Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning
This method addresses the challenge of solving PDEs, particularly on complex domains, for researchers and practitioners who need robust and flexible numerical solvers.
This paper introduces Friedrichs learning, a deep learning approach that finds weak solutions to partial differential equations (PDEs) by formulating the problem as a minimax optimization. The method parameterizes both the weak solution and test function as neural networks, demonstrating its ability to solve various PDEs on both regular and irregular domains where traditional methods might struggle.
This paper proposes Friedrichs learning as a novel deep learning methodology that can learn the weak solutions of PDEs via a minmax formulation, which transforms the PDE problem into a minimax optimization problem to identify weak solutions. The name "Friedrichs learning" is for highlighting the close relationship between our learning strategy and Friedrichs theory on symmetric systems of PDEs. 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.