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arxiv: 2604.20141 · v1 · submitted 2026-04-22 · 💻 cs.LG · math.DS

Fourier Weak SINDy: Spectral Test Function Selection for Robust Model Identification

Zhiheng Chen , Urban Fasel , Anastasia Bizyaeva This is my paper

Pith reviewed 2026-05-10 00:44 UTC · model grok-4.3

classification 💻 cs.LG math.DS
keywords weak-form SINDyFourier test functionsspectral estimationnoise-robust equation discoverysparse regressionchaotic ODEsderivative-free learning
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The pith

Fourier Weak SINDy selects sinusoidal test functions from spectral data to learn equations without derivatives even from noisy measurements.

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

The paper introduces Fourier Weak SINDy to make sparse equation discovery more robust to noise by choosing test functions automatically from the data's frequency content. It builds on weak-form methods that avoid differentiating noisy data by integrating against test functions, but replaces hand-chosen or random test functions with orthogonal sinusoids whose frequencies come from multitaper spectral estimation. This reduces the problem to regressing Fourier coefficients while preserving sparsity. Readers would care because many real datasets from physics or biology are noisy and chaotic, where standard SINDy and its variants struggle to identify the correct model. The unification of spectral analysis and weak-form learning offers a compact framework for such tasks.

Core claim

Fourier Weak SINDy combines weak-form sparse equation learning with spectral density estimation to select data-driven test functions. By employing orthogonal sinusoidal test functions, the weak-form sparse regression reduces to a regression over Fourier coefficients. Dominant frequencies are identified through multitaper estimation of the data's frequency spectrum, enabling robust and interpretable derivative-free model identification illustrated on chaotic and hyperchaotic ODEs.

What carries the argument

The central mechanism is the data-driven selection of orthogonal sinusoidal test functions via multitaper spectral estimation, which converts the weak-form regression into a Fourier coefficient problem for sparse recovery of the governing equations.

If this is right

  • Recovers sparse representations of chaotic dynamics without needing to compute numerical derivatives from noisy data.
  • Integrates frequency spectrum analysis directly into the equation learning process for automatic test function choice.
  • Maintains interpretability through sparsity while improving noise tolerance across multiple benchmark systems.
  • Provides a unified framework that links weak-form methods with classical spectral estimation techniques.

Where Pith is reading between the lines

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

  • This approach may generalize to systems where the dominant frequencies vary over time if the multitaper estimation is applied in sliding windows.
  • Real-world applications could benefit in fields like fluid dynamics or neuroscience where measurements are inherently noisy and the underlying equations are unknown.
  • Further work might explore whether other orthogonal bases, such as wavelets, could be selected similarly for non-periodic or localized dynamics.

Load-bearing premise

The measurements contain clear dominant frequencies that can be reliably extracted by multitaper spectral estimation to form test functions capable of recovering the sparse dynamics.

What would settle it

Running the algorithm on a known chaotic oscillator with moderate additive noise and checking whether the recovered equation matches the true sparse form would falsify the claim if the correct terms are not identified.

Figures

Figures reproduced from arXiv: 2604.20141 by Anastasia Bizyaeva, Urban Fasel, Zhiheng Chen.

Figure 1
Figure 1. Figure 1: Graphical summary of proposed Fourier Weak SINDy method for sparse equation learn￾ing described in detail in Section 3. First, multitaper spectral density estimation is used to identify dominant temporal frequencies in the observed noisy signals. Second, spec￾tral sparse regression problem is solved to learn model coefficients in a selected function dictionary, leveraging the identified dominant Fourier fr… view at source ↗
Figure 2
Figure 2. Figure 2: Lorenz system equation learning results with second-order polynomial library. Median and quartiles of the relative coefficient error and TPR of (a) four different SINDy and weak SINDy methods at varying noise levels, (b) Fourier weak SINDy with SDE and different numbers of dominant frequencies chosen, and (c) Fourier weak SINDy with different bandwidth settings (in Hz) for the multitaper SDE [PITH_FULL_IM… view at source ↗
Figure 3
Figure 3. Figure 3: Equation learning results of four different SINDy and weak SINDy methods for (a) Lotka￾Volterra equations, (b) hyperchaotic Lorenz system, and (c) hyperchaotic Jha system. cies to match maximally energetic modes in the data, presents an interpretable, robust, and effective approach to equation learning. We presented numerical evidence that Fourier weak SINDy shows either improved or comparable performance … view at source ↗
Figure 1
Figure 1. Figure 1: Equation learning results comparisons of (a) Lorenz system with fifth-order polynomial library, (b) hyperchaotic Lorenz system with third-order polynomial library, and (c) hyperchaotic Jha system with third-order polynomial library. Note that in the experiments on the bump-function weak SINDy for the hyperchaotic Jha and hyperchaotic Lorenz systems with the third-order polynomial library, we choose sequent… view at source ↗
Figure 2
Figure 2. Figure 2: Trajectory error comparisons for regular SINDy, bump-function weak SINDy, and Fourier weak SINDy. References William Gilpin. Chaos as an interpretable benchmark for forecasting and data-driven modelling, 2023. URL https://arxiv.org/abs/2110.05266. Samuel H Rudy, Steven L Brunton, Joshua L Proctor, and J Nathan Kutz. Data-driven discovery of partial differential equations. Science advances, 3(4):e1602614, 2… view at source ↗
read the original abstract

We introduce Fourier Weak SINDy, a minimal noise-robust and interpretable derivative-free equation learning method that combines weak-form sparse equation learning with spectral density estimation for data-driven test function selection. By using orthogonal sinusoidal test functions inspired by their prevalence in Modulating Function-based system identification, the weak-form sparse regression problem reduces to a regression over Fourier coefficients. Dominant frequencies are then selected via multitaper estimation of the frequency spectrum of the data. This formulation unifies weak-form learning and spectral estimation within a compact and flexible framework. We illustrate the effectiveness of this approach in numerical experiments across multiple chaotic and hyperchaotic ODE benchmarks.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces Fourier Weak SINDy, a derivative-free method for sparse equation discovery that reduces the weak-form regression problem to a Fourier-coefficient regression by selecting orthogonal sinusoidal test functions at dominant frequencies identified via multitaper spectral estimation of the observed trajectories. The approach is positioned as noise-robust and interpretable, unifying weak-form SINDy with spectral techniques, and is illustrated through numerical experiments on multiple chaotic and hyperchaotic ODE benchmarks.

Significance. If the central claims hold, the work offers a compact, parameter-light framework that leverages data-driven spectral information to select test functions, potentially improving robustness in weak-form model identification without requiring derivative estimation or manual tuning of modulating functions. The unification of spectral density estimation with sparse regression is a natural extension of prior weak-form and modulating-function literature and could aid interpretability in applications where frequency content is physically meaningful.

major comments (2)
  1. [§3] §3 (Method), around the multitaper frequency selection step: the argument that dominant frequencies recovered from the state spectrum yield a sufficient span for the weak-form library matrix in nonlinear systems is not secured. In chaotic ODEs the vector field produces higher harmonics and broadband content absent from the observed trajectories; the manuscript must show (via conditioning analysis or ablation) that the selected sinusoids still isolate the correct sparse coefficients rather than omitting modes required for the integrals.
  2. [§4] §4 (Numerical experiments): the effectiveness claims across chaotic benchmarks are stated without reported quantitative metrics (e.g., coefficient error, reconstruction error, success rate over noise levels) or direct baseline comparisons to standard weak SINDy or other spectral test-function methods. This leaves the noise-robustness advantage unquantified and prevents assessment of whether the frequency-selection step improves or degrades performance relative to fixed or random test functions.
minor comments (2)
  1. [§2] Notation for the Fourier coefficients and the reduced regression matrix should be introduced with an explicit equation rather than inline text to improve readability.
  2. [Abstract] The abstract and introduction would benefit from a single-sentence statement of the precise algorithmic complexity or number of free parameters retained after frequency selection.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and recommendation for major revision. We address each major comment below and will make the indicated changes to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Method), around the multitaper frequency selection step: the argument that dominant frequencies recovered from the state spectrum yield a sufficient span for the weak-form library matrix in nonlinear systems is not secured. In chaotic ODEs the vector field produces higher harmonics and broadband content absent from the observed trajectories; the manuscript must show (via conditioning analysis or ablation) that the selected sinusoids still isolate the correct sparse coefficients rather than omitting modes required for the integrals.

    Authors: We agree this point requires additional support. The manuscript selects dominant frequencies from the multitaper estimate of the observed trajectories on the grounds that these capture the primary energy content for the weak-form projections. However, we acknowledge that nonlinear vector fields may generate higher harmonics not prominent in the state spectrum. In the revision we will add a conditioning analysis of the library matrix and an ablation study on the chaotic benchmarks that compares coefficient recovery using the selected frequencies versus an augmented set that includes higher harmonics. revision: yes

  2. Referee: [§4] §4 (Numerical experiments): the effectiveness claims across chaotic benchmarks are stated without reported quantitative metrics (e.g., coefficient error, reconstruction error, success rate over noise levels) or direct baseline comparisons to standard weak SINDy or other spectral test-function methods. This leaves the noise-robustness advantage unquantified and prevents assessment of whether the frequency-selection step improves or degrades performance relative to fixed or random test functions.

    Authors: We accept that quantitative metrics and baseline comparisons are necessary to substantiate the claims. The current manuscript presents illustrative results on the benchmarks. In the revision we will add tables reporting coefficient errors, reconstruction errors, and success rates across noise levels, together with direct comparisons to standard weak-form SINDy and to variants that employ fixed or randomly chosen test functions. revision: yes

Circularity Check

0 steps flagged

No circularity: Fourier Weak SINDy reformulation and frequency selection are independent methodological steps

full rationale

The paper's core chain begins with the standard weak-form integral formulation of SINDy, then substitutes orthogonal sinusoidal test functions (inspired by modulating-function literature) to obtain an equivalent regression over Fourier coefficients of the data. Dominant frequencies for those test functions are extracted via multitaper spectral estimation of the observed trajectories. This selection is an explicit algorithmic design choice justified by noise-robustness goals and the prevalence of sinusoids in prior system-identification work; it does not define the target dynamics or force the sparse coefficients to match the spectrum by construction. The subsequent sparse regression on the resulting library matrix recovers the unknown vector field independently, and the paper validates the overall procedure on external chaotic ODE benchmarks rather than on any tautological reproduction of the input spectrum. No self-definitional loop, fitted-input-as-prediction, or load-bearing self-citation appears in the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard assumptions of sparse regression and spectral analysis without introducing new free parameters, axioms, or invented entities beyond those already present in weak SINDy and multitaper methods.

axioms (1)
  • domain assumption The observed data are generated by a dynamical system whose weak-form representation is sparse in a suitable basis.
    Implicit in all SINDy-style methods and required for the regression to recover the correct terms.

pith-pipeline@v0.9.0 · 5402 in / 1262 out tokens · 33885 ms · 2026-05-10T00:44:42.857780+00:00 · methodology

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Reference graph

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