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

2026-04-21

AI Summary

This work addresses the challenge of identifying differential equation models from noisy observational data when derivative information is unavailable. The authors propose a novel approach that integrates weak-form sparse regression with multitaper spectral estimation. By introducing orthogonal sinusoidal test functions, the problem of equation discovery is recast as a sparse regression on Fourier coefficients. Multitaper spectral estimation is leveraged to automatically select dominant frequencies for constructing optimal test functions. This framework uniquely unifies weak-form equation learning with spectral density estimation, yielding an interpretable, derivative-free, and noise-robust mechanism for adaptive test function selection. Numerical experiments on benchmark chaotic and hyperchaotic ordinary differential equations demonstrate that the method accurately recovers the underlying dynamical systems even under high levels of noise.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Other Foundations of Machine Learning
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• equation learning
• noise-robust identification
• weak-form formulation
• spectral estimation
• dynamic system modeling

Innovation

• Fourier Weak SINDy
• weak-form learning
• spectral density estimation
• multitaper estimation
• derivative-free equation learning