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.

翻译：伪哈密顿神经网络（PHNN）近年来被引入用于学习可由常微分方程建模的动力系统。本文将该方法扩展至偏微分方程。所建模型由至多三个神经网络组成，分别建模守恒项、耗散项和外力项，同时包含可学习或给定的离散卷积算子。我们通过数值实验证明了PHNN相比单一神经网络的基线模型具有更优性能。此外，由于PHNN模型由三个具有不同物理意义的子部分组成，可单独研究这些部分以深入了解系统特性，且所学模型在外力移除或改变后依然适用。