Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimization

תקציר מקורי באנגליתarXiv:2412.04596v3 Announce Type: replace Abstract: We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution operator by a multilayer perceptron (MLP) that takes problem data variables as input and gives a prediction of the finite element solution as output. The variables will typically correspond to parameters in a parametrization of input data such as boundary conditions, coefficients, and right-hand sides. The output will be an approximation of the corresponding finite element solution, thus enabling support and enhancement by the standard finite element method (FEM) both theoretically and practically. We focus on