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

arXiv: https://arxiv.org/abs/2409.16826 · HTML: https://mlsanity.com/paper/2409.16826/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 11 references that are in the map.
- First posted on arXiv 25 Sep 2024.

## Builds on (most cited first, 1 of 1)

- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)

## Abstract (arXiv)

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.