Nonlinear embeddings for conserving Hamiltonians and other quantities with Neural Galerkin schemes

TL;DR

This paper introduces Neural Galerkin schemes with nonlinear embeddings that ensure the exact conservation of physical quantities like Hamiltonians during PDE solution approximations with deep networks.

Contribution

It proposes a novel Neural Galerkin approach using explicit embeddings to guarantee conservation laws, overcoming limitations of continuous-time constraints.

Findings

Conserves quantities up to machine precision.

Compatible with explicit and implicit time schemes.

Addresses nonlinear dependence in parameterized PDE solutions.

Abstract

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.