  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-adjoint problems with complex geometries, and extends the application to a
broader range of non-self-adjoint time-dependent differential equations. In
addition, PFNN-2 introduces an overlapping domain decomposition strategy to
substantially 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 network
methods in various aspects such as numerical accuracy, convergence speed, and
parallel scalability.
