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Science China Mathematics 2018年第10期61卷 1733-1740页

作者： weinan e Qingcan Wang Department of Mathematics and PACM Princeton University Princeton NJ 08544 USA Center for Big Data Research Peking University Beijing 100871 China Beijing Institute of Big Data Research Beijing 100871 China Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA

We prove that for analytic functions in low dimension, the convergence rate of the deep neural network approximation is exponential.

关键词： neural networks approximation theory analytic functions

The elastic Continuum Limit of the Tight Binding Model

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Chinese Annals of Mathematics,Series B 2007年第6期28卷 665-676页

作者： weinan e Jianfeng LU Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA

The authors consider the simplest quantum mechanics model of solids, the tight binding model, and prove that in the continuum limit, the energy of tight binding model converges to that of the continuum elasticity model obtained using Cauchy-Born rule. The technique in this paper is based mainly on spectral perturbation theory for large matrices.

关键词： Continuum limit elasticity theory Cauchy-Born rule Tight binding model

Convergence of a Stochastic Method for the Modeling of Polymeric Fluids

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Acta Mathematicae Applicatae Sinica 2002年第4期18卷 529-536页

作者： weinan e Department of Mathematics and PACM Princeton University Princeton NJ 08544 USA

We present a convergence analysis of a stochastic method for numerical modeling of complex fluids using Brownian configuration fields (BCF) for shear flows. The analysis takes into account the special structure of the stochastic partial differential equations for shear flows. We establish the optimal rate of convergence. We also analyze the nature of the error by providing its leading order asymptotics.

关键词： Brownian Configuration Fields (BCF) convergence analysis dumbbell model

Machine learning from a continuous viewpoint,I

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Science China Mathematics 2020年第11期63卷 2233-2266页

作者： weinan e Chao Ma Lei Wu Department of Mathematics Princeton UniversityPrincetonNJ 08544USA The Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Beijing Institute of Big Data Research Beijing 100871China

We present a continuous formulation of machine learning,as a problem in the calculus of variations and differential-integral equations,in the spirit of classical numerical *** demonstrate that conventional machine learning models and algorithms,such as the random feature model,the two-layer neural network model and the residual neural network model,can all be recovered(in a scaled form)as particular discretizations of different continuous *** also present examples of new models,such as the flow-based random feature model,and new algorithms,such as the smoothed particle method and spectral method,that arise naturally from this continuous *** discuss how the issues of generalization error and implicit regularization can be studied under this framework.

关键词： machine learning continuous formulation flow-based model gradient flow particle approximation

Optimization-Based String Method for Finding Minimum energy Path

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Communications in Computational Physics 2013年第7期14卷 265-275页

作者： Amit Samanta weinan e Program in Applied and Computational Mathematics Princeton UniversityPrincetonNew JerseyUSA Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNew JerseyUSA Beijing International Center for Mathematical Research Peking UniversityBeijingChina

We present an efficient algorithm for calculating the minimum energy path(MeP)and energy barriers between local minima on a multidimensional potential energy surface(PeS).Such paths play a central role in the understanding of transition pathways between metastable *** method relies on the original formulation of the string method[***.B,66,052301(2002)],*** evolve a smooth curve along a direction normal to the *** algorithm works by performing minimization steps on hyperplanes normal to the *** the problem of finding MeP on the PeS is remodeled as a set of constrained minimization *** provides the flexibility of using minimization algorithms faster than the steepest descent method used in the simplified string method[***.,126(16),164103(2007)].At the same time,it provides a more direct analog of the finite temperature string *** applicability of the algorithm is demonstrated using various examples.

关键词： Rare events string method minimum energy path dislocation nucleation

Mean field limit of a dynamical model of polymer systems

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Science China Mathematics 2013年第12期56卷 2591-2598页

作者： e weinan SHeN Hao School of Mathematical Sciences and BICMR Peking University Department of Mathematics and PACM Princeton UniversityPrinceton NJ 08544 USA Program in Applied and Computational Mathematics Princeton UniversityPrinceton NJ 08544 USA

This paper provides a mathematically rigorous foundation for self-consistent mean field theory of the polymeric physics. We study a new model for dynamics of mono-polymer systems. every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces. every two points on the same string or on two different strings also interact under a pairwise potential V. The dynamics of the system is described by a system of N coupled stochastic partial differential equations （SPDes）. We show that the mean field limit as N -＋ c~ of the system is a self-consistent McKean-Vlasov type equation, under suitable assumptions on the initial and boundary conditions and regularity of V. We also prove that both the SPDe system of the polymers and the mean field limit equation are well-posed.

关键词： polymers mean field limit stochastic PDes McKean-Vlasov equation

A comparative analysis of optimization and generalization properties of two-layer neural network and random feature models under gradient descent dynamics

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Science China Mathematics 2020年第7期63卷 1235-1258页

作者： weinan e Chao Ma Lei Wu Department of Mathematics Princeton UniversityPrincetonNJ 08544USA The Program in Applied and Computational Mathematics Princeton UniversityNJ 08544USA Beijing Institute of Big Data Research Beijing 100871China

A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are *** initialization schemes as well as general regimes for the network width and training data size are *** the overparametrized regime,it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the *** addition,it is proved that throughout the training process the functions represented by the neural network model are uniformly close to those of a kernel *** general values of the network width and training data size,sharp estimates of the generalization error are established for target functions in the appropriate reproducing kernel Hilbert space.

关键词： two-layer neural network random feature model Gram matrix generalization error

The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems

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Communications in Mathematics and Statistics 2018年第1期6卷 1-12页

作者： weinan e Bing Yu The Beijing Institute of Big Data Research BeijingChina Department of Mathematics and PACM Princeton UniversityPrincetonNJUSA School of Mathematical Sciences and BICMR Peking UniversityBeijingChina School of Mathematical Sciences Peking UniversityBeijingChina

We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential *** Deep Ritz Method is naturally nonlinear,naturally adaptive and has the potential to work in rather high *** framework is quite simple and fits well with the stochastic gradient descent method used in deep *** illustrate the method on several problems including some eigenvalue problems.

关键词： Deep Ritz Method Variational problems PDe eigenvalue problems

effective Maxwell equations from Time-dependent Density Functional Theory

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Acta Mathematica Sinica,english Series 2011年第2期27卷 339-368页

作者： weinan e Jianfeng LU Xu YANG Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA Beijing International Center for Mathematical Research and School of Mathematical Sciences Peking University Beijing 100871 P. R. China Department of Mathematics Courant Institute of Mathematical Sciences New York University New York NY 10012 USA

The behavior of interacting electrons in a perfect crystal under macroscopic external electric and magnetic fields is studied. effective Maxwell equations for the macroscopic electric and magnetic fields are derived starting from time-dependent density functional theory. effective permittivity and permeability coefficients are obtained.

关键词： Time-dependent density functional theory Maxwell equations effective permittivity electromagnetic fields

Asymptotic Analysis of Quantum Dynamics in Crystals: the Bloch-Wigner Transform, Bloch Dynamics and Berry Phase

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Acta Mathematicae Applicatae Sinica 2013年第3期29卷 465-476页

作者： weinan e Jian-feng LU Xu YANG Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University School of Mathematical Sciences Peking University

We study the semi-classical limit of the Schro¨dinger equation in a crystal in the presence of an external potential and magnetic field. We first introduce the Bloch-Wigner transform and derive the asymptotic equations governing this transform in the semi-classical setting. For the second part, we focus on the appearance of the Berry curvature terms in the asymptotic equations. These terms play a crucial role in many important physical phenomena such as the quantum Hall effect. We give a simple derivation of these terms in different settings using asymptotic analysis.

关键词： semiclassical limit Bloch-Wigner transform Bloch dynamics Berry phase asymptotic analysis