Holomorphic Neural ODEs with Kolmogorov-Arnold Networks for Interpretable Discovery of Complex Dynamics

TL;DR

This paper introduces Holomorphic KAN-ODE, a parameter-efficient, interpretable neural network framework that models complex holomorphic dynamical systems with high accuracy, noise resilience, and symbolic equation discovery.

Contribution

It replaces traditional MLPs with Kolmogorov-Arnold Networks incorporating complex-analytic priors, enabling interpretable discovery of complex dynamics with fewer parameters.

Findings

Achieves velocity-field R^2 > 0.95 on six dynamical systems

Correctly identifies governing symbolic families and fractal boundaries

Outperforms MLPs in noise resilience and transfer learning

Abstract

Complex dynamical systems governed by holomorphic maps such as $z^2 + c$ exhibit fractal boundaries with extreme sensitivity to initial conditions. Accurately modelling these structures from data requires methods that respect the underlying complex-analytic geometry, yet Multi-Layer Perceptrons (MLPs) within Neural Ordinary Differential Equations (Neural ODEs) lack complex-analytic priors, violate the Cauchy--Riemann conditions, and function as opaque approximators incapable of yielding governing equations. We introduce Holomorphic KAN-ODE, a framework that replaces the MLP with a Kolmogorov-Arnold Network (KAN) whose learnable B-spline activations reside on network edges, and incorporates Cauchy--Riemann equations as a differentiable regularization to preserve holomorphic structure. We evaluate on six families of complex dynamical systems spanning polynomial and transcendental classes. With only 280 parameters ($16\times$ fewer than the MLP baseline), the network achieves velocity-field $R^2 > 0.95$ on all six systems, correctly identifies all six governing symbolic families through automatic spline-to-formula fitting, and reconstructs Julia set fractal boundaries with up to 98.0\% agreement. Crucially, the model exhibits only 4\% MSE degradation under 10\% observation noise versus $15.2\times$ for MLPs, and achieves 90.4\% improvement in transfer learning from quadratic to cubic dynamics. While the MLP attains lower pointwise reconstruction error due to its larger capacity, the KAN uniquely provides interpretable symbolic equations, enforced holomorphic structure, and superior noise resilience, capabilities that are entirely absent in black-box architectures. These results establish KANs as a parameter-efficient, interpretable alternative to MLPs for physics-informed discovery of holomorphic dynamics.