PFNN-2: A Domain Decomposed Penalty-Free Neural Network Method for Solving Partial Differential Equations (Communications in Computational Physics, 2022)
A new penalty-free neural network method, PFNN-2, is presented for solving partial differential equations, which is a subsequent improvement of our previously proposed PFNN method [1]. PFNN-2 inherits all advantages of PFNN in handling the smoothness constraints and essential boundary conditions of self-adjointproblems with complex geometries, and extends the application to a broader rangeof non-self-adjoint time-dependent differential equations. In addition, PFNN-2 introduces an overlapping domain decomposition strategy to substantially improve thetraining efficiency without sacrificing accuracy. Experiments results on a series ofpartial differential equations are reported, which demonstrate that PFNN-2 can outperform state-of-the-art neural network methods in various aspects such as numericalaccuracy, convergence speed, and parallel scalability.
Keywords: Neural network; penalty-free method; domain decomposition; initial-boundary value problem; partial differential equation.