A joint optimization approach to identifying sparse dynamics using least squares kernel collocation

TL;DR

This paper introduces a joint optimization framework combining sparse recovery and RKHS techniques to accurately learn ODE systems from limited, noisy data, improving robustness and efficiency over existing methods.

Contribution

It presents a novel all-at-once modeling approach that integrates sparse recovery with kernel methods for better system identification from scarce data.

Findings

Significant improvements in accuracy and robustness to noise.

Enhanced sample efficiency in learning ODEs.

Outperforms existing algorithms in various experiments.

Abstract

We develop an all-at-once modeling framework for learning systems of ordinary differential equations (ODE) from scarce, partial, and noisy observations of the states. The proposed methodology amounts to a combination of sparse recovery strategies for the ODE over a function library combined with techniques from reproducing kernel Hilbert space (RKHS) theory for estimating the state and discretizing the ODE. Our numerical experiments reveal that the proposed strategy leads to significant gains in terms of accuracy, sample efficiency, and robustness to noise, both in terms of learning the equation and estimating the unknown states. This work demonstrates capabilities well beyond existing and widely used algorithms while extending the modeling flexibility of other recent developments in equation discovery.