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

• Published in: Journal of computational physics Vol. 446; p. 110651
• Main Authors: Mitusch, Sebastian K., Funke, Simon W., Kuchta, Miroslav
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.12.2021; Elsevier Science Ltd
• Subjects: Artificial neural networks; Computational physics; Constraints; Data-driven scientific computing; Finite element analysis; Finite element method; Learning unknown physics; Machine learning; Mathematical models; Neural networks; Operators (mathematics); Optimization; Partial differential equations; Finite element method; Data-driven scientific computing; Partial differential equations; Learning unknown physics; Machine learning
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2021.110651

•An approach for training neural networks combined with partial differential equations.•Combines a finite element framework with a machine learning library.•Learns missing physics in the partial differential equation. 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.