Discovering Dynamics with Kolmogorov Arnold Networks: Linear Multistep Method-Based Algorithms and Error Estimation

TL;DR

This paper introduces a novel framework combining Kolmogorov Arnold Networks with linear multistep methods to accurately discover and approximate dynamical systems from data, with theoretical error bounds and numerical validation.

Contribution

It presents the integration of KANs with LMMs for dynamical systems discovery, providing error bounds and analysis of solution accuracy, which is a novel approach.

Findings

Error bounds for KANs approximating dynamical functions.

Total error constrained by step size and network approximation.

Numerical examples validate the effectiveness of the approach.

Abstract

Uncovering the underlying dynamics from observed data is a critical task in various scientific fields. Recent advances have shown that combining deep learning techniques with linear multistep methods (LMMs) can be highly effective for this purpose. In this work, we propose a novel framework that integrates Kolmogorov Arnold Networks (KANs) with LMMs for the discovery and approximation of dynamical systems' vector fields. Specifically, we begin by establishing precise error bounds for two-layer B-spline KANs when approximating the governing functions of dynamical systems. Leveraging the approximation capabilities of KANs, we demonstrate that for certain families of LMMs, the total error is constrained within a specific range that accounts for both the method's step size and the network's approximation accuracy. Additionally, we analyze the difference between the numerical solution obtained from solving the ordinary differential equations with the fitted vector fields and the true solution of the dynamical system. To validate our theoretical results, we provide several numerical examples that highlight the effectiveness of our approach.