关于检验PINN的两种变体在验证广义非线性薛定谔方程局域波解中的预测能力
On examining the predictive capabilities of two variants of PINN in validating localised wave solutions in the generalized nonlinear Schrödinger equation
AI总结：
本研究提出强约束理论引导神经网络SCTgNN，结合PINN与TgNN的概念，用于研究广义非线性薛定谔方程的多种局域波解，实验验证两种模型均能有效预测复杂非线性系统行为。
摘要：
我们提出了一种名为强约束理论引导神经网络（Strongly Constrained Theory-Guided Neural Network，SCTgNN）的新型神经网络结构，用于研究广义非线性薛定谔（nonlinear Schrödinger，NLS）方程局域解的行为。该方程包含四个具有物理意义的非线性演化方程，分别为：（i）NLS方程、Hirota方程、Lakshmanan-Porsezian-Daniel（LPD）方程和五阶NLS方程。广义NLS方程展现出高达五阶的非线性效应，在多个物理领域中呈现出丰富且复杂的动力学特性。\nSCTgNN结合了物理信息神经网络（Physics-Informed Neural Network，PINN）和理论引导神经网络（Theory-Guided Neural Network，TgNN）模型的概念，旨在加深我们对复杂现象的理解，尤其是针对那些不符合常规模式的非线性系统。首先，我们采用TgNN方法预测广义NLS方程中的局域波行为，包括孤子、怪波和呼吸子。随后，我们使用SCTgNN预测上述局域解，并计算SCTgNN和TgNN在预测这三种局域解时的均方误差。研究结果表明，两种模型均能出色地解析复杂行为，并可在多种场景下提供预测结果。
英文摘要：
We introduce a novel neural network structure called Strongly Constrained Theory-Guided Neural Network (SCTgNN), to investigate the behaviours of the localized solutions of the generalized nonlinear Schrödinger (NLS) equation. This equation comprises four physically significant nonlinear evolution equations, namely, (i) NLS equation, Hirota equation Lakshmanan-Porsezian-Daniel (LPD) equation and fifth-order NLS equation. The generalized NLS equation demonstrates nonlinear effects up to quintic order, indicating rich and complex dynamics in various fields of physics. By combining concepts from the Physics-Informed Neural Network (PINN) and Theory-Guided Neural Network (TgNN) models, SCTgNN aims to enhance our understanding of complex phenomena, particularly within nonlinear systems that defy conventional patterns. To begin, we employ the TgNN method to predict the behaviours of localized waves, including solitons, rogue waves, and breathers, within the generalized NLS equation. We then use SCTgNN to predict the aforementioned localized solutions and calculate the mean square errors in both SCTgNN and TgNN in predicting these three localized solutions. Our findings reveal that both models excel in understanding complex behaviours and provide predictions across a wide variety of situations.
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