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.

使用物理信息神经网络（PINN）来解决具有多尺度问题的新框架，通过重新构建损失函数并应用不同数量的幂运算到损失项上以使损失函数中的项具有相近的数量级，并且引入分组正则化策略以解决不同子域中变化显著的问题，该方法使得具有不同数量级的损失项可以同时优化，推动了 PINN 在多尺度问题中的应用。