Data-driven model discovery with Kolmogorov-Arnold networks

Abstract

Data-driven model discovery of complex dynamical systems can be 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 functions. Examples where sparse optimization fails abound, including the classic Ikeda or optical-cavity map in nonlinear dynamics, as well as a wide variety of ecosystems. Another approach is based on machine learning, e.g., deep neural networks, which excels at capturing system behavior from data but functions as black boxes, offering little insight into how inputs influence the outputs. We propose a general model-discovery framework based on the Kolmogorov-Arnold networks (KANs) that are not constrained by the sparsity condition. The KAN framework with a simple structure is capable of accurately capturing the complex behavior of dynamical systems that do not meet the sparsity requirement while offering greater interpretability compared to conventional neural networks. This interpretability provides insights into the dynamics generating the data, which are typically inaccessible in traditional black-box function approximation methods. We demonstrate nonuniqueness in that a large number of approximate models of the system can be found that generate the same invariant set with the true statistics such as the Lyapunov exponents and Kullback–Leibler divergence. An analogy to shadowing of numerical trajectories in chaotic systems is pointed out.

Topics & Concepts