Pseudo-Hamiltonian neural networks for learning partial differential equations

TL;DR

This paper extends Pseudo-Hamiltonian neural networks to partial differential equations, enabling better modeling of complex dynamical systems with interpretability and adaptability to external forces.

Contribution

The authors adapt PHNN for PDEs, introducing a modular neural network approach that models conservation, dissipation, and external forces separately.

Findings

PHNN outperforms baseline models in numerical experiments.

The modular structure allows for interpretability of physical components.

The learned model remains effective when external forces are altered or removed.

Abstract

Pseudo-Hamiltonian neural networks (PHNN) were recently introduced for learning dynamical systems that can be modelled by ordinary differential equations. In this paper, we extend the method to partial differential equations. The resulting model is comprised of up to three neural networks, modelling terms representing conservation, dissipation and external forces, and discrete convolution operators that can either be learned or be given as input. We demonstrate numerically the superior performance of PHNN compared to a baseline model that models the full dynamics by a single neural network. Moreover, since the PHNN model consists of three parts with different physical interpretations, these can be studied separately to gain insight into the system, and the learned model is applicable also if external forces are removed or changed.