by-point responses

Referee: §3.2, combined loss (Eq. 8): the weighting between the noise-separation term and the SINDy sparsity term is treated as a tunable hyperparameter; the manuscript should demonstrate that the reported Lorenz recovery is stable across a reasonable range of these weights rather than a single tuned value, as this directly affects the claim of reliable disentanglement.

Authors: We agree that explicit sensitivity analysis with respect to the loss weights is important to substantiate the reliability of the noise disentanglement. In the revised manuscript we will add results (in a new figure or table in §4 or an appendix) showing trajectory error, coefficient error, and noise-variance recovery for a range of weight ratios around the reported values. These additional experiments confirm that the Lorenz recovery remains stable and interpretable, thereby strengthening the claim. revision: yes
Referee: §4.2, Table 2 (noise variance recovery): the reported error bars are obtained from 10 random seeds, but the table does not include a baseline comparison against a standard SINDy-AE without the noise module on the same noisy trajectories; adding this would strengthen the central claim that the module is responsible for the improvement.

Authors: We thank the referee for this suggestion. A direct baseline comparison on identical noisy data is indeed the clearest way to isolate the benefit of the noise-separation module. We will expand Table 2 in the revised manuscript to include the corresponding metrics (with error bars from the same 10 seeds) for the original SINDy-AE architecture without the noise module, thereby directly supporting the central claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces an explicit architectural extension to the SINDy autoencoder by adding a noise-separation module, with claims resting on numerical validation for the Lorenz system using trajectory error, coefficient error, and noise variance metrics. No derivation chain exists that reduces predictions or results to inputs by construction, self-definition, or self-citation load-bearing steps. The method is presented as a new combination of existing ideas (SINDy regression plus autoencoders plus noise separation), and outcomes are not forced to match any fitted quantity; they are measured against ground-truth dynamics and noise levels in experiments. This is a standard empirical methods paper with independent content in the proposed architecture and benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The approach inherits the core SINDy sparsity assumption in latent space and standard neural-network training assumptions; the noise-separation module is the only new postulated component.

free parameters (1)

neural-network hyperparameters and loss weights
Typical for any autoencoder architecture; values are chosen to make training converge on the given data.

axioms (1)

domain assumption The underlying dynamics admit a sparse representation in some latent coordinate system.
This is the foundational premise of all SINDy methods and is carried over unchanged.

invented entities (1)

noise-separation module no independent evidence
purpose: To isolate measurement noise from the signal inside the autoencoder pipeline.
New architectural block introduced by the paper; no independent evidence outside the reported Lorenz experiment is supplied.

pith-pipeline@v0.9.0 · 5468 in / 1201 out tokens · 53134 ms · 2026-05-10T19:51:37.310786+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

Schmidt and H

M. Schmidt and H. Lipson. Distilling free-form natural laws from experimental data.Sci- ence, 324(5923):81–85, 2009

work page 2009
[2]

Daniels and I

B. Daniels and I. Nemenman. Automated adaptive inference of phenomenological dynam- ical models.Nature Communications, 6:8133, 2015

work page 2015
[3]

S. L. Brunton, J. L. Proctor, and J. N. Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems.Proceedings of the National Academy of Sciences, 113(15):3932–3937, 2016

work page 2016
[4]

N. M. Mangan, J. N. Kutz, S. L. Brunton, and J. L. Proctor. Model selection for dynamical systems via sparse regression and information criteria.Proceedings of the Royal Society A, 473(2204):20170009, 2017

work page 2017
[5]

Kaiser, J

E. Kaiser, J. N. Kutz, and S. L. Brunton. Sparse identification of nonlinear dynamics for model predictive control in the low-data limit.Proceedings of the Royal Society A, 474(2219):20180335, 2018

work page 2018
[6]

Fasel, J

U. Fasel, J. N. Kutz, B. W. Brunton, and S. L. Brunton. Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit.Proceedings of the Royal Society A, 478(2260):20210904, 2022

work page 2022
[7]

J. S. North, C. K. Wikle, and E. M. Schliep. A review of data-driven discovery for dynamic systems.International Statistical Review, 91(3):464–492, 2023

work page 2023
[8]

Champion, B

K. Champion, B. Lusch, J. N. Kutz, and S. L. Brunton. Data-driven discovery of coordinates and governing equations.Proceedings of the National Academy of Sciences, 116(45):22445– 22451, 2019

work page 2019
[9]

Fukami, R

K. Fukami, R. Taira, and K. Fukagata. Sparse identification of nonlinear dynamics with low-dimensionalized flow representations.Journal of Fluid Mechanics, 926:A10, 2021

work page 2021
[10]

F. J. Gonzalez and M. Balajewicz. Learning low-dimensional feature dynamics using deep convolutional recurrent autoencoders.arXiv preprint arXiv:1808.01346, 2018

work page arXiv 2018
[11]

K. T. Carlberg, A. Jameson, M. J. Kochenderfer, J. Morton, L. Peng, and F. D. Witherden. Recovering missing CFD data for high-order discretizations using deep neural networks and dynamics learning.arXiv preprint arXiv:1812.01177, 2018

work page arXiv 2018
[12]

A. O’Brien. Sparse identification of nonlinear dynamics in the presence of library and system uncertainty .arXiv preprint arXiv:2401.13099, 2024

work page arXiv 2024
[13]

K. Egan, W. Li, and R. Carvalho. Automatically discovering ordinary differential equations from data with sparse regression.Communications Physics, 7:20, 2024

work page 2024
[14]

S. H. Rudy , J. N. Kutz, and S. L. Brunton. Deep learning of dynamics and signal-noise decomposition with time-stepping constraints.Journal of Computational Physics, 396:483– 506, 2019

work page 2019
[15]

Carr.Applications of Centre Manifold Theory

J. Carr.Applications of Centre Manifold Theory. Springer-Verlag, New York, 1981

work page 1981
[16]

Y. A. Kuznetsov.Elements of Applied Bifurcation Theory. 3rd ed., Springer, New York, 2004

work page 2004
[17]

Goodfellow, Y

I. Goodfellow, Y. Bengio, and A. Courville.Deep Learning. MIT Press, Cambridge, MA, 2016. 27

work page 2016