Learning nonlinear finite element solution operators with MLPs and energy minimization yields small learning errors on parametric Poisson, Gaussian random field, and nonlinear elasticity cases, and speeds up Newton's method when the network prediction is used as an initial guess
The authors develop and evaluate a data-free, physics-informed method for learning solution operators: after a finite element discretization, a multilayer perceptron (MLP) maps problem data parameters (boundary conditions, coefficients, right-hand sides, etc.) to the finite element degrees of freedom, with a loss function given by the expected energy functional, plus a parallelizable training algorithm that can use random batches of mesh elements; on a parametric Poisson problem, a Gaussian random field coefficient problem, and a nonlinear neo-Hookean elastic beam, learning errors are small (e.g., for the parametric problem with the full mesh and a 4·512 network, mean relative energy error 2.0e-4, L2 0.0020, H1 0.