Global physics-informed neural networks (GPINNs): from local point-wise constraint to global nodal association

Recently, physics-informed neural networks (PINNs) and their variants have gained significant popularity as a scientific computing method for solving partial differential equations (PDEs), whereas accuracy is still its main shortcoming. Despite numerous development efforts, there is no literature demonstrating that these methods surpass classic numerical algorithms in solving the forward issue. In this paper, by analyzing the disparities between PINNs and traditional numerical methods based on mesh discretization, we investigate the underlying causes for the in adequate precision of PINNs and introduce a novel approach named global physics-informed neural networks (GPINNs). Inspired by the crucial concept of global nodal association in conventional numerical algorithms, GPINNs leverages the prior field distribution information from pre-trained PINNs to estimate the association weights between arbitrary nodes in space. GPINNs can not only be regarded as a meshless approach but also be demonstrated, both theoretically and in practical circumstances, to have the ability of second-order convergence when trained with equidistant nodes. Overall, GPINNs may be seen as an ideal approach to inheriting the merits of scientific machine learning (SciML) and conventional numerical computing, which also represent the first SciML algorithm to surpass standard numerical methods in terms of accuracy.