Finite element interpolated neural networks for solving forward and inverse problems

Abstract

We propose a general framework for solving forward and inverse problems constrained by partial differential equations, where we interpolate neural networks onto finite element spaces to represent the (partial) unknowns. The framework overcomes the challenges related to the imposition of boundary conditions, the choice of collocation points in physics-informed neural networks, and the integration of variational physics-informed neural networks. A numerical experiment set confirms the framework's capability of handling various forward and inverse problems. In particular, the trained neural network generalises well for smooth problems, beating finite element solutions by some orders of magnitude. We finally propose an effective one-loop solver with an initial data fitting step (to obtain a cheap initialisation) to solve inverse problems.

References in corpus (2)

• Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
• A comparison study of deep Galerkin method and deep Ritz method for elliptic problems with different boundary conditions

Cited by in corpus (8)

• Deep Learning in Deterministic Computational Mechanics
• Compatible finite element interpolated neural networks
• Finite Element Neural Network Interpolation. Part I: Interpretable and Adaptive Discretization for Solving PDEs
• Finite Element Neural Network Interpolation. Part II: Hybridisation with the Proper Generalised Decomposition for non-linear surrogate modelling
• The lowest-order Neural Approximated Virtual Element Method on polygonal elements
• An $r$-adaptive finite element method using neural networks for parametric self-adjoint elliptic problem
• Deep Ritz-Finite Element methods: Neural Network Methods trained with Finite Elements
• Unfitted finite element interpolated neural networks