报 告 人：陈勇 教授
报告题目：Lax pairs informed neural networks solving integrable systems
报告时间：2025年1月5日（周日）下午15:30
报告地点：静远楼1506学术报告厅
主办单位：数学与统计学院、数学研究院、科学技术研究院
报告人简介：
陈勇，华东师范大学数学学院教授，博士生导师，计算机理论所所长，上海市闵行区拔尖人才,长期从事非线性数学物理、可积系统、计算机代数及程序开发、可积深度学习算法，混沌理论、大气和海洋动力学等领域的研究工作。提出了一系列可以机械化实现非线性方程求解的方法,发展了李群理论并成功应用于大气海洋物理模型的研究。提出可积深度学习算法，开发出一系列可机械化实现的非线性发展方程的研究程序。已在 SCI 收录的国际学术期刊上发表 SCI 论文 300 余篇，引用 7000 余次。主持国家自然科学基金面上项目 4 项，国家自然科学基金重点项目 3 一参加人和项目负责人)、973 项目 1 项(骨干科学家)、国家自然科学基金长江团队项目 2 项(PD）。在AI方面的贡献：陈勇教授2019年开创性的提出了可积深度学习的概念和框架，带领其博士研究生团队，利用可积系统丰富的样本空间、对称、守恒律和Lax对等可积特性建立了一系列的深度学习算法，取得了突破性研究成果：在《Journal of Computer Physics》、《Physica D》、《Chaos》、《Chaos Solitons & Fractals》等国际重要学术刊物上发表了20余篇学术论文。研究成果获得了国内外学术界的认可，在国内外做了50余场专题报告，近期应美国布朗大学工程院院士Karniadakis教授之邀做了关于可积深度学习的报告。
报告摘要：
Lax pairs are one of the most important features of integrable system. In this talk, we propose the Lax pairs informed neural networks (LPINNs) tailored for integrable systems with Lax pairs by designing novel network architectures and loss functions, comprising LPINN-v1 and LPINN-v2. The most noteworthy advantage of LPINN-v1 is that it can transform the solving of complex integrable systems into the solving of a simpler Lax pairs to simplify the study of integrable systems, and it not only efficiently solves data-driven localized wave solutions, but also obtains spectral parameters and corresponding spectral functions in Lax pairs. On the basis of LPINN-v1, we additionally incorporate the compatibility condition/zero curvature equation of Lax pairs in LPINN-v2, its major advantage is the ability to solve and explore high-accuracy data-driven localized wave solutions and associated spectral problems for all integrable systems with Lax pairs. The numerical experiments in this work involve several important and classic low-dimensional and high-dimensional integrable systems, abundant localized wave solutions and their Lax pairs , including the soliton of the Korteweg-de Vries (KdV) equation and modified KdV equation, rogue wave solution of the nonlinear Schrodinger equation, kink solution of the sine-Gordon equation, non-smooth peakon solution of the Camassa-Holm equation and pulse solution of the short pulse equation, as well as the line-soliton solution Kadomtsev-Petviashvili equation and lump solution of high-dimensional KdV equation. The innovation of this work lies in the pioneering integration of Lax pairs informed of integrable systems into deep neural networks, thereby presenting a fresh methodology and pathway for investigating data-driven localized wave solutions and spectral problems of Lax pairs.
