Deep-Koopman-KANDy: Dictionary Discovery for Deep-Koopman Operators with Kolmogorov-Arnold Networks for Dynamics

TL;DR

Deep-Koopman-KANDy introduces a post-hoc symbolic dictionary readout for Deep-Koopman operators using Kolmogorov-Arnold Networks, enabling interpretable observables in data-driven dynamical systems.

Contribution

It combines Deep-Koopman modeling with KANDy for structured, interpretable observables, evaluated on multiple dynamical systems with promising results.

Findings

Recovers target dictionaries with high accuracy on Lorenz system.

Identifies analytical Fourier basis on standard map.

Successfully extracts foliation coordinate in Ikeda map.

Abstract

Symbolic library -- or Koopman dictionary -- selection is a fundamental challenge in data-driven dynamical systems. Extended Dynamic Mode Decomposition (EDMD), Sparse Identification of Nonlinear Dynamics (SINDy), and Kolmogorov--Arnold Networks for Dynamics (KANDy) all require the practitioner to commit to a function library at training time; Deep-Koopman Operators avoid this commitment but produce uninterpretable latent observables. We propose Deep-Koopman-KANDy, a structured approach to post-hoc symbolic dictionary readout that combines Deep-Koopman modeling with Kolmogorov-Arnold Networks for Dynamics (KANDy). The encoder and decoder of a Deep-Koopman Operator are replaced with two-layer Kolmogorov--Arnold Networks (KANs), and a level-set construction together with a chain-rule gradient identity exposes the compositional structure of the learned observables in a basis chosen \emph{after} training. We evaluate the method on the Lorenz system, the Chirikov standard map, the Ikeda map, and the Arnold cat map. On Lorenz it recovers the target dictionary $\{x,y,z,xy,xz\}$ with perfect recall and Jaccard score $0.79\pm0.06$; on the standard map it recovers a low-order Fourier basis matching the analytical structure; on Ikeda -- which has no sparse polynomial representation -- a misspecified polynomial readout still recovers the correct foliation coordinate $g\approx x^2+y^2$ together with a nontrivial outer function; and on the Arnold cat map -- used as a negative control because finite-dimensional Koopman closure is provably impossible -- the method fails to find a sparse closure, as expected.