Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning
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This paper proposes Friedrichs learning as a novel deep learning methodology that can learn the weak solutions of PDEs via Friedrichs' seminal minimax formulation, which transforms the PDE problem into a minimax optimization problem to identify weak solutions. The name "Friedrichs learning" is for Friedrichs' contribution to the minimax framework for PDEs in a weak form. The weak solution and the test function in the weak formulation are parameterized as deep neural networks in a mesh-free manner, which are alternately updated to approach the optimal solution networks approximating the weak solution and the optimal test function, respectively. Extensive numerical results indicate that our mesh-free method can provide reasonably good solutions to a wide range of PDEs defined on regular and irregular domains in various dimensions, where classical numerical methods such as finite difference methods and finite element methods may be tedious or difficult to be applied.
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