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.
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.