非线性长短波系统中的深度学习方法及应用

基于神经网络的深度学习是近些年备受关注的一种新型技术手段，其在给定的初始数据基础上直接研究符合一定统计学分布规律的物理现象，已经在许多领域有着成功的应用。在非线性偏微分方程的研究方面，最近提出的物理信息神经网络是指训练求解监督学习任务(其遵守模型描述的物理规律)的网络，主要用来解决两类问题：数据驱动求解和数学驱动挖掘方程。本项目将利用物理信息神经网络的深度学习研究长短波系统，主要构建可积的YO型和Newell型长短波系统对应的神经网络，通过数值实验模拟探索长波和短波之间的相互作用机制和动力学现象，并对耦合长短波模型的可积分类进行数据驱动挖掘等。这些研究将拓展深度学习在耦合微分系统中的应用，揭示更为丰富的非线性动力学现象，同时将从科学计算角度为可积性研究提供新的手段和途径。

Deep learning on the basis of neural network is a new approach that has attracted much attention in recent years, which studies physical phenomena in accordance with the certain statistical distribution laws directly from the given initial data and has been successfully applied in many fields. For the nonlinear partial differential equations, the recently proposed physics-informed neural networks (PINNs) refers to the network trained to solve supervised learning tasks which follow the given physical laws described by the model, and it is used to solve two main classes of problems: data-driven solution and data-driven discovery of the equation. This project will study the long wave-short wave system via the deep learning on the PINNs, which mainly contains the construction of integrable Yajima-Oikawa and Newell long wave-short wave system’s neural network, the exploration of the interaction mechanism between the long wave and short wave and the dynamic phenomena through numerical simulation, and the data-driven discovery of the integrable classification of the coupled long wave-short wave model. These studies will extend the application of deep learning into coupled differential systems, reveal more abundant nonlinear dynamics phenomena, and provide the new techniques and approaches for the integrability research from the perspective of scientific computation.