Nyström-Accelerated Primal LS-SVMs: Breaking the $O (a n^{3})$ Complexity Bottleneck for Scalable ODEs Learning

China Three Gorges University· Hubei Geological Disaster Prevention and Control Engineering Technology Research Center, Yichang· Hubei Key Laboratory of Tumor Microenvironment and Immunotherapy, Yichang

Closed-form approximate solution Nyström-accelerated learning theory least squares support vector machines (LS-SVMs)ordinary differential equations (ODEs)

⋅ NeurIPS ⋅ Poster ⋅OpenReview

Abstract

A major problem of kernel-based methods (e.g., least squares support vector machines, LS-SVMs) for solving linear/nonlinear ordinary differential equations (ODEs) is the prohibitive

O (a n^{3})

(

a = 1

for linear ODEs and 27 for nonlinear ODEs) part of their computational complexity with increasing temporal discretization points

n

We propose a novel Nyström-accelerated LS-SVMs framework that breaks this bottleneck by reformulating ODEs as primal-space constraints. Specifically, we derive for the first time an explicit Nyström-based mapping and its derivatives from one-dimensional temporal discretization points to a higher

m

-dimensional feature space (

1 < m \leq n

), enabling the learning process to solve linear/nonlinear equation systems with

m

-dependent complexity.

Numerical experiments on sixteen benchmark ODEs demonstrate:

$10 - 6000$ times faster computation than classical LS-SVMs and physics-informed neural networks (PINNs)
comparable accuracy to LS-SVMs ( $< 0.13%$ relative MAE, RMSE, and $∥ y - \overset{y}{^} ∥_{\infty}$ difference) while maximum surpassing PINNs by 72\% in RMSE
scalability to $n = 1 0^{4}$ time steps with $m = 50$ features.

This work establishes a new paradigm for efficient kernel-based ODEs learning without significantly sacrificing the accuracy of the solution.