The use of deep learning methods in scientific computing represents a potential paradigm shift in engineering problem solving. One of the most prominent developments is Physics-Informed Neural Networks (PINNs), in which neural networks are trained to satisfy partial differential equations (PDEs). While this method shows promise, the standard version has been shown to struggle in accurately predicting the dynamic behavior of time-dependent problems. To address this challenge, methods have been proposed that decompose the time domain into multiple segments, employing a distinct neural network in each segment and directly incorporating continuity between them in the loss function of the minimization problem. In this work we introduce a method to exactly enforce continuity between successive time segments via a solution ansatz. This hard constrained sequential PINN (HCS-PINN) method is simple to implement and eliminates the need for any loss terms associated with temporal continuity. The method is tested for a number of benchmark problems involving both linear and non-linear PDEs. Examples include various first order time dependent problems in which traditional PINNs struggle, namely advection, Allen–Cahn, and Korteweg–de Vries equations. Furthermore, second and third order time-dependent problems are demonstrated via wave and Jerky dynamics examples, respectively. Notably, the Jerky dynamics problem is chaotic, making the problem especially sensitive to temporal accuracy. The numerical experiments conducted with the proposed method demonstrated superior convergence and accuracy over both traditional PINNs and the soft-constrained counterparts.

中文翻译：

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在顺序物理信息神经网络中精确执行时间连续性


深度学习方法在科学计算中的使用代表了工程问题解决的潜在范式转变。最突出的发展之一是物理信息神经网络 (PINN)，其中神经网络经过训练以满足偏微分方程 (PDE)。虽然这种方法显示出希望，但标准版本已被证明难以准确预测与时间相关的问题的动态行为。为了解决这一挑战，人们提出了将时域分解为多个片段的方法，在每个片段中采用不同的神经网络，并将它们之间的连续性直接纳入最小化问题的损失函数中。在这项工作中，我们介绍了一种通过解决方案 ansatz 精确强制连续时间段之间的连续性的方法。这种硬约束顺序 PINN (HCS-PINN) 方法实现起来很简单，并且不需要与时间连续性相关的任何损失项。该方法针对涉及线性和非线性偏微分方程的许多基准问题进行了测试。例子包括传统 PINN 难以解决的各种一阶时间相关问题，即平流方程、Allen–Cahn 方程和 Korteweg–de Vries 方程。此外，二阶和三阶时间相关问题分别通过波和急动动力学示例进行了演示。值得注意的是，Jerky 动力学问题是混乱的，使得该问题对时间精度特别敏感。使用所提出的方法进行的数值实验表明，与传统 PINN 和软约束对应方法相比，具有更高的收敛性和准确性。