【廿周年院庆学术报告70】 · 【和山数学论坛第363期】

一、报告题目：PINN based on Miura transformations and discovery of new localized wave solutions

二、报告人： 陈勇 教授

三、时 间：2023年4月15日(周日) 下午 15:40-16:30

四、地 点：闻理园A4-218

报告摘要：We put forth two physics-informed neural network (PINN) schemes based on Miura transformations. The novelty of this research is the incorporation of Miura transformation constraints into neural networks to solve nonlinear PDEs, which is an implementation method of unsupervised learning. The most noteworthy advantage of our method is that we can simply exploit the initial–boundary data of a solution of a certain nonlinear equation to obtain the data-driven solution of another evolution equation with the aid of Miura transformations and PINNs. In the process, the Miura transformation plays an indispensable role of a bridge between solutions of two separate equations. It is tailored to the inverse process of the Miura transformation and can overcome the difficulties in solving solutions based on the implicit expression. Moreover, two schemes are applied to perform abundant computational experiments to effectively reproduce dynamic behaviors of solutions for the well-known KdV equation and mKdV equation. Significantly, new data-driven solutions are successfully simulated and one of the most important results is the discovery of a new localized wave solution: kink-bell type solution of the defocusing mKdV equation and it has not been previously observed and reported to our knowledge. It provides a possibility for new types of numerical solutions by fully leveraging the many-to-one relationship between solutions before and after Miura transformations. Performance comparisons in different cases as well as advantages and disadvantages analysis of two schemes are also discussed. On the basis of the performance of two schemes and no free lunch theorem, they both have their own merits and thus more appropriate one should be chosen according to specific cases.

报告人简介：陈勇，华东师范大学，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，可积深度学习算法的研究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论文280篇. 发表论文的SCI 引用4000余篇次，其中ESI热点论文2篇、高被引论文10篇。SCI一区、二区文章90余篇。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2项(第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创新团队项目2项(PI)。

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