KANDy: Kolmogorov-Arnold Networks and Dynamical System Discovery

TL;DR

The Kolmogorov-Arnold Network for Dynamics (KANDy) is introduced as a zero-depth, wide neural architecture capable of discovering governing equations in chaotic and complex dynamical systems and is positioned as an interpretable and effective alternative for data-driven modeling of nonlinear dynamical systems.

Abstract

We introduce the Kolmogorov-Arnold Network for Dynamics (KANDy) as a zero-depth, wide neural architecture capable of discovering governing equations in chaotic and complex dynamical systems. Building on the foundation of Kolmogorov-Arnold Networks (KANs), KANDy explicitly learns governing equations by replacing sparse regression with a KAN. The synthesis of KANs and sparse regression addresses the limitations of equation discovery for KANs applied to dynamical systems and overcomes cases where sparse regression is hindered by sparsity constraints. Additionally, we show that our model, applied to the Hopf Fibration, recovers topological structure, thereby improving coherence with attractor properties. We apply our model to discrete and continuous dynamical systems, as well as to chaotic partial differential equations (PDEs). These results position KANDy as an interpretable and effective alternative for data-driven modeling of nonlinear dynamical systems.

Figures (13)

• Figure 1: The KANDy model synthesizes sparse regression governing equation discovery with KANs. Reimplementing sparse regression as a KAN yields interpretable, closed-form governing laws directly, a width-only path that reinforces the model's transparency.
• Figure 1: Left: The four-dimensional sphere $S^3$ with two distinct Hopf fibers $h^{-1}(p)$ and $h^{-1}(q)$, shown as linked circles inside a dashed boundary representing $S^3$. Any two distinct fibers of the Hopf fibration are linked once, illustrating the nontrivial topology of the fibration. Right: The three-dimensional sphere $S^2$ as the base space, with two points $p$ and $q$. Under the Hopf map $h : S^3 \to S^2$, each point in $S^2$ corresponds to a circle (fiber) in $S^3$.
• Figure 1: Example of the network for the Lorenz system with zero-depth KAN architecture, including nonlinear and physics informed library terms.
• Figure 1: Left: Fibers of the Hopf fibration visualized by stereographic projection of $S^3$ into $\mathbb{R}^3$. Each closed curve corresponds to a $U(1)$ orbit under the group action $(z_1,z_2)\mapsto(e^{i\theta}z_1,e^{i\theta}z_2)$ and each closed curve is colored. Center: a deep KAN of two hidden layers, each with four terms. The learned embedding model scatters the fibres on the sphere. Right: Outputs of a trained KANDy model evaluated along the same fibers. Fibres of the closed curve are colored (left) correspond to their respective fibres (center and right).
• Figure 2: The KANDy architecture is equivalent to a commutative diagram where the composition of the estimated KAN composes with the lifting map $\Phi$.