NySALT: Nystr\"{o}m-type inference-based schemes adaptive to large time-stepping

TL;DR

NySALT is a novel inference-based integrator that adapts to large time steps in simulating Hamiltonian systems, improving stability and efficiency by learning optimal parameters from data.

Contribution

It introduces NySALT, a data-driven, adaptive Nyström-type scheme that optimizes parameters for each time step to enhance stability and accuracy in large time-stepping simulations.

Findings

Enlarges the maximal admissible step size for stability.

Quadruples the time step size of classical integrators like Störmer--Verlet.

Proven and verified convergence of the data-driven estimators.

Abstract

Large time-stepping is important for efficient long-time simulations of deterministic and stochastic Hamiltonian dynamical systems. Conventional structure-preserving integrators, while being successful for generic systems, have limited tolerance to time step size due to stability and accuracy constraints. We propose to use data to innovate classical integrators so that they can be adaptive to large time-stepping and are tailored to each specific system. In particular, we introduce NySALT, Nystr\"{o}m-type inference-based schemes adaptive to large time-stepping. The NySALT has optimal parameters for each time step learnt from data by minimizing the one-step prediction error. Thus, it is tailored for each time step size and the specific system to achieve optimal performance and tolerate large time-stepping in an adaptive fashion. We prove and numerically verify the convergence of the estimators as data size increases. Furthermore, analysis and numerical tests on the deterministic and stochastic Fermi-Pasta-Ulam (FPU) models show that NySALT enlarges the maximal admissible step size of linear stability, and quadruples the time step size of the St\"{o}rmer--Verlet and the BAOAB when maintaining similar levels of accuracy.