This work focuses on the conservation of quantities such as Hamiltonians, mass, and momentum when solution fields of partial differential equations are approximated with nonlinear parametrizations such as deep networks. The proposed approach builds on Neural Galerkin schemes that are based on the Dirac--Frenkel variational principle to train nonlinear parametrizations sequentially in time. We first show that only adding constraints that aim to conserve quantities in continuous time can be insufficient because the nonlinear dependence on the parameters implies that even quantities that are linear in the solution fields become nonlinear in the parameters and thus are challenging to discretize in time. Instead, we propose Neural Galerkin schemes that compute at each time step an explicit embedding onto the manifold of nonlinearly parametrized solution fields to guarantee conservation of quantities. The embeddings can be combined with standard explicit and implicit time integration schemes. Numerical experiments demonstrate that the proposed approach conserves quantities up to machine precision.

翻译：本文聚焦于当偏微分方程解场采用深度网络等非线性参数化近似时，哈密顿量、质量、动量等物理量的守恒问题。所提方法基于狄拉克-弗兰克尔变分原理构建神经Galerkin方案，实现对非线性参数化表示的时间序列训练。我们首先证明：仅添加旨在连续时间域守恒量的约束可能不足，因为参数的非线性依赖关系会导致解场中原本线性的量在参数空间呈现非线性特性，从而难以进行时间离散化。为此，我们提出神经Galerkin方案，在每个时间步通过显式嵌入到非线性参数化解流形，保障物理量的守恒性。该嵌入方法可与标准显式及隐式时间积分格式结合使用。数值实验表明，所提方法可将守恒量精度保持至机器精度水平。