Learning phase-space flows using time-discrete implicit Runge-Kutta PINNs

TL;DR

This paper introduces a high-order implicit Runge-Kutta Physics-Informed Neural Network framework to efficiently solve multidimensional phase-space equations of motion, especially for time-independent and periodic fields.

Contribution

It adapts existing IRK-PINNs to treat coordinates as functions, enabling effective solutions for particle dynamics in external fields.

Findings

Successfully solved equations of motion for particles in central force fields.

Demonstrated effectiveness in periodic electric fields.

Enhanced computational efficiency for complex dynamical systems.

Abstract

We present a computational framework for obtaining multidimensional phase-space solutions of systems of non-linear coupled differential equations, using high-order implicit Runge-Kutta Physics- Informed Neural Networks (IRK-PINNs) schemes. Building upon foundational work originally solving differential equations for fields depending on coordinates [J. Comput. Phys. 378, 686 (2019)], we adapt the scheme to a context where the coordinates are treated as functions. This modification enables us to efficiently solve equations of motion for a particle in an external field. Our scheme is particularly useful for explicitly time-independent and periodic fields. We apply this approach to successfully solve the equations of motion for a mass particle placed in a central force field and a charged particle in a periodic electric field.