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。

利用赝哈密顿神经网络学习偏微分方程

Pseudo-Hamiltonian neural networks for learning partial differential  equations

What happens if one pushes a cup sitting on a table toward the edge of the
table? How about pushing a desk against a wall? In this paper, we study the
problem of understanding the movements of objects as a result of applying
external forces to them. For a given force vector applied to a specific
location in an image, our goal is to predict long-term sequential movements
caused by that force. Doing so entails reasoning about scene geometry, objects,
their attributes, and the physical rules that govern the movements of objects.
We design a deep neural network model that learns long-term sequential
dependencies of object movements while taking into account the geometry and
appearance of the scene by combining Convolutional and Recurrent Neural
Networks. Training our model requires a large-scale dataset of object movements
caused by external forces. To build a dataset of forces in scenes, we
reconstructed all images in SUN RGB-D dataset in a physics simulator to
estimate the physical movements of objects caused by external forces applied to
them. Our Forces in Scenes (ForScene) dataset contains 10,335 images in which a
variety of external forces are applied to different types of objects resulting
in more than 65,000 object movements represented in 3D. Our experimental
evaluations show that the challenging task of predicting long-term movements of
objects as their reaction to external forces is possible from a single image.

本文介绍了一种使用深度神经网络结合物理模拟器对外力作用下物体运动进行预测的方法，并构建了一个大规模的物理运动数据集。实验表明，从单张图片中预测物体受外部力作用下的长期运动是可行的。