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Number of Solitons Emerged in the Initial Profile of Shallow Water Using Convolutional Neural Networks

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Journal of Systems Science & Complexity 2024年第2期37卷 463-479页

作者： WANG Zhen CUI Shikun school of mathematical sciences Dalian University of TechnologyDalian 116024China school of mathematical sciences Beihang UniversityBeijing 100191China

The soliton resolution conjecture proposes that the initial value problem can evolve into a dispersion part and a soliton part.However,the problem of determining the number of solitons that form in a given initial profile remains unsolved,except for a few specific cases.In this paper,the authors use the deep learning method to predict the number of solitons in a given initial value of the Korteweg-de Vries(KdV)equation.By leveraging the analytical relationship between Asech^(2)(x)initial values and the number of solitons,the authors train a Convolutional Neural Network(CNN)that can accurately identify the soliton count from spatio-temporal data.The trained neural network is capable of predicting the number of solitons with other given initial values without any additional assistance.Through extensive calculations,the authors demonstrate the effectiveness and high performance of the proposed method.

关键词： Convolutional neural network deep learning method KdV equation soliton

Rational Solutions of First Order Algebraic Ordinary Differential Equations

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Journal of Systems Science & Complexity 2024年第2期37卷 567-580页

作者： FENG Shuang SHEN Liyong school of Physical and mathematical sciences Nanjing Tech UniversityNanjing 211816China school of mathematical sciences University of Chinese Academy of SciencesBeijing 100049China

Let f(t,y,y')=∑ _(i=0)^(n )a_(i)(t,y)y'^(i)=0 be an irreducible first order ordinary differential equation with polynomial coefficients.Eremenko in 1998 proved that there exists a constant C such that every rational solution of f(t,y,y')=0 is of degree not greater than C.Examples show that this degree bound C depends not only on the degrees of f in t,y,y' but also on the coefficients of f viewed as the polynomial in t,y,y'.In this paper,the authors show that if f satisfies deg(f,y)0,then the degree bound C only depends on the degrees of f in t,y,y',and furthermore we present an explicit expression for C in terms of the degrees of f in t,y,y'.

关键词： Degree bound first order AODE height rational solution

Shock Formation for 2D Isentropic Euler Equations with Self-similar Variables

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Chinese Annals of Mathematics,Series B 2024年第3期45卷 349-412页

作者： Wenze SU school of mathematical sciences Fudan University

The author studies the 2D isentropic Euler equations with the ideal gas law.He exhibits a set of smooth initial data that give rise to shock formation at a single point near the planar symmetry.These solutions to the 2D isentropic Euler equations are associated with non-zero vorticity at the shock and have uniform-in-time■-H■lder bound.Moreover,these point shocks are of self-similar type and share the same profile,which is a solution to the 2D self-similar Burgers equation.The proof of the solutions,following the3D construction of Buckmaster,Shkoller and Vicol (in 2023),is based on the stable 2D self-similar Burgers profile and the modulation method.

关键词： 2D isentropic Euler equations Shock formation Self-similar solution 35Q31 35L67 35B44

Geometry of holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains

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Science China Mathematics 2024年第8期67卷 1827-1864页

作者： Xiaoshu Ge Chunping Zhong school of mathematical sciences Xiamen University

In this paper, a class of holomorphic invariant metrics is introduced on the irreducible classical domains of types Ⅰ-Ⅳ, which are strongly pseudoconvex complex Finsler metrics in the strict sense of Abate and Patrizio(1994). These metrics are of particular interest in several complex variables since they are holomorphic invariant complex Finsler metrics found in the literature which enjoy good regularity as well as strong pseudoconvexity and can be explicitly expressed to admit differential geometry studies. They are,however, not necessarily Hermitian quadratic as the Bergman metrics. These metrics are explicitly constructed via deformation of the corresponding Bergman metric on the irreducible classical domains of types Ⅰ-Ⅳ,respectively, and they are all proved to be complete K?hler-Berwald metrics. They enjoy very similar curvature properties as those of the Bergman metric on the irreducible classical domains, i.e., their holomorphic sectional curvatures are bounded between two negative constants and their holomorphic bisectional curvatures are always nonpositive and bounded below by negative constants, respectively. From the viewpoint of complex analysis,these metrics are analogs of Bergman metrics in complex Finsler geometry which do not necessarily have Hermitian quadratic restrictions in the sense of Chern(1996).

关键词： holomorphic invariant metric Kähler-Berwald metric irreducible classical domains

Markovian Quadratic BSDEs with an Unbounded Sub-quadratic Growth

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Chinese Annals of Mathematics,Series B 2024年第3期45卷 441-462页

作者： Jingnan JU Shanjian TANG Shanghai Center for mathematical sciences Fudan University Department of Finance and Control sciences School of Mathematical Sciences Fudan University

This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations(BSDEs for short) with bounded terminal conditions. The generator is allowed to have an unbounded sub-quadratic growth in the second unknown variable z. The existence and uniqueness results are given to these BSDEs. As an application, an existence result is given to a system of coupled forward-backward stochastic differential equations with measurable coefficients.

关键词： Markovian BSDE Quadratic growth Unbounded sub-quadratic term coefficients Coupled FBSDE 60H10

Data-Driven Direct Adaptive Risk-Sensitive Control of Stochastic Systems

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Journal of Systems Science & Complexity 2024年第4期37卷 1446-1469页

作者： QIAO Nan LI Tao school of mathematical sciences East China Normal University Shanghai Key Laboratory of Pure Mathematics and mathematical Practice School of Mathematical SciencesEast China Normal University

The authors propose a data-driven direct adaptive control law based on the adaptive dynamic programming(ADP) algorithm for continuous-time stochastic linear systems with partially unknown system dynamics and infinite horizon quadratic risk-sensitive indices. The authors use online data of the system to iteratively solve the generalized algebraic Riccati equation(GARE) and to learn the optimal control law directly. For the case with measurable system noises, the authors show that the adaptive control law approximates the optimal control law as time goes on. For the case with unmeasurable system noises, the authors use the least-square solution calculated only from the measurable data instead of the real solution of the regression equation to iteratively solve the GARE. The authors also study the influences of the intensity of the system noises, the intensity of the exploration noises, the initial iterative matrix, and the sampling period on the convergence of the ADP algorithm.Finally, the authors present two numerical simulation examples to demonstrate the effectiveness of the proposed algorithms.

关键词： Adaptive dynamic programming direct adaptive control generalized algebraic Riccati equation risk-sensitive control

On Stability Discrimination of Limit Cycles for Piecewise Smooth Systems

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Acta Mathematica Sinica,English Series 2024年第7期40卷 1785-1803页

作者： Mao An HAN Xia Yu ZHOU school of mathematical sciences Zhejiang Normal University

This paper is concerned with the problem of stability discrimination of limit cycles for piecewise smooth systems. We first establish the Poincar′e map near a periodic orbit, and deduce the first order derivative of the map for general piecewise smooth systems on the plane. Then, we obtain a sufficient condition for determining the stability of limit cycles for these systems.

关键词： Piecewise smooth system limit cycle stability 34C07 34C23

Global well-posedness for 2D inhomogeneous asymmetric fluids with large initial data

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Science China Mathematics 2024年第3期67卷 527-556页

作者： Chenyin Qian Beibei He Ting Zhang school of mathematical sciences Zhejiang Normal UniversityJinhua 321004China school of mathematical sciences Zhejiang UniversityHangzhou 310058China

In this paper, by using time-weighted global estimates and the Lagrangian approach, we first investigate the global existence and uniqueness of the solution for the 2D inhomogeneous incompressible asymmetric fluids wi... 详细信息

关键词： inhomogeneous asymmetric fluids Littlewood-Paley theory Besov spaces global well-posedness

Partial regularity and nonlinear potential estimates for Stokes systems with super-quadratic growth

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Science China Mathematics 2024年第7期67卷 1525-1554页

作者： Lingwei Ma Zhenqiu Zhang school of mathematical sciences Tianjin Normal UniversityTianjin300387China school of mathematical sciences and LPMC Nankai UniversityTianjin300071China

This paper builds a bridge between partial regularity theory and nonlinear potential theory for the following generalized stationary Stokes system with super-quadratic growth and continuous coefficients:-div A(x,Du)+■π=f,div u=0,where Du is the symmetric part of the gradient■u.We first establish anε-regularity criterion involving both the excess functional of the symmetric gradient Du and Wolff potentials of the nonhomogeneous term f to guarantee the local vanishing mean oscillation(VMO)-regularity of Du in an open subset Ω_(u) of Ω with full measure.Such anε-regularity criterion leads to a pointwise Wolff potential estimate of Du,which immediately infers that Du is partially C^(0)-regular under appropriate assumptions.Finally,we give a local continuous modulus estimate of Du.

关键词： nonlinear Stokes system partial regularity Wolff potential estimates continuous modulus estimates