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A new penalty-free neural network method, PFNN-2, is presented for solvingpartial differential equations, which is a subsequent improvement of ourpreviously proposed PFNN method [1]. PFNN-2 inherits all advantages of PFNN inhandling the smoothness constraints and essential boundary conditions ofself-adjoint problems with complex geometries, and extends the application to abroader range of non-self-adjoint time-dependent differential equations. Inaddition, PFNN-2 introduces an overlapping domain decomposition strategy tosubstantially improve the training efficiency without sacrificing accuracy.Experiments results on a series of partial differential equations are reported,which demonstrate that PFNN-2 can outperform state-of-the-art neural networkmethods in various aspects such as numerical accuracy, convergence speed, andparallel scalability.

PFNN-2: A Domain Decomposed Penalty-Free Neural Network Method for Solving Partial Differential Equations