Data-driven model discovery with Kolmogorov-Arnold networks

TL;DR

This paper introduces Kolmogorov-Arnold networks as a new framework for data-driven discovery of dynamical systems, overcoming the limitations of sparse optimization especially in non-sparse systems.

Contribution

It develops a general model-discovery approach applicable to non-sparse systems, demonstrating non-uniqueness and statistical equivalence of multiple models.

Findings

Applicable to non-sparse dynamical systems.

Multiple models can generate the same invariant set.

Models match key statistical properties like Lyapunov exponents.

Abstract

Data-driven model discovery of complex dynamical systems is typically done using sparse optimization, but it has a fundamental limitation: sparsity in that the underlying governing equations of the system contain only a small number of elementary mathematical terms. Examples where sparse optimization fails abound, such as the classic Ikeda or optical-cavity map in nonlinear dynamics and a large variety of ecosystems. Exploiting the recently articulated Kolmogorov-Arnold networks, we develop a general model-discovery framework for any dynamical systems including those that do not satisfy the sparsity condition. In particular, we demonstrate non-uniqueness in that a large number of approximate models of the system can be found which generate the same invariant set with the correct statistics such as the Lyapunov exponents and Kullback-Leibler divergence. An analogy to shadowing of numerical trajectories in chaotic systems is pointed out.