Variational training of neural network approximations of solution maps for physical models

Simulation of physical models has been one of main driven forces for scientific computing. Physical phenomena at different scales, e.g., macroscopic scale, microscopic scale, etc., are characterized by Newton's laws of motion, Darcy's law, Maxwell's equations, Schrödinger equation, etc. Solving these equations efficiently, especially those nonlinear ones, has challenged computational scientists for decades and led to remarkable development in algorithms and in computing hardware. As the rise of machine learning, particularly deep learning, many researchers have been attempting to adopt artificial neural networks (NN) to represent the high-dimensional solutions or the low-dimensional solution maps. This paper proposes a variational training framework for solving the solution map of low-dimensional physical models via NNs. Here we emphasize solving a solution map in contrast with fitting a solution map, where solving can be to some extent viewed as unsupervised learning with input functions only and fitting refers to supervised learning with both input functions and the corresponding solutions. Solving the solution map for physical models is feasible due to an intrinsic difference between the physical problems and other data-driven problems, e.g., handwriting recognition, speech recognition, spam detection, etc.