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

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