用于可积非局部方程的PT对称PINN：正问题与反问题
$PT$ Symmetric PINN for integrable nonlocal equations: Forward and inverse problems
AI总结：
本文提出PTS-PINN方法，通过将非局部项局部化并在损失函数中嵌入PT对称物理信息，有效求解可积非局部方程的正反问题，并能学习大时空尺度怪波。
摘要：
由于PT对称非局部方程包含PT对称的物理信息，将PT对称的物理信息嵌入到PINN的损失函数中是非常合适的，该方法被命名为PTS-PINN。对于一般的PT对称非局部方程，尤其是那些涉及非局部项求导的方程，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将面临一定挑战。这一问题可以通过PTS-PINN方法解决，具体体现在两个方面。首先，我们将方程的非局部项视为一个新的局部分量，从而使方程在此时实现耦合。通过这种方式，我们成功避免了在神经网络中对非局部项进行求导。另一方面，为了提高精度，我们进行了第二项改进，即将PT对称的物理信息嵌入到损失函数中。通过一系列独立的数值实验，我们评估了PTS-PINN在处理非局部非线性薛定谔（NLS）方程、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正问题和反问题中的有效性。数值实验表明，PTS-PINN具有良好的性能。特别是，PTS-PINN还展示了学习非局部方程大时空尺度怪波的非凡能力。
英文摘要：
Since the $PT$-symmetric nonlocal equations contain the physical information of the $PT$-symmetric, it is very appropriate to embed the physical information of the $PT$-symmetric into the loss function of PINN, named PTS-PINN. For general $PT$-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 $PT$-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 Schrödinger (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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
