Deep learning with physics-informed neural networks (PINNs) has emerged as a highly popular and effective approach for solving partial differential equations(PDEs). In this paper, we first investigate the extrapolation capability of the PINN method for time-dependent PDEs. Taking advantage of this extrapolation property, we can generalize the training result obtained in the time subinterval to the large interval 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 the 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 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 proposed method.

翻译：物理信息神经网络（PINNs）的深度学习方法已成为求解偏微分方程（PDEs）的一种极为流行且有效的途径。本文首先研究了PINN方法在时间相关偏微分方程中的外推能力。利用这一外推特性，我们可通过在子区间网络参数中添加修正项，将子区间获得的训练结果推广至大区间。该修正项通过在新增子区间内使用采样点进行进一步训练来确定。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相结合，我们构建了一种新的神经网络架构，其网络参数与时间变量相耦合，我们称之为外推驱动网络架构。基于此架构，使用单一神经网络即可获得整个区域的完整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相关的偏微分方程。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。数值实验验证了所提方法的性能。