Nonlinear embeddings for conserving Hamiltonians and other quantities with Neural Galerkin schemes
This addresses a key issue in physics-informed machine learning for researchers and practitioners, offering an incremental improvement to existing Neural Galerkin methods.
The paper tackles the challenge of conserving quantities like Hamiltonians when approximating PDE solutions with deep networks, proposing an explicit embedding method within Neural Galerkin schemes that achieves conservation up to machine precision in numerical experiments.
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.