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

被引:1
作者
Schwerdtner, Paul [1 ]
Schulze, Philipp [2 ]
Berman, Jules [1 ]
Peherstorfer, Benjamin [1 ]
机构
[1] NYU, Courant Inst Math Sci, New York, NY 10012 USA
[2] Tech Univ Berlin, D-10623 Berlin, Germany
基金
美国国家科学基金会;
关键词
model reduction; Hamiltonian systems; conservation of quantities; deep networks; structure preservation; Neural Galerkin schemes; Dirac-Frenkel variational principle; PRESERVING MODEL-REDUCTION; SYSTEMS; INTEGRATION; STABILITY; DYNAMICS; POD;
D O I
10.1137/23M1607799
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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.
引用
收藏
页码:C583 / C607
页数:25
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