A deep learning framework for multi-scale models based on physics-informed neural networks

Physics-informed neural networks (PINN) combine deep neural networks with the solution of partial differential equations (PDEs), creating a new and promising research area for numerically solving PDEs. Faced with a class of multi-scale problems that include loss terms of different orders of magnitude in the loss function, it is challenging for standard PINN methods to obtain an available prediction. In this paper, we propose a new framework for solving multi-scale problems by reconstructing the loss function. The framework is based on the standard PINN method, and it modifies the loss function of the standard PINN method by applying different numbers of power operations to the loss terms of different magnitudes, so that the individual loss terms composing the loss function have approximately the same order of magnitude among themselves. In addition, we give a grouping regularization strategy, and this strategy can deal well with the problem which varies significantly in different subdomains. The proposed method enables loss terms with different magnitudes to be optimized simultaneously, and it advances the application of PINN for multi-scale problems.