Discovery of Nonlinear Dynamics with Automated Basis Function Generation

2026-05-10

AI Summary

This work addresses the limitations of traditional methods for discovering governing equations from observational data, which often rely on manually prescribed basis functions and fail due to missing terms or complex dynamics, as well as pure symbolic regression approaches that are sensitive to noise and yield redundant expressions. The authors propose AutoSINDy, a novel three-stage automated framework that synergistically integrates symbolic regression with sparse identification. It leverages PySR to generate candidate functions from resampled data, applies collinearity-aware decomposition and selection to refine expressions, and employs SINDy for sparse regression—eliminating the need for predefined basis functions. Evaluated across diverse nonlinear systems, AutoSINDy recovers the true equations with 92.8% accuracy, significantly outperforming baseline methods while achieving superior predictive accuracy, enhanced generalization under high noise, and reduced symbolic complexity.

Abstract

Discovering governing equations from observational data remains a fundamental challenge in scientific modeling, particularly when the underlying mathematical structure is unknown. Traditional sparse identification methods like SINDy excel at discovering parsimonious models but require researchers to specify candidate basis functions a priori, a limitation that often leads to model failure when critical terms are omitted or when systems exhibit unconventional dynamics. Purely symbolic regression approaches offer unlimited flexibility but struggle with noise sensitivity and frequently produce overly complex, unstable equations. We present AutoSINDy, a hybrid Discovery-then-Solve framework that combines the exploratory power of symbolic regression with the robust sparsity-promoting capabilities of SINDy. Our method operates in three stages: (1) PySR-based symbolic regression discovers candidate functional forms from bootstrapped data chunks; (2) a curation pipeline decomposes, expands, and filters these expressions using collinearity analysis to construct a minimal yet comprehensive library; and (3) SINDy identifies sparse governing equations from this custom-tailored library. Extensive experiments across canonical nonlinear systems demonstrate that AutoSINDy consistently recovers ground-truth equations even under high observational noise, achieving a ground-truth recovery rate of 92.8% across all trials. Compared with standard SINDy using enriched libraries and standalone symbolic regression, AutoSINDy achieves higher predictive accuracy, superior generalization to unseen trajectories, and substantially lower symbolic complexity.

Problem

• nonlinear dynamics
• governing equations
• symbolic regression
• sparse identification
• basis functions

Innovation

• AutoSINDy
• symbolic regression
• SINDy
• nonlinear dynamics
• basis function generation