Yanzhong Yao

Yong Wang, Yanzhong Yao, Jiawei Guo et al.

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.

Yushi Zhang, Shuai Su, Yong Wang et al.

This paper develops a novel deep learning approach for solving evolutionary equations, which integrates sequential learning strategies with an enhanced hard constraint strategy featuring trainable parameters, addressing the low computational accuracy of standard Physics-Informed Neural Networks (PINNs) in large temporal domains.Sequential learning strategies divide a large temporal domain into multiple subintervals and solve them one by one in a chronological order, which naturally respects the principle of causality and improves the stability of the PINN solution. The improved hard constraint strategy strictly ensures the continuity and smoothness of the PINN solution at time interval nodes, and at the same time passes the information from the previous interval to the next interval, which avoids the incorrect/trivial solution at the position far from the initial time. Furthermore, by investigating the requirements of different types of equations on hard constraints, we design a novel influence function with trainable parameters for hard constraints, which provides theoretical and technical support for the effective implementations of hard constraint strategies, and significantly improves the universality and computational accuracy of our method. In addition, an adaptive time-domain partitioning algorithm is proposed, which plays an important role in the application of the proposed method as well as in the improvement of computational efficiency and accuracy. Numerical experiments verify the performance of the method. The data and code accompanying this paper are available at https://github.com/zhizhi4452/HCS.