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.

该研究论文关注利用深度网络等非线性参数化近似解偏微分方程时，保持如 Hamiltonians、质量和动量等数量的守恒。研究提出的方法基于基于 Dirac-Frenkel 变分原理的神经 Galerkin 方案，在时间上逐步训练非线性参数化。论文首先表明仅添加连续时间内旨在保持数量守恒的约束可能不足够，因为参数的非线性依赖意味着即使对解场线性的数量在参数上也变为非线性，因此在时间上离散化变得困难。相反，论文提出了神经 Galerkin 方案，它在每个时间步计算出非线性参数化解场在流形上的显式嵌入，以确保数量的守恒。这些嵌入可以与标准的显式和隐式时间积分方案相结合。数值实验表明，所提出的方法可以保持数量守恒直至机器精度。