For multi-scale problems, the conventional physics-informed neural networks (PINNs) face some challenges in obtaining available predictions. In this paper, based on PINNs, we propose a practical deep learning framework for multi-scale problems by reconstructing the loss function and associating it with special neural network architectures. New PINN methods derived from the improved PINN framework differ from the conventional PINN method mainly in two aspects. First, the new methods use a novel loss function by modifying the standard loss function through a (grouping) regularization strategy. The regularization strategy implements a different power operation on each loss term so that all loss terms composing the loss function are of approximately the same order of magnitude, which makes all loss terms be optimized synchronously during the optimization process. Second, for the multi-frequency or high-frequency problems, in addition to using the modified loss function, new methods upgrade the neural network architecture from the common fully-connected neural network to special network architectures such as the Fourier feature architecture, and the integrated architecture developed by us. The combination of the above two techniques leads to a significant improvement in the computational accuracy of multi-scale problems. Several challenging numerical examples demonstrate the effectiveness of the proposed methods. The proposed methods not only significantly outperform the conventional PINN method in terms of computational efficiency and computational accuracy, but also compare favorably with the state-of-the-art methods in the recent literature. The improved PINN framework facilitates better application of PINNs to multi-scale problems.

翻译：针对多尺度问题，传统物理信息神经网络在获取有效预测方面面临挑战。本文基于PINN，通过重构损失函数并将其与特定神经网络架构相关联，提出了一种适用于多尺度问题的实际化深度学习框架。源于改进PINN框架的新型PINN方法与传统方法主要在两方面存在差异：首先，新方法采用一种基于（分组）正则化策略的改进损失函数，通过对标准损失函数进行修正而实现。该正则化策略对每个损失项施加不同的幂运算，使构成损失函数的所有损失项量级大致相同，从而在优化过程中实现所有损失项的同步优化。其次，针对多频率或高频率问题，除使用改进损失函数外，新方法将神经网络架构从常见的全连接网络升级为特殊架构，例如傅里叶特征架构及我们所提出的集成架构。上述两种技术的结合显著提升了多尺度问题的计算精度。多个具有挑战性的数值算例验证了所提出方法的有效性。该方法不仅在计算效率与计算精度上显著优于传统PINN方法，而且与近年文献中的最新方法相比也展现出竞争力。改进的PINN框架有助于PINN更好地应用于多尺度问题。