学术报告报告题目1：可积深度学习(Integrable Deep Learning )---PINN based on Miura transformations and discovery of new localized wave solutions
报 告人1：陈勇教授(华东师范大学)
报告时间：2023年4月8日（星期六）上午 8：10-8：50
报告题目2：Defocusing NLS equation with nonzero background: Painleve asymptotics in transition region
报 告人2：范恩贵 教授 (复旦大学 )
报告时间：2023年4月8日（星期六）上午 8：50-9：30
报告题目3：Gradient-optimized physics-informed neural networks (GOPINNs): a deep learning method for solving the complex modified KdV equation
报 告人3：李彪教授 (宁波大学)
报告时间：2023年4月8日（星期六）上午 9：30-10：00
报告题目4：可压Navier-Stokes方程中的可积 Ermakov结构、薛定谔关系、 Lax对和涡旋解
报 告人4：安红利教授 (南京农业大学)
报告时间：2023年4月8日（星期六）上午 10：20-10：50
报告题目5：Pseudopotentials and Nonlinear Waves
报 告人5：杨云青教授 (浙江科技学院)
报告时间：2023年4月8日（星期六）上午 10：50-11：20
报告地点：数学学院4318
邀 请人：凌黎明教授、张晓恩博士后
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数学学院
2023年4月7日
报告1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1：陈勇，华东师范大学数学科学学院教授、博士生导师，计算机理论所所长，上海市闵行区拔尖人才。从事可积系统、计算机代数及程序开发、可积深度学习算法的研究。提出了可积深度学习理论框架，发展了李群理论，开发了一系列可机械化实现的可积系统的研究程序。已在SCI收录的国际学术期刊上发表论文300余篇，引用7000余篇次。主持和参加了国家自然科学基金重点项目、973项目、国家自然科学基金面上项目，长江创新团队项目等。
报告2摘要：Consider to the Cauchy problem of the defocusing NLS equation with a nonzero background. With the Deift-Zhou, Dbar steepest descent method and double scaling limit, we obtain Painleve asymptotics in two transitions region.
报告人简介2：范恩贵，复旦大学教授、博士生导师，主要研究方向是孤立子理论、可积系统、Riemann-Hilbert问题、正交多项式和随机矩阵理论；主持国家自然科学基金、上海曙光计划等多项研究课题。 在 《Adv. Math. 》、 《SIAM J. Math. Anal.》、《J. Diff. Equ.》等国际重要期刊发表论文100余篇，被SCI刊源他引3000余次。应邀访问美国密苏里大学、日本京都大学等。上海市曙光学者、曾获教育部自然科学二等奖、上海市自然科学二等奖、复旦大学谷超豪数学奖。
报告3摘要：Recently, the physics-informed neural networks (PINNs) have received more and more attention because of their ability to solve nonlinear partial differential equations via only a small amount of data to quickly obtain data-driven solutions with high accuracy. However, despite their remarkable promise in the early stage, their unbalanced back-propagation gradient calculation leads to drastic oscillations in the gradient value during model training, which is prone to unstable prediction accuracy. Based on this, we develop a gradient optimization algorithm, which proposes a
new neural network structure and balances the interaction between different terms in the loss function during model training by means of gradient statistics, so that the newly proposed network architecture is more robust to gradient fluctuations. In this paper, we take the complex modified KdV equation as an example and use the gradient-optimized PINNs (GOPINNs) deep learning method to obtain data-driven rational wave solution and soliton molecules solution. Numerical results show that the GOPINNs method effectively smooths the gradient fluctuations and reproduces the dynamic behavior of these data-driven solutions better than the original PINNs method. In summary, our work provides new insights for optimizing the learning performance of neural networks and improves the prediction accuracy by a factor of 10 to 30 when solving the complex modified KdV equation.
报告人简介3：李彪宁波大学数学与统计学院教授，博导。主要从事非线性数学物理，可积系统及应用，深度学习等方面的研究。主持完成国家自然科学基金4项、省部级项目3项； 参与完成国家自然科学基金重点项目2项；现主持国家自然科学基金面上项目1项和参加国家自然科学基金重点项目1项。发表论文SCI论文100余篇，他引3千多次。
报告4摘要：In this paper, we investigate the 2+1-dimensional compressible Navier-Stokes equation with density-dependent viscosity coefficients. We introduce a novel power-type elliptic vortex ansatz and thereby obtain a finite-dimensional nonlinear dynamical system. The latter is shown to not only have an underlying integrable Ermakov structure of Hamiltonian type, but also admit a Lax pair formulation and associated stationary nonlinear Schr\odinger connection. In addition, we construct a class of elliptical vortex solutions termed pulsrodons corresponding to pulsating elliptic warm core eddies and discuss their dynamical behaviors. These solutions have recently found applications in geography, oceanic and atmospheric dynamics.
报告人简介4：安红利，南京农业大学教授，博士生导师，南京农业大学钟山学者—学术新秀，江苏高校“青蓝工程”优秀骨干青年教师。主要研究方向是数学物理方程、可积系统理论和混沌同步。目前在国际期刊《Phys. Review E》,《Stud. Appl. Math.》,《J. Phys. A》和《J. Math. Phys.》等发表学术论文近50篇。主持了国家自然科学基金面上项目、江苏省自然科学基金面上项目、留学人员科技活动择优资助项目（优秀类）和中央高校专项基金重点项目等9项课题。
报告5摘要：In this talk, we first introduce the pseudopotentials of integrable systems, from which some integrable properties and Darboux transformations are derived. Secondly, the concepts of pseudopotentials are generalized to the nonlocal and discrete integrable systems, form which various types of nonlinear localized wave solutions on constant background and their corresponding dynamical properties are investigated. Finally, nonlinear wave solutions on constant background are generalized to the nonconstant background, and some interesting nonlinear wave solutions including soliton, breather and rogue wave solutions on two types of periodic backgrounds are constructed, and the corresponding dynamics are studied.
报告人简介5：杨云青，教授，博士，硕士生导师，舟山市“111人才工程第三层次”人选。2011年博士毕业于华东师范大学，2013年至2015年于中国科学院数学与系统科学研究院数学机械化重点实验室从事博士后研究工作，2018年至2019年访问台湾大学、香港教育大学与日本大阪大学。主要从事可积系统、孤立子、复杂非线性波等非线性数学物理领域的科研工作。主持完成国家自然科学基金3项，浙江省自然科学基金2项，中国博士后基金1项，参与国家自然科学基金与浙江省自然科学基金多项。在《Journal of Mathematical Physics》、《Chaos》《Applied Mathematics Letters》与《Nonlinear Dynamics》等国内外学术期刊上发表论文30余篇，他引700余次。