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NSNO:Neumann Series Neural Operator for Solving Helmholtz Equations in Inhomogeneous Medium

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Journal of Systems Science & Complexity 2024年第2期37卷 413-440页

作者： CHEN Fukai LIU Ziyang LIN Guochang CHEN Junqing SHI Zuoqiang Department of mathematical Sciences Tsinghua UniversityBeijing 100084China Yau mathematical Sciences center Tsinghua UniversityBeijing 100084China Yanqi Lake Beijing Institute of mathematical Sciences and Applications Beijing 101408China

In this paper,the authors propose Neumann series neural operator(NSNO)to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to *** equation is a crucial partial differential equation(PDE)with applications in various scientific and engineering ***,efficient solver of Helmholtz equation is still a big challenge especially in the case of high ***,deep learning has shown great potential in solving PDEs especially in learning solution *** by Neumann series in Helmholtz equation,the authors design a novel network architecture in which U-Net is embedded inside to capture the multiscale *** experiments show that the proposed NSNO significantly outperforms the state-of-the-art FNO with at least 60%lower relative L^(2)-error,especially in the large wavenumber case,and has 50%lower computational cost and less data ***,NSNO can be used as the surrogate model in inverse scattering *** tests show that NSNO is able to give comparable results with traditional finite difference forward solver while the computational cost is reduced tremendously.

关键词： Helmholtz equation inverse problem neumann series neural network solution operator

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Trace inequalities, isocapacitary inequalities, and regularity of the complex Hessian equations

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Science China Mathematics 2024年第3期67卷 557-576页

作者： Jiaxiang Wang Bin Zhou Beijing International center for mathematical Research Peking UniversityBeijing 100871China School of mathematical Sciences Peking UniversityBeijing 100871China

In this paper, we study the relations between trace inequalities(Sobolev and Moser-Trudinger types), isocapacitary inequalities, and the regularity of the complex Hessian and Monge-Amp`ere equations with respect to a general nonnegative Borel measure. We obtain a quantitative characterization for these relations through the properties of the capacity-minimizing functions.

关键词： complex Monge-Ampère equations plurisubharmonic functions Sobolev inequality Moser-Trudinger inequality

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 UniversityShanghai 200433China Department of Finance and Control Sciences School of Mathematical SciencesFudan UniversityShanghai 200433China

This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations(BSDEs for short)with bounded terminal *** generator is allowed to have an unbounded sub-quadratic growth in the second unknown variable *** existence and uniqueness results are given to these *** 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 coeficients Coupled FBSDE

On the Bicanonical Map of Algebraic Threefolds of General Type

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Acta Mathematica Sinica,English Series 2024年

作者： Meng CHEN Chen JIANG Jian Shi YAN School of mathematical Sciences & Shanghai center for mathematical Sciences Fudan University Shanghai center for mathematical Sciences & School of mathematical Sciences Fudan University Department of Mathematics Northeastern University

We answer an open problem of the first author and Zhang(see Open Problem 6.4(3) in Math. Z. 258(2008), 565–585) and prove that, for any nonsingular projective 3-fold of general type with the geometric genus greater t...

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Oscillatory Dynamics of Heterogeneous Stem Cell Regeneration

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Communications on Applied Mathematics and Computation 2024年第1期6卷 431-453页

作者： Xiyin Liang Jinzhi Lei School of mathematical Sciences Center for Applied MathematicsTiangong UniversityTianjin300387China

Stem cell regeneration is an essential biological process in the maintenance of tissue homeostasis;dysregulation of stem cell regeneration may result in dynamic diseases that show oscillations in cell *** heterogeneity and plasticity are necessary for the dynamic equilibrium of tissue homeostasis;however,how these features may affect the oscillatory dynamics of the stem cell regeneration process remains poorly ***,based on a mathematical model of heterogeneous stem cell regeneration that includes cell heterogeneity and random transition of epigenetic states,we study the conditions to have oscillation solutions through bifurcation analysis and numerical *** results show various model system dynamics with changes in different parameters associated with kinetic rates,cellular heterogeneity,and *** show that introducing heterogeneity and plasticity to cells can result in oscillation dynamics,as we have seen in the homogeneous stem cell regeneration ***,increasing the cell heterogeneity and plasticity intends to maintain tissue homeostasis under certain *** current study is an initiatory investigation of how cell heterogeneity and plasticity may affect stem cell regeneration dynamics,and many questions remain to be further studied both biologically and mathematically.

关键词： Stem cell regeneration Heterogenous Hopf bifurcation Hematopoietic dynamical disease

A general mirror equivalence theorem for coset vertex operator algebras

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Science China Mathematics 2024年

作者： Robert McRae Yau mathematical Sciences center Tsinghua University

We prove a general mirror duality theorem for a subalgebra U of a simple conformal vertex algebra A and its commutant V=ComA（U）.Specifically,we assume that A≌⊕i∈IUi×Uias a U×V-module,where the U-modules Uiare simple and distinct and are objects of a semisimple braided ribbon category of U-modules,and the V-modules Viare semisimple and contained in a （not necessarily rigid） braided tensor category of *** also assume U=ComA（V）.Under these conditions,we construct a braid-reversed tensor equivalenceτ:UA→VA,where UAis the semisimple category of U-modules with simple objects Ui,i∈I,and VAis the category of V-modules whose objects are finite direct sums of *** particular,the V-modules Viare simple and distinct,and VAis a rigid tensor *** an application,we find a rigid semisimple tensor subcategory of modules for the Virasoro algebra at central charge 13+6p+6p-1,p∈Z 2,which is braided tensor equivalent to an abelian 3-cocycle twist of the category of finite-dimensional ***,the Virasoro vertex operator algebra at central charge 13+6p+6p-1is the PSL2（C）-fixed-point subalgebra of a simple conformal vertex algebra W（-p）,analogous to the realization of the Virasoro vertex operator algebra at central charge 13-6p-6p-1as the PSL2（C）-fixed-point subalgebra of the triplet algebra W（p）.

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Weak type(1, 1) of the Riesz transform on some direct product manifolds with exponential volume growth

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Science China Mathematics 2024年第7期67卷 1599-1622页

作者： Hong-Quan Li Jie-Xiang Zhu School of mathematical Sciences Fudan UniversityShanghai200433China center for Applied Mathematics Tianjin UniversityTianjin300072China

In this paper, we are concerned with the Riesz transform on the direct product manifold H^(n)× M,where H^(n) is the n-dimensional real hyperbolic space, and M is a connected complete non-compact Riemannian manifold satisfying the volume doubling property and generalized Gaussian or sub-Gaussian upper estimates for the heat kernel. We establish its weak type(1, 1) property. In addition, we obtain the weak type(1, 1) of the heat maximal operator in the same setting. Our arguments also work for a large class of direct product manifolds with exponential volume growth. Particularly, we provide a simpler proof of weak type(1, 1) boundedness of some operators considered in the work of Li et al.(2016).

关键词： Riesz transform heat kernel direct product real hyperbolic space

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UNIFORM ERROR BOUNDS OF A CONSERVATIVE COMPACT FINITE DIFFERENCE METHOD FOR THE QUANTUM ZAKHAROV SYSTEM IN THE SUBSONIC LIMIT REGIME

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Journal of Computational Mathematics 2024年第1期42卷 289-312页

作者： Gengen Zhang Chunmei Su School of Mathematics and Statistics Yunnan UniversityKunming 650504China Yau mathematical Sciences center Tsinghua UniversityBeijing 100084China

In this paper,we consider a uniformly accurate compact finite difference method to solve the quantum Zakharov system(QZS)with a dimensionless parameter 0<ε≤1,which is inversely proportional to the acoustic *** the subsonic limit regime,i.e.,when 0<ε?1,the solution of QZS propagates rapidly oscillatory initial layers in time,and this brings significant difficulties in devising numerical algorithm and establishing their error estimates,especially as 0<ε?*** solvability,the mass and energy conservation laws of the scheme are also *** on the cut-off technique and energy method,we rigorously analyze two independent error estimates for the well-prepared and ill-prepared initial data,respectively,which are uniform in both time and space forε∈(0,1]and optimal at the fourth order in *** results are reported to verify the error behavior.

关键词： Quantum Zakharov system Subsonic limit Compact finite difference method Uniformly accurate Error estimate

Two Applications of the ■■-Hodge Theory

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Chinese Annals of Mathematics,Series B 2024年第1期45卷 137-150页

作者： Dingchang WEI Shengmao ZHU center of mathematical Sciences Zhejiang UniversityHangzhou 310027China Department of Mathematics Zhejiang Normal UniversityJinhua 321004ZhejiangChina

Using Hodge theory and Banach fixed point theorem,Liu and Zhu developed a global method to deal with various problems in deformation *** this note,the authors generalize Liu-Zhu's method to treat two deformation problems for non-Kahler *** apply the ■■-Hodge theory to construct a deformation formula for(p,q)-forms of compact complex manifold under deformations,which can be used to study the Hodge number of complex manifold under *** the second part of this note,by using the ■■-Hodge theory,they provide a simple proof of the unobstructed deformation theorem for the non-Kahler Calabi-Yau ■■-manifolds.

关键词： Complex structures Deformations (p,q)-Forms Non-Kahler Calabi-Yaumanifolds