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.

本文将伪哈密顿神经网络方法扩展到偏微分方程，生成包含三个神经网络的模型，分别模拟代表守恒、耗散和外力的项，并包括可学习的离散卷积算子，数值结果表明 PHNN 的卓越性能。此外，由于 PHNN 模型由三个有不同物理解释的部分组成，反演时可以分别研究每个部分，具有良好的泛化能力。