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

TLDR

The study proposes a hybrid neural network–finite element framework that embeds PDE constraints to train models while respecting physical laws. The method trains neural networks by enforcing PDE constraints as hard constraints during optimization, discretizing the models with FEM and implementing the approach as an extension of FEniCS and dolfin‑adjoint. Experiments show the approach can recover missing PDE terms, outperform alternative methods, and successfully model a complex cardiac cell problem using deep neural networks.

Abstract

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.