  Learning solutions of partial differential equations (PDEs) with
Physics-Informed Neural Networks (PINNs) is an attractive alternative approach
to traditional solvers due to its flexibility and ease of incorporating
observed data. Despite the success of PINNs in accurately solving a wide
variety of PDEs, the method still requires improvements in terms of
computational efficiency. One possible improvement idea is to optimize the
generation of training point sets. Residual-based adaptive sampling and
quasi-uniform sampling approaches have been each applied to improve the
training effects of PINNs, respectively. To benefit from both methods, we
propose the Residual-based Adaptive Node Generation (RANG) approach for
efficient training of PINNs, which is based on a variable density nodal
distribution method for RBF-FD. The method is also enhanced by a memory
mechanism to further improve training stability. We conduct experiments on
three linear PDEs and three nonlinear PDEs with various node generation
methods, through which the accuracy and efficiency of the proposed method
compared to the predominant uniform sampling approach is verified numerically.
