Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning

This paper proposes Friedrichs learning as a novel deep methodology that can learn the weak solutions of PDEs via minmax formulation, which transforms PDE problem into minimax optimization to identify solutions. The name "Friedrichs learning" is for highlighting close relationship between our strategy and theory on symmetric systems PDEs. solution test function in formulation are parameterized neural networks mesh-free manner, alternately updated approach optimal approximating function, respectively. Extensive numerical results indicate method provide reasonably good wide range defined regular irregular domains various dimensions, where classical methods such finite difference element may be tedious or difficult applied.