Data-driven model discovery with Kolmogorov-Arnold networks
This addresses the limitation of sparse optimization in model discovery for complex systems like nonlinear dynamics and ecosystems, offering a more general approach, though it appears incremental as it builds on existing network concepts.
The authors tackled the problem of data-driven model discovery for complex dynamical systems where sparse optimization fails, by developing a framework using Kolmogorov-Arnold networks that can handle systems not satisfying sparsity conditions, resulting in the discovery of multiple approximate models that reproduce invariant sets and correct statistics like Lyapunov exponents and Kullback-Leibler divergence.
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.