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Discovering Dynamics with Kolmogorov Arnold Networks: Linear Multistep Method-Based Algorithms and Error Estimation

Jintao Hu, Hongjiong Tian, Qian Guo

TL;DR

This work proposes a novel framework that integrates Kolmogorov Arnold Networks (KANs) with LMMs for the discovery and approximation of dynamical systems' vector fields and establishes precise error bounds for two-layer B-spline KANs when approximating the governing functions of dynamical systems.

Abstract

Uncovering the underlying dynamics from observed data is a critical task in various scientific fields. Recent advances have shown that combining deep learning techniques with linear multistep methods (LMMs) can be highly effective for this purpose. In this work, we propose a novel framework that integrates Kolmogorov Arnold Networks (KANs) with LMMs for the discovery and approximation of dynamical systems' vector fields. Specifically, we begin by establishing precise error bounds for two-layer B-spline KANs when approximating the governing functions of dynamical systems. Leveraging the approximation capabilities of KANs, we demonstrate that for certain families of LMMs, the total error is constrained within a specific range that accounts for both the method's step size and the network's approximation accuracy. Additionally, we analyze the difference between the numerical solution obtained from solving the ordinary differential equations with the fitted vector fields and the true solution of the dynamical system. To validate our theoretical results, we provide several numerical examples that highlight the effectiveness of our approach.

Discovering Dynamics with Kolmogorov Arnold Networks: Linear Multistep Method-Based Algorithms and Error Estimation

TL;DR

This work proposes a novel framework that integrates Kolmogorov Arnold Networks (KANs) with LMMs for the discovery and approximation of dynamical systems' vector fields and establishes precise error bounds for two-layer B-spline KANs when approximating the governing functions of dynamical systems.

Abstract

Uncovering the underlying dynamics from observed data is a critical task in various scientific fields. Recent advances have shown that combining deep learning techniques with linear multistep methods (LMMs) can be highly effective for this purpose. In this work, we propose a novel framework that integrates Kolmogorov Arnold Networks (KANs) with LMMs for the discovery and approximation of dynamical systems' vector fields. Specifically, we begin by establishing precise error bounds for two-layer B-spline KANs when approximating the governing functions of dynamical systems. Leveraging the approximation capabilities of KANs, we demonstrate that for certain families of LMMs, the total error is constrained within a specific range that accounts for both the method's step size and the network's approximation accuracy. Additionally, we analyze the difference between the numerical solution obtained from solving the ordinary differential equations with the fitted vector fields and the true solution of the dynamical system. To validate our theoretical results, we provide several numerical examples that highlight the effectiveness of our approach.
Paper Structure (16 sections, 14 theorems, 81 equations, 8 figures, 3 tables)

This paper contains 16 sections, 14 theorems, 81 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. B-Spline KANs Approximation
  3. Kolmogorov-Arnold theorem
  4. B-Spline KANs framework
  5. B-Spline KANs approximation
  6. Lower bound approximation error and VC-dimension
  7. Dynamical systems and discovery of dynamics
  8. LMMs: Notation and concepts
  9. Discovery of dynamics
  10. Convergence Analysis
  11. Network-based discovery method
  12. Error estimates
  13. Numerical experiments
  14. Linear ode system
  15. Glycolytic oscillator
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $f: [0,1]^{d} \rightarrow \mathbb{R}$ be an arbitrary multivariate continuous function. Then it has the representation with continuous one-dimensional outer and inner functions $\phi_q$ and $\psi_{q, p}$. All these functions $\phi_q$, $\psi_{q, p}$ are defined on the real line. The inner functions $\psi_{q, p}$ are independent of the function $f$.

Figures (8)

  • Figure 2.1: The activation function is parameterized by a set of B-spline basis functions.
  • Figure 4.1: Schematic representation of the network architecture employing LMMs in conjunction with KANs for dynamical system discovery.
  • Figure 5.1: Left panel: The black dashed vertical line at $t=1$ demarcates the boundary between the training interval $[0,1]$ and the prediction interval $[1,10]$. Right panel: The black dashed vertical line at $t=1$ demarcates the boundary between the training interval $[0,1]$ and the extended prediction interval $[1,20]$.
  • Figure 5.2: Influence of B-Spline KANs parameters: $e_\mathcal{NN}(k, G, N)$ as a function of degree $k$ and node number $G$.
  • Figure 5.3: The $L^\infty$-norm error between $\boldsymbol{x}(t)$ and $\boldsymbol{x}_{\mathcal{NN}}(t)$ over the interval $[0,T].$
  • ...and 3 more figures

Theorems & Definitions (29)

  • Lemma 2.1: Kolmogorov-Arnold Representation theorem BraunGriebel2009
  • Definition 2.1: Modulus of continuity
  • Definition 2.2: Hölder continuous functions
  • Lemma 2.2: Extension of Continuous Functions ShenYangZhang2020
  • Lemma 2.3: B-spline function error approximation deBoor1978
  • Theorem 2.1: Upper bound of two-layer B-spline KANs
  • proof
  • Corollary 2.1
  • proof
  • Theorem 2.2
  • ...and 19 more