The neural network basis method for nonlinear partial differential equations and its Gauss–Newton optimizer

This paper focuses on designing the neural network basis method (NNBM) and its optimizer for solving systems of nonlinear partial differential equations (PDEs) in two/three dimensions. We first discretize the underlying problem in terms of a set of neural network basis functions from ELM-type methods combined with the collocation method, so as to produce a nonlinear least squares method (which is named as the NNBM specifically). Then, we elaborate on the implementation procedure of the Gauss–Newton method for the previous minimization problem. Moreover, it is proved by mathematical induction that this method is equivalent to the Newton-LLSQ method proposed by Dong and Li in 2021. Further, we use the method to numerically solve two typical nonlinear PDEs in mechanics. The numerical results show the proposed method is efficient and accurate if the exact solution is sufficiently smooth.