Hybrid FEM-NN models: Combining artificial neural networks with the finite element method

JOURNAL OF COMPUTATIONAL PHYSICS

卷:446

Article

Mitusch, Sebastian K.¹  Funke, Simon W.¹  Kuchta, Miroslav¹ 

[1] Simula Res Lab, N-1364 Fornebu, Norway

关键词: Data-driven scientific computing;    Partial differential equations;    Learning unknown physics;    Machine learning;    Finite element method;   

DOI  :  10.1016/j.jcp.2021.110651

来源: Elsevier

【 摘 要 】

We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs). The approach allows to train neural networks while respecting the PDEs as a strong constraint in the optimisation as apposed to making them part of the loss function. The resulting models are discretised in space by the finite element method (FEM). The method applies to both stationary and transient as well as linear/nonlinear PDEs. We describe implementation of the approach as an extension of the existing FEM framework FEniCS and its algorithmic differentiation tool dolfin-adjoint. Through series of examples we demonstrate capabilities of the approach to recover coefficients and missing PDE operators from observations. Further, the proposed method is compared with alternative methodologies, namely, physics informed neural networks and standard PDE-constrained optimisation. Finally, we demonstrate the method on a complex cardiac cell model problem using deep neural networks. (C) 2021 The Authors. Published by Elsevier Inc.

【 授权许可】

Free