Hard constraint learning approaches with trainable influence functions for evolutionary equations

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.

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.