Current PINN implementations with sequential learning strategies often experience some weaknesses, such as the failure to reproduce the previous training results when using a single network, the difficulty to strictly ensure continuity and smoothness at the time interval nodes when using multiple networks, and the increase in complexity and computational overhead. To overcome these shortcomings, we first investigate the extrapolation capability of the PINN method for time-dependent PDEs. Taking advantage of this extrapolation property, we generalize the training result obtained in a specific time subinterval to larger intervals by adding a correction term to the network parameters of the subinterval. The correction term is determined by further training with the sample points in the added subinterval. Secondly, by designing an extrapolation control function with special characteristics and combining it with a correction term, we construct a new neural network architecture whose network parameters are coupled with the time variable, which we call the extrapolation-driven network architecture. Based on this architecture, using a single neural network, we can obtain the overall PINN solution of the whole domain with the following two characteristics: (1) it completely inherits the local solution of the interval obtained from the previous training, (2) at the interval node, it strictly maintains the continuity and smoothness that the true solution has. The extrapolation-driven network architecture allows us to divide a large time domain into multiple subintervals and solve the time-dependent PDEs one by one in a chronological order. This training scheme respects the causality principle and effectively overcomes the difficulties of the conventional PINN method in solving the evolution equation on a large time domain. Numerical experiments verify the performance of our method.

翻译：当前采用顺序学习策略的PINN实现常存在一些弱点，例如使用单一网络时无法复现先前训练结果、使用多个网络时难以严格保证时间区间节点处的连续性与光滑性，以及复杂度和计算开销的增加。为克服这些缺陷，我们首先研究了PINN方法对时间依赖偏微分方程的外推能力。利用这一外推特性，我们通过向子区间网络参数添加修正项，将特定时间子区间获得的训练结果推广至更大区间。修正项通过新增子区间样本点的进一步训练确定。其次，通过设计具有特殊性质的外推控制函数并将其与修正项结合，我们构建了一种网络参数与时间变量耦合的新型神经网络架构，称为外推驱动网络架构。基于该架构，使用单一神经网络即可获得全域的整体PINN解，该解具有以下两个特征：（1）完全继承先前训练所得区间的局部解；（2）在区间节点处严格保持真实解所具有的连续性与光滑性。外推驱动网络架构允许我们将大时间域划分为多个子区间，并按时间顺序逐个求解时间依赖偏微分方程。这种训练方案遵循因果律原则，有效克服了传统PINN方法在大时间域上求解演化方程的困难。数值实验验证了本方法的性能。