Sparse reconstruction of ordinary differential equations with inference

TL;DR

This paper introduces a statistical inference approach using bias-corrected Lasso, ridge, and SEMMS methods for sparse reconstruction of ODEs, improving the identification of governing equations from data.

Contribution

It applies advanced statistical inference techniques to sparse regression for ODEs, enabling uncertainty quantification and more accurate discovery of system dynamics.

Findings

Better recovery of true functional terms in simulations

Outperforms existing methods that lack uncertainty quantification

Demonstrates effectiveness in identifying governing equations

Abstract

Sparse regression has emerged as a popular technique for learning dynamical systems from temporal data, beginning with the SINDy (Sparse Identification of Nonlinear Dynamics) framework proposed by arXiv:1509.03580. Quantifying the uncertainty inherent in differential equations learned from data remains an open problem, thus we propose leveraging recent advances in statistical inference for sparse regression to address this issue. Focusing on systems of ordinary differential equations (ODEs), SINDy assumes that each equation is a parsimonious linear combination of a few candidate functions, such as polynomials, and uses methods such as sequentially-thresholded least squares or the Lasso to identify a small subset of these functions that govern the system's dynamics. We instead employ bias-corrected versions of the Lasso and ridge regression estimators, as well as an empirical Bayes variable selection technique known as SEMMS, to estimate each ODE as a linear combination of terms that are statistically significant. We demonstrate through simulations that this approach allows us to recover the functional terms that correctly describe the dynamics more often than existing methods that do not account for uncertainty.