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

TL;DR

This paper introduces a hybrid modeling approach that integrates neural networks with finite element methods to solve PDEs, ensuring physical constraints are strongly enforced during training, applicable to various PDE types.

Contribution

The paper presents a novel methodology combining neural networks with FEM to enforce PDE constraints strongly, extending existing FEM frameworks and demonstrating effectiveness on complex biological models.

Findings

Successfully recovers PDE coefficients and operators from data

Outperforms physics-informed neural networks and standard PDE optimization

Applicable to complex, nonlinear, and transient PDE problems

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.