NONLINEAR EMBEDDINGS FOR CONSERVING HAMILTONIANS AND OTHER QUANTITIES WITH NEURAL GALERKIN SCHEMES

Paul Schwerdtner, Philipp Schulze, Jules Berman, Benjamin Peherstorfer

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)C583-C607
JournalSIAM Journal on Scientific Computing
Volume46
Issue number5
DOIs
StatePublished - Oct 2024

Keywords

  • conservation of quantities
  • deep networks
  • Dirac-Frenkel variational principle
  • Hamiltonian systems
  • model reduction
  • Neural Galerkin schemes
  • structure preservation

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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