A major problem of kernel-based methods (e.g., least squares support vector machines, LS-SVMs) for solving the ordinary differential equations (ODEs) governing structural dynamics (e.g., vibration response analysis) is the prohibitive O(an³) (a = 1 for linear systems and a > 1 for nonlinear systems) part of their computational complexity with increasing temporal discretization points n. We propose a novel Nyström-accelerated LS-SVMs framework that breaks this bottleneck for structural dynamics problems by reformulating the governing 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 efficient learning of structural dynamic responses with m-dependent complexity. Numerical experiments on sixteen benchmark problems, including linear and nonlinear structural systems, 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 and RMSE) in predicting dynamic responses while maximum surpassing PINNs by 72% in RMSE, and 3) scalability to n = 10⁴ time steps with m = 50 features, facilitating long-duration simulation. This work establishes a new paradigm for efficient kernel-based learning in structural dynamics without significantly sacrificing the accuracy of the solution.
 

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