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.

翻译：本文提出了一种新颖的深度学习方法来求解演化方程，该方法将序列学习策略与具有可训练参数的增强型硬约束策略相结合，以解决标准物理信息神经网络（PINNs）在较大时间域内计算精度不足的问题。序列学习策略将大时间域划分为多个子区间，并按时间顺序逐一求解，这自然遵循了因果律原理，并提高了PINN解的稳定性。改进的硬约束策略严格保证了PINN解在时间区间节点处的连续性和光滑性，同时将前一区间的信息传递至后一区间，从而避免了远离初始时刻位置处出现错误解或平凡解的问题。此外，通过研究不同类型方程对硬约束的要求，我们为硬约束设计了一种具有可训练参数的新型影响函数，这为硬约束策略的有效实施提供了理论与技术支持，并显著提升了本方法的普适性与计算精度。另外，本文还提出了一种自适应时间域划分算法，该算法在所述方法的应用以及计算效率与精度的提升中均起到重要作用。数值实验验证了该方法的性能。本文附带的数据与代码可在 https://github.com/zhizhi4452/HCS 获取。