Nyström-Accelerated Primal LS-SVMs: Breaking the O(an3) Complexity Bottleneck for Scalable ODEs Learning

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(an3) (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 n), enabling the learning process to solve linear/nonlinear equation systems with m-dependent complexity. Numerical experiments on sixteen benchmark ODEs demonstrate: 1) 10 6000 times faster computation than classical LS-SVMs and physics-informed neural networks (PINNs), 2) comparable accuracy to LS-SVMs (< 0.13% relative MAE, RMSE, and y ˆy difference) while maximum surpassing PINNs by 72% in RMSE, and 3) scalability to n = 104 time steps with m = 50features. This work establishes a new paradigm for efficient kernel-based ODEs learning without significantly sacrificing the accuracy of the solution.