Physics-informed neural networks (PINNs) has been shown to be an effective tool for solving partial differential equations (PDEs). PINNs incorporate the PDEs residual into the loss function, seamlessly integrating it as part of the neural network architecture. This novel methodology has exhibited success in tackling a wide range of both forward and inverse PDE problems. However, a limitation of the first generation of PINNs is that they usually have limited accuracy even with many training points. Here, we propose two advanced methodologies: PINNs utilizing an loss function and parallel physics-informed neural networks (P-PINNs). The former involves modifying the loss function with the norm, while the latter entails solving the coupled Schrödinger–Korteweg–de Vries (Sch–KdV) equation separately with two parallel networks. These advancements are designed to enhance both accuracy and training efficiency. We have extensively tested these two advanced methods through a series of experiments and demonstrated the accuracy and effectiveness in approximation of the Sch–KdV system.

中文翻译：

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用于耦合薛定谔-KdV 方程数值近似的高级物理神经网络


物理信息神经网络 (PINN) 已被证明是求解偏微分方程 (PDE) 的有效工具。 PINN 将偏微分方程残差合并到损失函数中，将其无缝集成为神经网络架构的一部分。这种新颖的方法在解决各种正向和逆偏微分方程问题方面取得了成功。然而，第一代 PINN 的局限性在于，即使有很多训练点，它们的准确度通常也有限。在这里，我们提出了两种先进的方法：利用损失函数的 PINN 和并行物理信息神经网络 (P-PINN)。前者涉及用范数修改损失函数，而后者则需要用两个并行网络分别求解耦合的薛定谔-科特韦格-德弗里斯（Sch-KdV）方程。这些进步旨在提高准确性和训练效率。我们通过一系列实验对这两种先进方法进行了广泛的测试，并证明了 Sch-KdV 系统逼近的准确性和有效性。