报告题目:The International Mini Workshop on Nonlinear Science and its Applications

报告人：Yiming Ying；Baofeng Feng；Wenxiu Ma；Qianlian Wu；Haijing Xu；Longkun Tao；Li Tang；Yanli Zhang

报告时间：8：30-17：00

报告地点：浙江工业大学屏峰校区理A二楼活动中心

New Learning Framework with Average Top-k Loss

Yiming Ying

New York University-Albany

Abstract： The central task of Machine Learning (ML) is to seek models that minimize the loss over all training data, which is referred to as the aggregate loss constructing from the individual loss for each sample. While there are impressive advances in ML research on new algorithms and theory in designing the individual loss for the purpose of statistical robustness or efficient optimization, the construction of the aggregate loss from the individual losses has largely been overlooked.

In this talk, I will present a new learning framework of minimizing the average of k-largest individual losses, to which we refer as the Average Top-k model (ATk). This general learning framework instantiates both the average loss (k=n) for standard empirical risk minimization (ERM) in learning theory, and the recently proposed max loss (k=1).  The formulation of ATk is reformulated as a convex optimization problem which can be solved efficiently.  In particular, we show that ATk with the hinge loss as an individual loss is equivalent to the well-known nu-SVM in ML community, when the data is normalized. I will then present Learning Theory foundations for this new learning model, which gives theoretical guidances on how to choose the critical model parameter k.  Finally, some experimental results will be shown to validate the effectiveness of ATk.  This is joint work with Siwei Lyu and Yanbo Fan.

The dark and breather solutions for the Fokas-Lenells equation

Baofeng Feng

University of Texas-RGV

Abstract： In this talk, we will construct the dark and breather solutions to the Fokas-Lenells(FL)equation by the KP reduction method. The FL equation gives a set of three bilinear equations. However, one of these bilinear equations cannot be obtained directly from the bilinear equations of the KP-Toda hierarchy. Instead, we have to introduce an auxiliary tau function and start from four bilinear equations to reduce to this particular bilinear equations. The rogue wave solution can be constructed in a similar way.

Hirota direct method and soliton equations

Wenxiu Ma

University of South Florida

Abstract： We will talk about the Hirota direct method in soliton theory. The Hirota conditions for N-soliton solutions will be discussed and applications to soliton equations in both (1+1)-dimensions and (2+1)-dimensions will be presented. A few open questions regarding generalized bilinear equations will be given.

Practical Stability of Generalized Ordinary Differential Equations and Applications

Qianlian Wu

Zhejiang University of Technology

Abstract：In this talk, we establish some criteria for the practical stability of trivial solution for generalized ordinary differential equations. As an application, we prove our results concerning the practical stability of trivial solution for measure differential equations and dynamic equations on time scales. Here the conditions concerning the functions are more general than the classical ones.

Local and nonlocal reductions of two non-isospectral Ablowitz-Kaup-Newell-Segur equations and solutions

Haijing Xu

Zhejiang University of Technology

Abstract：In this talk, local and nonlocal reductions of two nonisospectral Ablowitz-Kaup-Newell-Segur equations, third order nonisospectral Ablowitz-Kaup-Newell-Segur equation and negative order nonisospectral Ablowitz-Kaup-Newell-Segur equation, are studied. By imposing constraint conditions on the double Wronskian solutions of the aforesaid nonisospectral Ablowitz-Kaup-Newell-Segur equations, various solutions for the local and nonlocal nonisospectral modified Korteweg-de Vries equation and local and nonlocal nonisospectral sine-Gordon equation are derived, including soliton solutions and Jordan block solutions. Dynamics of some obtained solutions are analyzed and illustrated by asymptotic analysis.

Model and prediction of COVID-19 with age structure susceptibility

Longkun Tao

Zhejiang University of Technology

Abstract：This paper mainly proposes two models. One is the SEAI model with age structure susceptibility and immigration of new individuals into the susceptible, exposed, symptomatic infectious and asymptomatic infectious classes. The global asymptotic stability of the unique positive equilibrium can be proved by a Lyapunov function. The second is an SEI model with an age-sensitive structure. The stability of the disease-free equilibrium point is discussed and the model don't have a positive equilibrium point.

Hardy and Hardy-Rellich type inequalities for Dunkl operators

Li Tang

Zhejiang University of Technology

Abstract：In this paper, we obtained the Dunkl analogy of classical L^p Hardy inequality for p > N + 2γ, where 2γ is the degree of weight function associated with Dunkl operators, and L^p Hardy inequalities with distant function in some G-invariant domains. Moreover we proved two Hardy-Rellich type inequalities for Dunkl operators.

Spatially discrete Hirota equation: rational and breather solution, gauge equivalence, and continuous limit

Yanli Zhang

Zhejiang University of Technology

Abstract：In this paper, we focus on an integrable spatially discrete Hirota equation. The new Lax pair, Darboux transformation, rational wave solution, rogue wave solution, breather solutions and gauge equivalent structure of the spatially discrete Hirota equation are investigated.