Edit Profile (opens in new tab) E, Weinan Co-Author Distance Author ID: e.weinan Published as: E, Weinan; E, W.; É, Weinan more...less Further Spellings: 鄂维南 Homepage: https://web.math.princeton.edu/~weinan/ External Links: MGP · ORCID · Wikidata · Google Scholar · Math-Net.Ru · dblp · GND · IdRef Documents Indexed: 204 Publications since 1984, including 3 Books and 12 Additional arXiv Preprints 8 Contributions as Editor · 2 Further Contributions Software Indexed: 2 Packages Co-Authors: 130 Co-Authors with 191 Joint Publications 6,290 Co-Co-Authors all top 5 Co-Authors 23 single-authored 22 Vanden-Eijnden, Eric 21 Lu, Jianfeng 13 Han, Jiequn 11 Liu, Jianguo 11 Ren, Weiqing 10 Engquist, Bjorn E. 10 Zhang, Pingwen 8 Li, Qianxiao 8 Ma, Chao 7 Jentzen, Arnulf 7 Li, Tiejun 7 Lin, Lin 7 Shu, Chi-Wang 6 Li, Xiantao 6 Ying, Lexing 5 Ming, Pingbing 5 Sinaĭ, Yakov Grigor’evich 5 Wojtowytsch, Stephan 5 Wu, Lei 4 García-Cervera, Carlos J. 4 Long, Jihao 4 Zhou, Xiang 3 Abdulle, Assyr 3 Chen, Jingrun 3 Hu, Wei 3 Jiang, Song 3 Liu, Di 3 Shen, Hao 3 Tai, Cheng 3 Tang, Tao 3 Wang, Xiaoping 3 Yue, Xingye 3 Zhang, Linfeng 2 Avellaneda, Marco 2 Car, Roberto 2 Chen, Falai 2 Chen, Shanqin 2 Cheng, Xiuyuan 2 Constantin, Peter 2 Huang, Hongci 2 Hutzenthaler, Martin 2 Kruse, Thomas 2 Li, Dong 2 Ma, Zheng 2 Mattingly, Jonathan Christopher 2 Osher, Stanley Joel 2 Palffy-Muhoray, Peter 2 Rykov, Yurii G. 2 Wan, Xiaoliang 2 Wang, Jianchun 2 Wang, Qingcan 2 Xu, Zhi-Qin John 2 Yang, Chao 2 Yang, Xu 2 Yip, Nung Kwan 2 Yu, Haijun 2 Zang, Yaohua 2 Zhang, Xuanxi 2 Zhang, Yaoyu 1 Bai, Zhaojun 1 Beck, Christian 1 Bertozzi, Andrea Louise 1 Burger, Martin 1 Caffarelli, Luis Ángel 1 Caflisch, Russ 1 Cai, Zhiqiang 1 Chan, Tony Fan-Cheong 1 Chazelle, Bernard 1 Chen, Long 1 Chen, Nan 1 Chen, Shiyi 1 Chen, Weijie 1 Chen, Zhengting 1 Chen, Zhiming 1 Cheng, Li-Tien 1 Chi, Xurong 1 Dai, Yu-Hong 1 Fefferman, Charles Louis 1 Friesecke, Gero 1 Gao, Weiguo 1 Gimbutas, Zydrunas 1 Han, Weimin 1 Hang, Liangkai 1 Hou, Thomas Yizhao 1 Hu, Dan 1 Huang, Zhongyi 1 Keller, Gerta 1 Khanin, Konstantin M. 1 Khouider, Boualem 1 Klein, Rupert 1 Kohn, Robert Vita 1 Kramer, Peter Roland 1 Li, Shangzhi 1 Lin, Ling 1 Lions, Pierre-Louis 1 Liu, Yunxian 1 Luo, Tao 1 Luo, Yixin 1 Malek-Madani, Reza 1 Mazel, Alexander E. ...and 51 more Co-Authors all top 5 Serials 22 Journal of Computational Physics 13 Communications on Pure and Applied Mathematics 13 Communications in Mathematical Sciences 9 Communications in Computational Physics 7 Physics of Fluids 6 Communications in Mathematical Physics 6 Journal of Statistical Physics 6 Science China. Mathematics 5 Nonlinearity 5 SIAM Journal on Numerical Analysis 5 Proceedings of the National Academy of Sciences of the United States of America 4 Journal of Computational Mathematics 4 Physica D 4 Notices of the American Mathematical Society 4 Journal of Machine Learning Research (JMLR) 4 Communications in Mathematics and Statistics 4 Research in the Mathematical Sciences 3 Archive for Rational Mechanics and Analysis 3 Acta Mathematicae Applicatae Sinica. English Series 3 Methods and Applications of Analysis 3 Discrete and Continuous Dynamical Systems 3 Annals of Applied Mathematics 2 Journal of Fluid Mechanics 2 Russian Mathematical Surveys 2 Mathematics of Computation 2 Chinese Annals of Mathematics. Series B 2 Journal of Nonlinear Science 2 Acta Mathematica Sinica. English Series 1 Computer Physics Communications 1 International Journal for Numerical Methods in Fluids 1 Journal of Mathematical Physics 1 Journal of the Mechanics and Physics of Solids 1 Physics of Fluids, A 1 ACM Transactions on Mathematical Software 1 Memoirs of the American Mathematical Society 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Studies in Applied Mathematics 1 Mathematica Numerica Sinica 1 Applied Numerical Mathematics 1 Constructive Approximation 1 Communications in Applied Numerical Methods 1 Journal of the American Mathematical Society 1 Asymptotic Analysis 1 Journal of Scientific Computing 1 European Journal of Applied Mathematics 1 Communications in Partial Differential Equations 1 Journal of Non-Newtonian Fluid Mechanics 1 SIAM Journal on Applied Mathematics 1 SIAM Journal on Scientific Computing 1 Calculus of Variations and Partial Differential Equations 1 European Series in Applied and Industrial Mathematics (ESAIM): Proceedings 1 Annals of Mathematics. Second Series 1 Discrete and Continuous Dynamical Systems. Series B 1 Physical Review E. Third Series 1 Multiscale Modeling & Simulation 1 Acta Numerica 1 Oberwolfach Reports 1 Graduate Studies in Mathematics 1 IAS/Park City Mathematics Series 1 East Asian Journal on Applied Mathematics 1 The Journal of Chemical Physics 1 Pure and Applied Functional Analysis 1 Communications on Applied Mathematics and Computation 1 SN Partial Differential Equations and Applications all top 5 Fields 71 Numerical analysis (65-XX) 70 Partial differential equations (35-XX) 68 Fluid mechanics (76-XX) 43 Statistical mechanics, structure of matter (82-XX) 35 Computer science (68-XX) 29 Probability theory and stochastic processes (60-XX) 27 Mechanics of deformable solids (74-XX) 14 Quantum theory (81-XX) 13 Dynamical systems and ergodic theory (37-XX) 12 General and overarching topics; collections (00-XX) 10 Ordinary differential equations (34-XX) 10 Calculus of variations and optimal control; optimization (49-XX) 9 Statistics (62-XX) 6 History and biography (01-XX) 6 Functional analysis (46-XX) 5 Approximations and expansions (41-XX) 5 Biology and other natural sciences (92-XX) 4 Optics, electromagnetic theory (78-XX) 4 Operations research, mathematical programming (90-XX) 4 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 Mechanics of particles and systems (70-XX) 3 Classical thermodynamics, heat transfer (80-XX) 2 Real functions (26-XX) 2 Systems theory; control (93-XX) 2 Mathematics education (97-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Potential theory (31-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Integral equations (45-XX) 1 Operator theory (47-XX) 1 Algebraic topology (55-XX) 1 Geophysics (86-XX) 1 Information and communication theory, circuits (94-XX) Publications by Year all cited Publications top 5 cited Publications Citations contained in zbMATH Open 182 Publications have been cited 7,112 times in 4,741 Documents Cited by ▼ Year ▼ Solving high-dimensional partial differential equations using deep learning. Zbl 1416.35137 Han, Jiequn; Jentzen, Arnulf; E, Weinan 470 2018 The heterogeneous multiscale methods. Zbl 1093.35012 E, Weinan; Engquist, Bjorn 459 2003 The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Zbl 1392.35306 E, Weinan; Yu, Bing 420 2018 Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Zbl 1382.65016 E, Weinan; Han, Jiequn; Jentzen, Arnulf 294 2017 Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Zbl 0818.35085 Constantin, Peter; E, Weinan; Titi, Edriss S. 289 1994 Heterogeneous multiscale methods: a review. Zbl 1164.65496 E, Weinan; Engquist, Bjorn; Li, Xiantao; Ren, Weiqing; Vanden-Eijnden, Eric 222 2007 Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Zbl 0852.35097 E, Weinan; Rykov, Yu. G.; Sinai, Ya. G. 209 1996 The heterogeneous multiscale method. Zbl 1255.65224 Abdulle, Assyr; E, Weinan; Engquist, Björn; Vanden-Eijnden, Eric 198 2012 Blowup of solutions of the unsteady Prandtl’s equation. Zbl 0908.35099 E, Weinan; Engquist, Bjorn 146 1997 Invariant measures for Burgers equation with stochastic forcing. Zbl 0972.35196 E, Weinan; Khanin, K.; Mazel, A.; Sinai, Ya. 143 2000 A proposal on machine learning via dynamical systems. Zbl 1380.37154 E, Weinan 129 2017 Minimum action method for the study of rare events. Zbl 1050.60068 E, Weinan; Ren, Weiqing; Vanden-Eijnden, Eric 128 2004 Analysis of the heterogeneous multiscale method for elliptic homogenization problems. Zbl 1060.65118 E, Weinan; Ming, Pingbing; Zhang, Pingwen 128 2005 Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. Zbl 1442.91116 Beck, Christian; E, Weinan; Jentzen, Arnulf 126 2019 Projection method. I: Convergence and numerical boundary layers. Zbl 0842.76052 E, Weinan; Liu, Jian-Guo 122 1995 Principles of multiscale modeling. Zbl 1238.00010 E, Weinan 116 2011 Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation. Zbl 0961.35101 E, Weinan 108 2000 Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Zbl 0994.60065 E, Weinan; Mattingly, J. C.; Sinai, Ya. 97 2001 Cauchy-Born rule and the stability of crystalline solids: Static problems. Zbl 1106.74019 E, Weinan; Ming, Pingbing 84 2007 Analysis of multiscale methods for stochastic differential equations. Zbl 1080.60060 E, Weinan; Liu, Di; Vanden-Eijnden, Eric 83 2005 Small-scale structures in Boussinesq convection. Zbl 0822.76087 E, Weinan; Shu, Chi-Wang 81 1994 Vorticity boundary condition and related isssues for finite difference schemes. Zbl 0847.76050 E, Weinan; Liu, Jian-Guo 79 1996 On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations. Zbl 1418.65149 E, Weinan; Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas 71 2019 Boundary conditions for the moving contact line problem. Zbl 1146.76513 Ren, Weiqing; E, Weinan 65 2007 Numerical methods for the Landau–Lifshitz equation. Zbl 0988.65079 E, Weinan; Wang, Xiao-Ping 65 2000 Essentially compact schemes for unsteady viscous incompressible flows. Zbl 0853.76045 E, Weinan; Liu, Jian-Guo 64 1996 Gauge method for viscous incompressible flows. Zbl 1160.76329 E, Weinan; Liu, Jian-Guo 63 2003 Multiscale modeling and computation. Zbl 1032.65013 E, Weinan; Engquist, Bjorn 63 2003 Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity. Zbl 0814.34039 E, Weinan 61 1994 Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite-dimensional approximation. Zbl 1024.76012 E, Weinan; Mattingly, Jonathan C. 57 2001 Towards a theory of transition paths. Zbl 1101.82016 E, Weinan; Vanden-Eijnden, Eric 56 2006 Homogenization of linear and nonlinear transport equations. Zbl 0794.35014 E, Weinan 53 1992 A continuum and molecular dynamics hybrid method for micro- and nano-fluid flow. Zbl 1059.76060 Nie, X. B.; Chen, S. Y.; E, Weinan; Robbins, M. O. 48 2004 A mean-field optimal control formulation of deep learning. Zbl 1421.49021 E, Weinan; Han, Jiequn; Li, Qianxiao 48 2019 The gentlest ascent dynamics. Zbl 1237.37052 E, Weinan; Zhou, Xiang 47 2011 A Gauss-Seidel projection method for micromagnetics simulations. Zbl 1012.82002 Wang, Xiao-Ping; García-Cervera, Carlos J.; E, Weinan 47 2001 Maximum principle based algorithms for deep learning. Zbl 1467.68156 Li, Qianxiao; Chen, Long; Tai, Cheng; E, Weinan 46 2018 A dynamic atomistic–continuum method for the simulation of crystalline materials. Zbl 1013.82022 E, Weinan; Huang, Zhongyi 45 2002 Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning. Zbl 1490.60202 E, Weinan; Han, Jiequn; Jentzen, Arnulf 44 2022 On the crystallization of 2D hexagonal lattices. Zbl 1180.82191 E, Weinan; Li, Dong 43 2009 Well-posedness for the dumbbell model of polymeric fluids. Zbl 1060.35169 E, Weinan; Li, Tiejun; Zhang, Pingwen 42 2004 Exponential convergence of the deep neural network approximation for analytic functions. Zbl 1475.65007 E, Weinan; Wang, Qingcan 42 2018 A priori estimates of the population risk for two-layer neural networks. Zbl 1427.68277 E, Weinan; Ma, Chao; Wu, Lei 41 2019 Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. Zbl 1162.80003 E, Weinan; Liu, Di; Vanden-Eijnden, Eric 40 2007 The local microscale problem in the multiscale modeling of strongly heterogeneous media: Effects of boundary conditions and cell size. Zbl 1158.74541 Yue, Xingye; E, Weinan 40 2007 Analysis of the heterogeneous multiscale method for ordinary differential equations. Zbl 1088.65552 E, Weinan 40 2003 Finite difference heterogeneous multi-scale method for homogenization problems. Zbl 1034.65067 Abdulle, Assyr; E, Weinan 40 2003 Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. Zbl 1143.76541 Ren, Weiqing; E, Weinan 38 2005 Multiscale modeling of the dynamics of solids at finite temperature. Zbl 1120.74313 Li, Xiantao; E, Weinan 37 2005 Continuum models for the contact line problem. Zbl 1308.76082 Ren, Weiqing; Hu, Dan; E, Weinan 37 2010 Solving many-electron Schrödinger equation using deep neural networks. Zbl 1457.81010 Han, Jiequn; Zhang, Linfeng; E, Weinan 35 2019 Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework. I: Total energy calculation. Zbl 1251.82008 Lin, Lin; Lu, Jianfeng; Ying, Lexing; E, Weinan 34 2012 Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems. Zbl 1182.65072 Lin, Lin; Lu, Jianfeng; Ying, Lexing; Car, Roberto; E, Weinan 34 2009 The Barron space and the flow-induced function spaces for neural network models. Zbl 1490.65020 E, Weinan; Ma, Chao; Wu, Lei 34 2022 Aubry-Mather theory and periodic solutions of the forced Burgers equation. Zbl 0916.35099 E, Weinan 30 1999 A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow. Zbl 0790.76055 E, Weinan; Shu, Chi-Wang 30 1994 Statistical properties of shocks in Burgers turbulence. Zbl 0844.35144 Avellaneda, Marco; E, Weinan 29 1995 Self-induced stochastic resonance in excitable systems. Zbl 1109.34040 Muratov, Cyrill B.; Vanden-Eijnden, Eric; E, Weinan 28 2005 Stochastic modified equations and dynamics of stochastic gradient algorithms. I: Mathematical foundations. Zbl 1484.62106 Li, Qianxiao; Tai, Cheng; E, Weinan 28 2019 SelInv – an algorithm for selected inversion of a sparse symmetric matrix. Zbl 1365.65069 Lin, Lin; Yang, Chao; Meza, Juan C.; Lu, Jianfeng; Ying, Lexing; E, Weinan 28 2011 The heterogeneous multi-scale method for homogenization problems. Zbl 1086.65521 E, Weinan; Engquist, Björn 27 2005 Uniformly accurate machine learning-based hydrodynamic models for kinetic equations. Zbl 1431.76100 Han, Jiequn; Ma, Chao; Ma, Zheng; E, Weinan 27 2019 Multilevel Picard iterations for solving smooth semilinear parabolic heat equations. Zbl 1476.65273 E, Weinan; Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas 26 2021 Cauchy-Born rule and the stability of crystalline solids: dynamic problems. Zbl 1138.35063 E, Weinan; Ming, Pingbing 25 2007 Effectiveness of implicit methods for stiff stochastic differential equations. Zbl 1199.65010 Li, Tiejun; Abdulle, Assyr; E, Weinan 25 2008 Projection method. III: Spatial discretization on the staggered grid. Zbl 1058.76043 E, Weinan; Liu, Jian-Guo 25 2002 Machine learning from a continuous viewpoint. I. Zbl 1472.68136 E, Weinan; Ma, Chao; Wu, Lei 25 2020 Projection method. II: Godunov-Ryabenki analysis. Zbl 0920.76056 E, Weinan; Liu, Jian-Guo 24 1996 Study of noise-induced transitions in the Lorenz system using the minimum action method. Zbl 1202.34104 Zhou, Xiang; E, Weinan 24 2010 Finite difference methods for 3D viscous incompressible flows in the vorticity-vector potential formulation on nonstaggered grids. Zbl 0901.76046 E, Weinan; Liu, Jian-Guo 23 1997 Nonlinear continuum theory of smectic-\(A\) liquid crystals. Zbl 0891.76009 E, Weinan 22 1997 Generalized flows, intrinsic stochasticity, and turbulent transport. Zbl 0967.76038 E, Weinan; Vanden Eijnden, Eric 22 2000 Convergence of spectral methods for Burgers’ equation. Zbl 0764.65057 E, Weinan 22 1992 Machine learning and computational mathematics. Zbl 1473.00035 E, Weinan 22 2020 The electronic structure of smoothly deformed crystals: Wannier functions and the Cauchy-Born Rule. Zbl 1253.74025 E, Weinan; Lu, Jianfeng 21 2011 Derivation of continuum models for the moving contact line problem based on thermodynamic principles. Zbl 1414.76019 Ren, Weiqing; E, Weinan 21 2011 A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit. Zbl 1131.76010 E, Weinan; Zhang, Pingwen 21 2006 Some recent progress in multiscale modeling. Zbl 1419.74252 E, Weinan; Li, Xiantao; Vanden-Eijnden, Eric 21 2004 Convergence of Fourier methods for the Navier-Stokes equations. Zbl 0776.76021 E, Weinan 21 1993 A general strategy for designing seamless multiscale methods. Zbl 1280.76038 E, Weinan; Ren, Weiqing; Vanden-Eijnden, Eric 21 2009 Model reduction with memory and the machine learning of dynamical systems. Zbl 1473.35450 Ma, Chao; Wang, Jianchun; E, Weinan 21 2019 Deep potential: a general representation of a many-body potential energy surface. Zbl 1488.81038 Han, Jiequn; Zhang, Linfeng; Car, Roberto; E, Weinan 19 2018 Study of the noise-induced transition and the exploration of the phase space for the Kuramoto-Sivashinsky equation using the minimum action method. Zbl 1185.82044 Wan, X.; Zhou, X.; E, W. 18 2010 The initial-value problem for measure-valued solutions of a canonical \(2 \times 2\) system with linearly degenerate fields. Zbl 0784.35059 E, Weinan; Kohn, Robert V. 18 1991 The Kohn-Sham equation for deformed crystals. Zbl 1435.74001 E, Weinan; Lu, Jianfeng 17 2013 Finite difference schemes for incompressible flows in the velocity-impulse density formulation. Zbl 0870.76048 E, Weinan; Liu, Jian-Guo 17 1997 Homogenization of scalar conservation laws with oscillatory forcing terms. Zbl 0755.35068 E, Weinan 17 1992 Energy landscapes and rare events. Zbl 1023.65004 E, Weinan; Ren, Weiqing; Vanden-Eijnden, Eric 17 2002 Metastability, conformation dynamics, and transition pathways in complex systems. Zbl 1141.82312 E, Weinan; Vanden-Eijnden, Eric 16 2004 Continuum theory of epitaxial crystal growth. I. Zbl 1074.76504 E, Weinan; Yip, Nung Kwan 16 2001 Electronic structure of smoothly deformed crystals: Cauchy-Born rule for the nonlinear tight-binding model. Zbl 1277.81117 E, Weinan; Lu, Jianfeng 15 2010 Statistical theory for the stochastic Burgers equation in the inviscid limit. Zbl 1037.76013 E, Weinan; Vanden Eijnden, Eric 15 2000 Gibbsian dynamics and invariant measures for stochastic dissipative PDEs. Zbl 1023.60057 E, Weinan; Liu, Di 15 2002 Large time behavior and homogenization of solutions of two-dimensional conservation laws. Zbl 0797.35114 Engquist, Bjorn; E, Weinan 15 1993 Contact line dynamics on heterogeneous surfaces. Zbl 1308.76212 Ren, Weiqing; E, Weinan 15 2011 Representation formulas and pointwise properties for Barron functions. Zbl 1482.41013 E, Weinan; Wojtowytsch, Stephan 15 2022 A comparative analysis of optimization and generalization properties of two-layer neural network and random feature models under gradient descent dynamics. Zbl 1453.68163 E, Weinan; Ma, Chao; Wu, Lei 15 2020 A fast parallel algorithm for selected inversion of structured sparse matrices with application to 2D electronic structure calculations. Zbl 1230.65039 Lin, Lin; Yang, Chao; Lu, Jianfeng; Ying, Lexing; E, Weinan 14 2011 Simple finite element method in vorticity formulation for incompressible flows. Zbl 0977.76048 Liu, Jian-Guo; E, Weinan 14 2001 Thermodynamically driven incompressible fluid mixtures. Zbl 1110.76307 Otto, Felix; E, Weinan 14 1997 The random feature method for time-dependent problems. Zbl 1527.65084 Chen, Jing-Run; E, Weinan; Luo, Yi-Xin 2 2023 Dynamical systems and optimal control approach to deep learning. Zbl 1507.68277 E, Weinan; Han, Jiequn; Li, Qianxiao 2 2023 Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning. Zbl 1490.60202 E, Weinan; Han, Jiequn; Jentzen, Arnulf 44 2022 The Barron space and the flow-induced function spaces for neural network models. Zbl 1490.65020 E, Weinan; Ma, Chao; Wu, Lei 34 2022 Representation formulas and pointwise properties for Barron functions. Zbl 1482.41013 E, Weinan; Wojtowytsch, Stephan 15 2022 MOD-Net: a machine learning approach via model-operator-data network for solving PDEs. Zbl 1497.35088 Zhang, Lulu; Luo, Tao; Zhang, Yaoyu; E, Weinan; Xu, Zhi-Qin John; Ma, Zheng 9 2022 Approximation and optimization theory for linear continuous-time recurrent neural networks. Zbl 07625195 Li, Zhong; Han, Jiequn; E, Weinan; Li, Qianxiao 4 2022 Generalization error of GAN from the discriminator’s perspective. Zbl 1527.62075 Yang, Hongkang; E, Weinan 1 2022 Multilevel Picard iterations for solving smooth semilinear parabolic heat equations. Zbl 1476.65273 E, Weinan; Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas 26 2021 Kolmogorov width decay and poor approximators in machine learning: shallow neural networks, random feature models and neural tangent kernels. Zbl 1538.68042 E, Weinan; Wojtowytsch, Stephan 9 2021 The dawning of a new era in applied mathematics. Zbl 1471.00008 E, Weinan 8 2021 Connections between deep learning and partial differential equations. Zbl 1479.35002 Burger, M.; E, W.; Ruthotto, L.; Osher, S. J. 2 2021 Machine learning from a continuous viewpoint. I. Zbl 1472.68136 E, Weinan; Ma, Chao; Wu, Lei 25 2020 Machine learning and computational mathematics. Zbl 1473.00035 E, Weinan 22 2020 A comparative analysis of optimization and generalization properties of two-layer neural network and random feature models under gradient descent dynamics. Zbl 1453.68163 E, Weinan; Ma, Chao; Wu, Lei 15 2020 Rademacher complexity and the generalization error of residual networks. Zbl 1467.62158 E, Weinan; Ma, Chao; Wang, Qingcan 4 2020 DP-GEN: a concurrent learning platform for the generation of reliable deep learning based potential energy models. Zbl 1522.81011 Zhang, Yuzhi; Wang, Haidi; Chen, Weijie; Zeng, Jinzhe; Zhang, Linfeng; Wang, Han; E, Weinan 3 2020 The generalization error of the minimum-norm solutions for over-parameterized neural networks. Zbl 1524.68273 E, Weinan; Ma, Chao; Wu, Lei 1 2020 Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. Zbl 1442.91116 Beck, Christian; E, Weinan; Jentzen, Arnulf 126 2019 On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations. Zbl 1418.65149 E, Weinan; Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas 71 2019 A mean-field optimal control formulation of deep learning. Zbl 1421.49021 E, Weinan; Han, Jiequn; Li, Qianxiao 48 2019 A priori estimates of the population risk for two-layer neural networks. Zbl 1427.68277 E, Weinan; Ma, Chao; Wu, Lei 41 2019 Solving many-electron Schrödinger equation using deep neural networks. Zbl 1457.81010 Han, Jiequn; Zhang, Linfeng; E, Weinan 35 2019 Stochastic modified equations and dynamics of stochastic gradient algorithms. I: Mathematical foundations. Zbl 1484.62106 Li, Qianxiao; Tai, Cheng; E, Weinan 28 2019 Uniformly accurate machine learning-based hydrodynamic models for kinetic equations. Zbl 1431.76100 Han, Jiequn; Ma, Chao; Ma, Zheng; E, Weinan 27 2019 Model reduction with memory and the machine learning of dynamical systems. Zbl 1473.35450 Ma, Chao; Wang, Jianchun; E, Weinan 21 2019 Applied stochastic analysis. Zbl 1430.60002 E, Weinan; Li, Tiejun; Vanden-Eijnden, Eric 13 2019 Machine learning: mathematical theory and scientific applications. Zbl 1439.68019 E, Weinan 2 2019 Solving high-dimensional partial differential equations using deep learning. Zbl 1416.35137 Han, Jiequn; Jentzen, Arnulf; E, Weinan 470 2018 The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Zbl 1392.35306 E, Weinan; Yu, Bing 420 2018 Maximum principle based algorithms for deep learning. Zbl 1467.68156 Li, Qianxiao; Chen, Long; Tai, Cheng; E, Weinan 46 2018 Exponential convergence of the deep neural network approximation for analytic functions. Zbl 1475.65007 E, Weinan; Wang, Qingcan 42 2018 Deep potential: a general representation of a many-body potential energy surface. Zbl 1488.81038 Han, Jiequn; Zhang, Linfeng; Car, Roberto; E, Weinan 19 2018 Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Zbl 1382.65016 E, Weinan; Han, Jiequn; Jentzen, Arnulf 294 2017 A proposal on machine learning via dynamical systems. Zbl 1380.37154 E, Weinan 129 2017 Noisy Hegselmann-Krause systems: phase transition and the \(2R\)-conjecture. Zbl 1393.82013 Wang, Chu; Li, Qianxiao; E, Weinan; Chazelle, Bernard 10 2017 Optimal convergence rate of the universal estimation error. Zbl 1361.62024 E, Weinan; Wang, Yao 1 2017 Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations. Zbl 1342.60094 E, Weinan; Jentzen, Arnulf; Shen, Hao 13 2016 Multiscale adaptive representation of signals. I: The basic framework. Zbl 1391.94406 Tai, Cheng; E, Weinan 3 2016 A thermodynamic study of the two-dimensional pressure-driven channel flow. Zbl 1333.65141 E, Weinan; Wang, Jianchun 1 2016 On the distinguished limits of the Navier slip model of the moving contact line problem. Zbl 1327.76019 Ren, Weiqing; Trinh, Philippe H.; E, Weinan 11 2015 Model the nonlinear instability of wall-bounded shear flows as a rare event: a study on two-dimensional Poiseuille flow. Zbl 1316.35246 Wan, Xiaoliang; Yu, Haijun; E, Weinan 8 2015 The free action of nonequilibrium dynamics. Zbl 1327.82060 Li, Qianxiao; E, Weinan 1 2015 The Kohn-Sham equation for deformed crystals. Zbl 1435.74001 E, Weinan; Lu, Jianfeng 17 2013 Asymptotic analysis of quantum dynamics in crystals: the Bloch-Wigner transform, Bloch dynamics and Berry phase. Zbl 1278.81084 E, Weinan; Lu, Jian-feng; Yang, Xu 6 2013 Optimization-based string method for finding minimum energy path. Zbl 1373.37062 Samanta, Amit; E, Weinan 6 2013 Mean field limit of a dynamical model of polymer systems. Zbl 1301.60071 E, Weinan; Shen, Hao 5 2013 Efficient iterative method for solving the Dirac-Kohn-Sham density functional theory. Zbl 1349.65168 Lin, Lin; Shao, Sihong; E, Weinan 1 2013 The heterogeneous multiscale method. Zbl 1255.65224 Abdulle, Assyr; E, Weinan; Engquist, Björn; Vanden-Eijnden, Eric 198 2012 Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework. I: Total energy calculation. Zbl 1251.82008 Lin, Lin; Lu, Jianfeng; Ying, Lexing; E, Weinan 34 2012 Optimized local basis set for Kohn-Sham density functional theory. Zbl 1250.82002 Lin, Lin; Lu, Jianfeng; Ying, Lexing; E, Weinan 7 2012 Subcritical bifurcation in spatially extended systems. Zbl 1275.60054 E, Weinan; Zhou, Xiang; Cheng, Xiuyuan 4 2012 Stability and the continuum limit of the spin-polarized Thomas-Fermi-Dirac-von Weizsäcker model. Zbl 1278.82054 E, Weinan; Lu, Jianfeng 1 2012 Principles of multiscale modeling. Zbl 1238.00010 E, Weinan 116 2011 The gentlest ascent dynamics. Zbl 1237.37052 E, Weinan; Zhou, Xiang 47 2011 SelInv – an algorithm for selected inversion of a sparse symmetric matrix. Zbl 1365.65069 Lin, Lin; Yang, Chao; Meza, Juan C.; Lu, Jianfeng; Ying, Lexing; E, Weinan 28 2011 The electronic structure of smoothly deformed crystals: Wannier functions and the Cauchy-Born Rule. Zbl 1253.74025 E, Weinan; Lu, Jianfeng 21 2011 Derivation of continuum models for the moving contact line problem based on thermodynamic principles. Zbl 1414.76019 Ren, Weiqing; E, Weinan 21 2011 Contact line dynamics on heterogeneous surfaces. Zbl 1308.76212 Ren, Weiqing; E, Weinan 15 2011 A fast parallel algorithm for selected inversion of structured sparse matrices with application to 2D electronic structure calculations. Zbl 1230.65039 Lin, Lin; Yang, Chao; Lu, Jianfeng; Ying, Lexing; E, Weinan 14 2011 Effective Maxwell equations from time-dependent density functional theory. Zbl 1210.81030 E, Weinan; Lu, Jianfeng; Yang, Xu 5 2011 The heterogeneous multiscale methods with application to combustion. Zbl 1380.76098 E, Weinan; Engquist, Björn; Sun, Yi 2 2011 Continuum models for the contact line problem. Zbl 1308.76082 Ren, Weiqing; Hu, Dan; E, Weinan 37 2010 Study of noise-induced transitions in the Lorenz system using the minimum action method. Zbl 1202.34104 Zhou, Xiang; E, Weinan 24 2010 Study of the noise-induced transition and the exploration of the phase space for the Kuramoto-Sivashinsky equation using the minimum action method. Zbl 1185.82044 Wan, X.; Zhou, X.; E, W. 18 2010 Electronic structure of smoothly deformed crystals: Cauchy-Born rule for the nonlinear tight-binding model. Zbl 1277.81117 E, Weinan; Lu, Jianfeng 15 2010 Localized bases of eigensubspaces and operator compression. Zbl 1205.15017 E, Weinan; Li, Tiejun; Lu, Jianfeng 14 2010 A multiscale coupling method for the modeling of dynamics of solids with application to brittle cracks. Zbl 1423.74958 Li, Xiantao; Yang, Jerry Z.; E, Weinan 13 2010 A numerical method for the study of nucleation of ordered phases. Zbl 1329.82041 Lin, Ling; Cheng, Xiuyuan; E, Weinan; Shi, An-Chang; Zhang, Pingwen 10 2010 On the crystallization of 2D hexagonal lattices. Zbl 1180.82191 E, Weinan; Li, Dong 43 2009 Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems. Zbl 1182.65072 Lin, Lin; Lu, Jianfeng; Ying, Lexing; Car, Roberto; E, Weinan 34 2009 A general strategy for designing seamless multiscale methods. Zbl 1280.76038 E, Weinan; Ren, Weiqing; Vanden-Eijnden, Eric 21 2009 Pole-based approximation of the Fermi-Dirac function. Zbl 1188.41007 Lin, Lin; Lu, Jianfeng; Ying, Lexing; E, Weinan 13 2009 Orbital minimization with localization. Zbl 1170.46313 Gao, Weiguo; E, Weinan 2 2009 Effectiveness of implicit methods for stiff stochastic differential equations. Zbl 1199.65010 Li, Tiejun; Abdulle, Assyr; E, Weinan 25 2008 The Andersen thermostat in molecular dynamics. Zbl 1148.82019 E, Weinan; Li, Dong 12 2008 Optimal partition and effective dynamics of complex networks. Zbl 1205.68031 E, Weinan; Li, Tiejun; Vanden-Eijnden, Eric 12 2008 Breakdown of homogenization for the random Hamilton-Jacobi equations. Zbl 1207.70018 E, Weinan; Wehr, Jan; Xin, Jack 1 2008 Sequential multiscale modeling using sparse representation. Zbl 1364.76156 Garcia-Cervera, Carlos J.; Ren, Weiqing; Lu, Jianfeng; E, Weinan 1 2008 Heterogeneous multiscale methods: a review. Zbl 1164.65496 E, Weinan; Engquist, Bjorn; Li, Xiantao; Ren, Weiqing; Vanden-Eijnden, Eric 222 2007 Cauchy-Born rule and the stability of crystalline solids: Static problems. Zbl 1106.74019 E, Weinan; Ming, Pingbing 84 2007 Boundary conditions for the moving contact line problem. Zbl 1146.76513 Ren, Weiqing; E, Weinan 65 2007 Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. Zbl 1162.80003 E, Weinan; Liu, Di; Vanden-Eijnden, Eric 40 2007 The local microscale problem in the multiscale modeling of strongly heterogeneous media: Effects of boundary conditions and cell size. Zbl 1158.74541 Yue, Xingye; E, Weinan 40 2007 Cauchy-Born rule and the stability of crystalline solids: dynamic problems. Zbl 1138.35063 E, Weinan; Ming, Pingbing 25 2007 A discontinuous Galerkin implementation of a domain decomposition method for kinetic-hydrodynamic coupling multiscale problems in gas dynamics and device simulations. Zbl 1343.76041 Chen, Shanqin; E, Weinan; Liu, Yunxian; Shu, Chi-Wang 10 2007 The elastic continuum limit of the tight binding model. Zbl 1136.74003 E, Weinan; Lu, Jianfeng 6 2007 Seamless multiscale modeling via dynamics on fiber bundles. Zbl 1133.65029 E, Weinan; Lu, Jianfeng 6 2007 A sub-linear scaling algorithm for computing the electronic structure of materials. Zbl 1133.35429 García-Cervera, Carlos J.; Lu, Jianfeng; E, Weinan 5 2007 Towards a theory of transition paths. Zbl 1101.82016 E, Weinan; Vanden-Eijnden, Eric 56 2006 A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit. Zbl 1131.76010 E, Weinan; Zhang, Pingwen 21 2006 Variational boundary conditions for molecular dynamics simulations of solids at low temperature. Zbl 1114.74326 Li, Xiantao; E, Weinan 14 2006 Simulations of 3-D domain wall structures in thin films. Zbl 1099.35153 Wang, Xiao-Ping; Wang, Ke; E, Weinan 3 2006 Analysis of the heterogeneous multiscale method for elliptic homogenization problems. Zbl 1060.65118 E, Weinan; Ming, Pingbing; Zhang, Pingwen 128 2005 Analysis of multiscale methods for stochastic differential equations. Zbl 1080.60060 E, Weinan; Liu, Di; Vanden-Eijnden, Eric 83 2005 Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. Zbl 1143.76541 Ren, Weiqing; E, Weinan 38 2005 Multiscale modeling of the dynamics of solids at finite temperature. Zbl 1120.74313 Li, Xiantao; E, Weinan 37 2005 Self-induced stochastic resonance in excitable systems. Zbl 1109.34040 Muratov, Cyrill B.; Vanden-Eijnden, Eric; E, Weinan 28 2005 The heterogeneous multi-scale method for homogenization problems. Zbl 1086.65521 E, Weinan; Engquist, Björn 27 2005 Analysis of the local quasicontinuum method. Zbl 1188.74019 E, Weinan; Ming, Pingbing 12 2005 ...and 82 more Documents all cited Publications top 5 cited Publications all top 5 Cited by 6,222 Authors 73 E, Weinan 52 Lu, Jianfeng 48 Chung, Tsz Shun Eric 46 Abdulle, Assyr 38 Jentzen, Arnulf 37 Efendiev, Yalchin R. 36 Vanden-Eijnden, Eric 35 Liu, Jianguo 31 Wang, Cheng 27 Karniadakis, George Em 26 Cui, Junzhi 26 Lin, Lin 26 Ortner, Christoph 24 Chen, Jingrun 24 Zhang, Pingwen 23 Leung, Wingtat 23 Ren, Weiqing 22 Ying, Lexing 21 Hou, Thomas Yizhao 21 Shen, Jie 21 Yang, Xiaofeng 20 Masmoudi, Nader 20 Yang, Haizhao 20 Zhang, Lei 19 Du, Qiang 19 Duan, Jinqiao 19 Li, Xiantao 19 Mishra, Siddhartha 19 Schwab, Christoph 18 Hutzenthaler, Martin 18 Vicol, Vlad C. 17 Ming, Pingbing 17 Titi, Edriss Saleh 17 Zhang, Zhifei 16 Engquist, Bjorn E. 16 Eyink, Gregory L. 16 Han, Jiequn 16 Kevrekidis, Ioannis George 16 Schütte, Christof 16 Wang, Xiaoming 16 Yang, Hanchun 16 Zhang, Lei 15 Constantin, Peter 15 Gao, Yuan 15 Jiang, Lijian 15 Ma, Qiang 15 Yang, Jerry Zhijian 15 Zhang, Zhiwen 14 Lelièvre, Tony 14 Liu, Cheng-Jie 14 Liu, Wing Kam 14 Stefanelli, Ulisse 14 Stuart, Andrew M. 14 Székelyhidi, László jun. 14 Wu, Jiahong 14 Yang, Zhiqiang 14 Zhang, Yanyan 14 Zhang, Yu 13 De Lellis, Camillo 13 Jin, Shi 13 Li, Qianxiao 13 Liu, Chun 13 Liu, Wei 13 Luskin, Mitchell 13 Majda, Andrew J. 13 Osher, Stanley Joel 13 Rykov, Yurii G. 13 Samaey, Giovanni 13 Vasilyeva, Maria V. 13 Wang, Yanqing 13 Yamazaki, Kazuo 13 Yan, Zhenya 13 Yang, Tong 12 Bakhtin, Yuri Yu. 12 Cameron, Maria Kourkina 12 Friedrich, Manuel 12 Grafke, Tobias 12 Mattingly, Jonathan Christopher 12 Ohlberger, Mario 12 Owhadi, Houman 12 Pavliotis, Grigorios A. 12 Sun, Yi 12 Tang, Shaoqiang 12 Zhou, Tao 11 Bétermin, Laurent 11 Isett, Philip 11 Kuksin, Sergeĭ B. 11 Lai, Mingchih 11 Li, Shihu 11 Nersesyan, Vahagn 11 Sun, Xiaobin 11 Valentin, Frédéric 11 Wan, Xiaoliang 11 Ye, Yulin 11 Yue, Xingye 11 Zhang, Zhen 11 Zhou, Xiang 10 De Rosa, Luigi 10 Degond, Pierre 10 Dong, Suchuan ...and 6,122 more Authors all top 5 Cited in 442 Serials 511 Journal of Computational Physics 212 Computer Methods in Applied Mechanics and Engineering 128 Multiscale Modeling & Simulation 123 Journal of Scientific Computing 110 Journal of Differential Equations 107 Physica D 106 Archive for Rational Mechanics and Analysis 101 SIAM Journal on Scientific Computing 95 Journal of Computational and Applied Mathematics 87 Communications in Computational Physics 72 Communications in Mathematical Physics 69 Computers & Mathematics with Applications 56 Journal of Statistical Physics 55 SIAM Journal on Mathematical Analysis 54 Communications on Pure and Applied Mathematics 54 International Journal for Numerical Methods in Engineering 49 Journal of Fluid Mechanics 49 Mathematics of Computation 48 Journal of Mathematical Analysis and Applications 46 Journal of Mathematical Physics 44 Nonlinearity 42 Computers and Fluids 42 Applied Numerical Mathematics 42 Computational Mechanics 41 Journal of Nonlinear Science 40 SIAM Journal on Numerical Analysis 38 Applied Mathematics and Computation 36 Mathematical Methods in the Applied Sciences 33 Stochastic Processes and their Applications 33 Discrete and Continuous Dynamical Systems. 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Series B 15 Neural Networks 15 Analysis and Applications (Singapore) 15 Acta Numerica 15 Journal of Physics A: Mathematical and Theoretical 14 Advances in Mathematics 14 BIT 14 Transactions of the American Mathematical Society 14 Applied Mathematical Modelling 14 SN Partial Differential Equations and Applications 13 Numerical Methods for Partial Differential Equations 13 SIAM Journal on Applied Dynamical Systems 13 Communications on Applied Mathematics and Computation 12 Journal of the Mechanics and Physics of Solids 12 Acta Applicandae Mathematicae 12 Mathematical Problems in Engineering 12 Foundations of Computational Mathematics 12 Stochastics and Dynamics 12 Communications on Pure and Applied Analysis 12 Numerical Mathematics: Theory, Methods and Applications 12 Annals of PDE 11 Inverse Problems 11 Quarterly of Applied Mathematics 11 SIAM Journal on Control and Optimization 11 European Journal of Applied Mathematics 11 SIAM Review 11 Computational and Applied Mathematics 11 Computational Geosciences 11 Oberwolfach Reports 11 Analysis & PDE 11 SIAM Journal on Financial Mathematics 10 Japan Journal of Industrial and Applied Mathematics 10 European Journal of Mechanics. 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Solids 10 Journal of the European Mathematical Society (JEMS) 10 Journal of Machine Learning Research (JMLR) 10 Journal of Hyperbolic Differential Equations ...and 342 more Serials all top 5 Cited in 53 Fields 2,211 Partial differential equations (35-XX) 1,982 Numerical analysis (65-XX) 1,607 Fluid mechanics (76-XX) 793 Probability theory and stochastic processes (60-XX) 744 Computer science (68-XX) 523 Mechanics of deformable solids (74-XX) 521 Statistical mechanics, structure of matter (82-XX) 265 Dynamical systems and ergodic theory (37-XX) 251 Calculus of variations and optimal control; optimization (49-XX) 181 Biology and other natural sciences (92-XX) 165 Ordinary differential equations (34-XX) 133 Systems theory; control (93-XX) 127 Statistics (62-XX) 120 Quantum theory (81-XX) 114 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 107 Optics, electromagnetic theory (78-XX) 102 Approximations and expansions (41-XX) 85 Operations research, mathematical programming (90-XX) 80 Mechanics of particles and systems (70-XX) 75 Classical thermodynamics, heat transfer (80-XX) 75 Geophysics (86-XX) 40 Functional analysis (46-XX) 38 Linear and multilinear algebra; matrix theory (15-XX) 38 Real functions (26-XX) 36 Global analysis, analysis on manifolds (58-XX) 35 Operator theory (47-XX) 30 Harmonic analysis on Euclidean spaces (42-XX) 28 Information and communication theory, circuits (94-XX) 26 Integral equations (45-XX) 18 General and overarching topics; collections (00-XX) 17 Differential geometry (53-XX) 14 Measure and integration (28-XX) 12 Combinatorics (05-XX) 12 Astronomy and astrophysics (85-XX) 9 Special functions (33-XX) 9 Difference and functional equations (39-XX) 7 Potential theory (31-XX) 7 Relativity and gravitational theory (83-XX) 6 Convex and discrete geometry (52-XX) 5 History and biography (01-XX) 5 Functions of a complex variable (30-XX) 3 Number theory (11-XX) 2 Algebraic geometry (14-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Manifolds and cell complexes (57-XX) 1 Mathematical logic and foundations (03-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Category theory; homological algebra (18-XX) 1 Group theory and generalizations (20-XX) 1 Topological groups, Lie groups (22-XX) 1 Abstract harmonic analysis (43-XX) 1 Integral transforms, operational calculus (44-XX) 1 General topology (54-XX) Citations by Year Wikidata Timeline The data are displayed as stored in Wikidata under a Creative Commons CC0 License. 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