Hybrid FEM-NN models: Combining artificial neural networks with the finite element method
This work provides a new approach for researchers and engineers working with physical simulations to integrate neural networks while strictly adhering to physical laws, potentially improving the accuracy and physical consistency of data-driven models.
This paper introduces a methodology that combines neural networks with physical principle constraints, specifically partial differential equations (PDEs), by enforcing PDEs as strong constraints during optimization rather than as part of the loss function. The method is discretized using the finite element method (FEM) and can recover coefficients and missing PDE operators from observations, demonstrated on various PDE types and a complex cardiac cell model.
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.