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 prior knowledge. 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）近期被提出用于学习由常微分方程建模的动力系统。本文将该方法推广至偏微分方程。所构建的模型由最多三个神经网络组成，分别建模代表守恒、耗散和外力的项，以及可学习或基于先验知识的离散卷积算子。我们通过数值实验证明了PHNN相较于由单一神经网络建模完整动力学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模块构成，可分别对其研究以深入理解系统特性，并且当外力项被移除或改变时，该学习模型仍具适用性。