P T Symmetric PINN for integrable nonlocal equations: Forward and inverse problems
这是一篇使用 PINN 和 PT 对称解决可积非局部方程逆问题和正问题的论文。
我们先来看看这篇论文的摘要：
摘要
英文原文：
Since the P T -symmetric nonlocal equations contain the physical information of the P T -symmetric, it is very appropriate to embed the physical information of the P T -symmetric into the loss function of PINN, named PTS-PINN. For general P T -symmetric nonlocal equations, especially those equations involving the derivation of nonlocal terms, due to the existence of nonlocal terms, directly using the original PINN method to solve such nonlocal equations will face certain challenges. This problem can be solved by the PTS-PINN method which can be illustrated in two aspects. First, we treat the nonlocal term of the equation as a new local component, so that the equation is coupled at this time. In this way, we successfully avoid differentiating nonlocal terms in neural networks. On the other hand, in order to improve the accuracy, we make a second improvement, which is to embed the physical information of the P T -symmetric into the loss function. Through a series of independent numerical experiments, we evaluate the efficacy of PTS-PINN in tackling the forward and inverse problems for the nonlocal nonlinear Schr ̈odinger (NLS) equation, the nonlocal derivative NLS equation, the nonlocal (2+1)-dimensional NLS equation, and the nonlocal three wave interaction systems. The numerical experiments demonstrate that PTS-PINN has good performance. In particular, PTS-PINN has also demonstrated an extraordinary ability to learn large space-time scale rogue waves for nonlocal equations.
翻译：
由于 PT -对称非局部方程包含 PT -对称的物理信息,因此将 PT -对称的物理信息嵌入 PINN 的损耗函数 PTS-PINN 中非常合适。对于一般的 PT -对称非局部方程,尤其是涉及非局部项推导的方程,由于非局部项的存在,直接使用原始 PINN 方法求解此类非局部方程会面临一定挑战。这个问题可以通过 PTS-PINN 方法解决,方法可以通过两个方面说明。首先,我们将方程的非局部项视为新的局部分量,因此此时方程是耦合的。通过这种方式,我们成功地避免了在神经网络中区分非局部项。另一方面,为了提高准确性,我们做了第二项改进,即将 P、T 对称的物理信息嵌入到损失函数中。通过一系列独立数值实验,我们评估了 PTS-PINN 在解决非局部非线性施尔·奥丁格(NLS)、非局部导数 NLS 方程、非局部(2+1)维 NLS 方程以及非局部三波相互作用系统的前向和逆问题方面的效能。数值实验表明 PTS-PINN 表现良好。特别是,PTS-PINN 还展现出了学习非局部方程中大规模时空尺度流氓波的非凡能力。