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Author ID: e.weinan Recent zbMATH articles by "E, Weinan"
Published as: E, Weinan; Weinan, E.; E., Weinan; E, W.; Veinan, I.; É, Weinan
Further Spellings: И Вейнан
Homepage: https://web.math.princeton.edu/~weinan/
External Links: MGP · ORCID · Wikidata · Google Scholar · Math-Net.Ru · dblp · GND · IdRef
all top 5

Co-Authors

22 single-authored
22 Lu, Jianfeng
21 Vanden-Eijnden, Eric
17 Han, Jiequn
12 Ma, Chao
11 Liu, Jianguo
11 Ren, Weiqing
9 Engquist, Bjorn E.
9 Li, Qianxiao
9 Wu, Lei
8 Lin, Lin
7 Jentzen, Arnulf
7 Li, Tiejun
7 Shu, Chi-Wang
7 Wojtowytsch, Stephan
7 Zhang, Pingwen
6 Li, Xiantao
6 Ying, Lexing
5 Long, Jihao
5 Ming, Pingbing
5 Sinaĭ, Yakov Grigor’evich
5 Zhang, Linfeng
5 Zhang, Yaoyu
4 García-Cervera, Carlos J.
4 Hu, Wei
4 Xu, Zhi-Qin John
4 Zhang, Tianhan
4 Zhou, Xiang
3 Abdulle, Assyr
3 Car, Roberto
3 Liu, Di
3 Shen, Hao
3 Tai, Cheng
3 Wang, Qingcan
3 Wang, Xiaoping
3 Yue, Xingye
2 Avellaneda, Marco
2 Chen, Falai
2 Chen, Jingrun
2 Chen, Shanqin
2 Cheng, Xiuyuan
2 Huang, Hongci
2 Hutzenthaler, Martin
2 Kruse, Thomas
2 Li, Dong
2 Ma, Zheng
2 Mattingly, Jonathan Christopher
2 Palffy-Muhoray, Peter
2 Rykov, Yurii G.
2 Wan, Xiaoliang
2 Wang, Jianchun
2 Yang, Chao
2 Yang, Xu
2 Yip, Nung Kwan
2 Yu, Haijun
2 Zang, Yaohua
1 Bai, Zhaojun
1 Beck, Christian
1 Burger, Martin
1 Caffarelli, Luis Ángel
1 Cai, Zhiqiang
1 Chan, Tony Fan-Cheong
1 Chazelle, Bernard
1 Chen, Shiyi
1 Chen, Weijie
1 Chen, Zhengting
1 Chen, Zhiming
1 Cheng, Li-Tien
1 Constantin, Peter
1 Dai, Yu-Hong
1 Feng, Jiashi
1 Friesecke, Gero
1 Gao, Weiguo
1 Gimbutas, Zydrunas
1 Han, Weimin
1 Hou, Thomas Yizhao
1 Hu, Dan
1 Khanin, Konstantin M.
1 Kohn, Robert Vita
1 Lei, Huan
1 Li, Shangzhi
1 Lin, Ling
1 Liu, Yunxian
1 Luo, Tao
1 Luo, Yixin
1 Mazel, Alexander E.
1 Meza, Juan C.
1 Mu, Mo
1 Muratov, Cyrill B.
1 Nie, Xiaobo
1 Osher, Stanley Joel
1 Otto, Felix
1 Pettifor, David
1 Robbins, Mark O.
1 Ruthotto, Lars
1 Ryan, Reade
1 Samanta, Amit
1 Schulze, Tim P.
1 Serre, Denis
1 Shao, Sihong
1 Shi, An-Chang
...and 27 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 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 Notices of the American Mathematical Society
3 Methods and Applications of Analysis
3 Discrete and Continuous Dynamical Systems
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 SN Partial Differential Equations and Applications

Publications by Year

Citations contained in zbMATH Open

177 Publications have been cited 5,816 times in 3,942 Documents Cited by Year
The heterogeneous multiscale methods. Zbl 1093.35012
E, Weinan; Engquist, Bjorn
397
2003
Solving high-dimensional partial differential equations using deep learning. Zbl 1416.35137
Han, Jiequn; Jentzen, Arnulf; E, Weinan
338
2018
The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems. Zbl 1392.35306
E, Weinan; Yu, Bing
243
2018
Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Zbl 0818.35085
Constantin, Peter; E, Weinan; Titi, Edriss S.
228
1994
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
214
2017
Heterogeneous multiscale methods: a review. Zbl 1164.65496
E, Weinan; Engquist, Bjorn; Li, Xiantao; Ren, Weiqing; Vanden-Eijnden, Eric
196
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.
189
1996
The heterogeneous multiscale method. Zbl 1255.65224
Abdulle, Assyr; Weinan, E.; Engquist, Björn; Vanden-Eijnden, Eric
166
2012
Invariant measures for Burgers equation with stochastic forcing. Zbl 0972.35196
E, Weinan; Khanin, K.; Mazel, A.; Sinai, Ya.
126
2000
Blowup of solutions of the unsteady Prandtl’s equation. Zbl 0908.35099
E, Weinan; Engquist, Bjorn
117
1997
Analysis of the heterogeneous multiscale method for elliptic homogenization problems. Zbl 1060.65118
E, Weinan; Ming, Pingbing; Zhang, Pingwen
116
2005
Minimum action method for the study of rare events. Zbl 1050.60068
E, Weinan; Ren, Weiqing; Vanden-Eijnden, Eric
116
2004
Projection method. I: Convergence and numerical boundary layers. Zbl 0842.76052
E, Weinan; Liu, Jian-Guo
105
1995
Principles of multiscale modeling. Zbl 1238.00010
E, Weinan
95
2011
Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation. Zbl 0961.35101
E, Weinan
94
2000
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
93
2019
Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Zbl 0994.60065
E, Weinan; Mattingly, J. C.; Sinai, Ya.
89
2001
A proposal on machine learning via dynamical systems. Zbl 1380.37154
E, Weinan
81
2017
Cauchy-Born rule and the stability of crystalline solids: Static problems. Zbl 1106.74019
E, Weinan; Ming, Pingbing
80
2007
Analysis of multiscale methods for stochastic differential equations. Zbl 1080.60060
E, Weinan; Liu, Di; Vanden-Eijnden, Eric
74
2005
Vorticity boundary condition and related isssues for finite difference schemes. Zbl 0847.76050
E, Weinan; Liu, Jian-Guo
73
1996
Small-scale structures in Boussinesq convection. Zbl 0822.76087
E, Weinan; Shu, Chi-Wang
69
1994
Essentially compact schemes for unsteady viscous incompressible flows. Zbl 0853.76045
E, Weinan; Liu, Jian-Guo
61
1996
Gauge method for viscous incompressible flows. Zbl 1160.76329
E, Weinan; Liu, Jian-Guo
58
2003
Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity. Zbl 0814.34039
E, Weinan
57
1994
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
56
2019
Multiscale modeling and computation. Zbl 1032.65013
E, Weinan; Engquist, Bjorn
54
2003
Numerical methods for the Landau–Lifshitz equation. Zbl 0988.65079
E, Weinan; Wang, Xiao-Ping
54
2000
Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite-dimensional approximation. Zbl 1024.76012
E, Weinan; Mattingly, Jonathan C.
53
2001
Homogenization of linear and nonlinear transport equations. Zbl 0794.35014
E, Weinan
49
1992
Towards a theory of transition paths. Zbl 1101.82016
E, Weinan; Vanden-Eijnden, Eric
49
2006
Boundary conditions for the moving contact line problem. Zbl 1146.76513
Ren, Weiqing; E, Weinan
45
2007
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.
44
2004
A dynamic atomistic–continuum method for the simulation of crystalline materials. Zbl 1013.82022
E, Weinan; Huang, Zhongyi
43
2002
Well-posedness for the dumbbell model of polymeric fluids. Zbl 1060.35169
E, Weinan; Li, Tiejun; Zhang, Pingwen
42
2004
On the crystallization of 2D hexagonal lattices. Zbl 1180.82191
E, Weinan; Li, Dong
42
2009
Finite difference heterogeneous multi-scale method for homogenization problems. Zbl 1034.65067
Abdulle, Assyr; E, Weinan
41
2003
The gentlest ascent dynamics. Zbl 1237.37052
E, Weinan; Zhou, Xiang
41
2011
A Gauss-Seidel projection method for micromagnetics simulations. Zbl 1012.82002
Wang, Xiao-Ping; García-Cervera, Carlos J.; E, Weinan
39
2001
Analysis of the heterogeneous multiscale method for ordinary differential equations. Zbl 1088.65552
E, Weinan
39
2003
Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. Zbl 1162.80003
E, Weinan; Liu, Di; Vanden-Eijnden, Eric
38
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
37
2007
Multiscale modeling of the dynamics of solids at finite temperature. Zbl 1120.74313
Li, Xiantao; E, Weinan
37
2005
Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. Zbl 1143.76541
Ren, Weiqing; E, Weinan
36
2005
A mean-field optimal control formulation of deep learning. Zbl 1421.49021
E, Weinan; Han, Jiequn; Li, Qianxiao
33
2019
A priori estimates of the population risk for two-layer neural networks. Zbl 1427.68277
E, Weinan; Ma, Chao; Wu, Lei
32
2019
Exponential convergence of the deep neural network approximation for analytic functions. Zbl 1475.65007
E, Weinan; Wang, Qingcan
31
2018
Continuum models for the contact line problem. Zbl 1308.76082
Ren, Weiqing; Hu, Dan; E, Weinan
31
2010
A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow. Zbl 0790.76055
E, Weinan; Shu, Chi-Wang
29
1994
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
28
2009
Statistical properties of shocks in Burgers turbulence. Zbl 0844.35144
Avellaneda, Marco; E, Weinan
28
1995
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
28
2012
Effectiveness of implicit methods for stiff stochastic differential equations. Zbl 1199.65010
Li, Tiejun; Abdulle, Assyr; E, Weinan
24
2008
Self-induced stochastic resonance in excitable systems. Zbl 1109.34040
Muratov, Cyrill B.; Vanden-Eijnden, Eric; E, Weinan
24
2005
Maximum principle based algorithms for deep learning. Zbl 1467.68156
Li, Qianxiao; Chen, Long; Tai, Cheng; E, Weinan
23
2018
The heterogeneous multi-scale method for homogenization problems. Zbl 1086.65521
E, Weinan; Engquist, Björn
23
2005
Projection method. II: Godunov-Ryabenki analysis. Zbl 0920.76056
E, Weinan; Liu, Jian-Guo
23
1996
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
Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning. Zbl 1490.60202
E, Weinan; Han, Jiequn; Jentzen, Arnulf
23
2022
A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit. Zbl 1131.76010
E, Weinan; Zhang, Pingwen
22
2006
Cauchy-Born rule and the stability of crystalline solids: dynamic problems. Zbl 1138.35063
E, Weinan; Ming, Pingbing
22
2007
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
22
2011
Convergence of Fourier methods for the Navier-Stokes equations. Zbl 0776.76021
E, Weinan
21
1993
Some recent progress in multiscale modeling. Zbl 1419.74252
E, Weinan; Li, Xiantao; Vanden-Eijnden, Eric
21
2004
Solving many-electron Schrödinger equation using deep neural networks. Zbl 1457.81010
Han, Jiequn; Zhang, Linfeng; E, Weinan
21
2019
Uniformly accurate machine learning-based hydrodynamic models for kinetic equations. Zbl 1431.76100
Han, Jiequn; Ma, Chao; Ma, Zheng; E, Weinan
20
2019
Nonlinear continuum theory of smectic-\(A\) liquid crystals. Zbl 0891.76009
E, Weinan
20
1997
Projection method. III: Spatial discretization on the staggered grid. Zbl 1058.76043
E, Weinan; Liu, Jian-Guo
20
2002
Generalized flows, intrinsic stochasticity, and turbulent transport. Zbl 0967.76038
E, Weinan; Vanden Eijnden, Eric
20
2000
Aubry-Mather theory and periodic solutions of the forced Burgers equation. Zbl 0916.35099
E, Weinan
20
1999
Study of noise-induced transitions in the Lorenz system using the minimum action method. Zbl 1202.34104
Zhou, Xiang; E, Weinan
20
2010
Derivation of continuum models for the moving contact line problem based on thermodynamic principles. Zbl 1414.76019
Ren, Weiqing; E, Weinan
19
2011
The electronic structure of smoothly deformed crystals: Wannier functions and the Cauchy-Born Rule. Zbl 1253.74025
E, Weinan; Lu, Jianfeng
19
2011
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
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
Machine learning from a continuous viewpoint. I. Zbl 1472.68136
E, Weinan; Ma, Chao; Wu, Lei
18
2020
The Barron space and the flow-induced function spaces for neural network models. Zbl 1490.65020
E, Weinan; Ma, Chao; Wu, Lei
17
2022
Convergence of spectral methods for Burgers’ equation. Zbl 0764.65057
E, Weinan
17
1992
Finite difference schemes for incompressible flows in the velocity-impulse density formulation. Zbl 0870.76048
E, Weinan; Liu, Jian-Guo
17
1997
A general strategy for designing seamless multiscale methods. Zbl 1280.76038
E, Weinan; Ren, Weiqing; Vanden-Eijnden, Eric
17
2009
Energy landscapes and rare events. Zbl 1023.65004
E, Weinan; Ren, Weiqing; Vanden-Eijnden, Eric
17
2002
Stochastic modified equations and dynamics of stochastic gradient algorithms. I: Mathematical foundations. Zbl 1484.62106
Li, Qianxiao; Tai, Cheng; E, Weinan
17
2019
The Kohn-Sham equation for deformed crystals. Zbl 1435.74001
E, Weinan; Lu, Jianfeng
16
2013
Homogenization of scalar conservation laws with oscillatory forcing terms. Zbl 0755.35068
E, Weinan
16
1992
Multilevel Picard iterations for solving smooth semilinear parabolic heat equations. Zbl 1476.65273
E, Weinan; Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas
16
2021
Statistical theory for the stochastic Burgers equation in the inviscid limit. Zbl 1037.76013
E, Weinan; Vanden Eijnden, Eric
15
2000
Electronic structure of smoothly deformed crystals: Cauchy-Born rule for the nonlinear tight-binding model. Zbl 1277.81117
E, Weinan; Lu, Jianfeng
15
2010
Continuum theory of epitaxial crystal growth. I. Zbl 1074.76504
E, Weinan; Yip, Nung Kwan
15
2001
Metastability, conformation dynamics, and transition pathways in complex systems. Zbl 1141.82312
E, Weinan; Vanden-Eijnden, Eric
15
2004
Large time behavior and homogenization of solutions of two-dimensional conservation laws. Zbl 0797.35114
Engquist, Bjorn; E, Weinan
14
1993
Deep potential: a general representation of a many-body potential energy surface. Zbl 1488.81038
Han, Jiequn; Zhang, Linfeng; Car, Roberto; E, Weinan
14
2018
Model reduction with memory and the machine learning of dynamical systems. Zbl 1473.35450
Ma, Chao; Wang, Jianchun; E, Weinan
14
2019
Machine learning and computational mathematics. Zbl 1473.00035
E, Weinan
14
2020
Phase separation in incompressible systems. Zbl 1110.82304
E, Weinan; Palffy-Muhoray, P.
13
1997
Variational boundary conditions for molecular dynamics simulations of solids at low temperature. Zbl 1114.74326
Li, Xiantao; E, Weinan
13
2006
Localized bases of eigensubspaces and operator compression. Zbl 1205.15017
E, Weinan; Li, Tiejun; Lu, Jianfeng
13
2010
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
13
2020
Contact line dynamics on heterogeneous surfaces. Zbl 1308.76212
Ren, Weiqing; E, Weinan
12
2011
A class of homogenization problems in the calculus of variations. Zbl 0773.49007
E, Weinan
12
1991
Gibbsian dynamics and invariant measures for stochastic dissipative PDEs. Zbl 1023.60057
E, Weinan; Liu, Di
12
2002
Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning. Zbl 1490.60202
E, Weinan; Han, Jiequn; Jentzen, Arnulf
23
2022
The Barron space and the flow-induced function spaces for neural network models. Zbl 1490.65020
E, Weinan; Ma, Chao; Wu, Lei
17
2022
Representation formulas and pointwise properties for Barron functions. Zbl 1482.41013
E, Weinan; Wojtowytsch, Stephan
9
2022
Approximation and optimization theory for linear continuous-time recurrent neural networks. Zbl 07625195
Li, Zhong; Han, Jiequn; E, Weinan; Li, Qianxiao
3
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
2
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
16
2021
Kolmogorov width decay and poor approximators in machine learning: shallow neural networks, random feature models and neural tangent kernels. Zbl 07307666
E, Weinan; Wojtowytsch, Stephan
8
2021
The dawning of a new era in applied mathematics. Zbl 1471.00008
E, Weinan
2
2021
Connections between deep learning and partial differential equations. Zbl 1479.35002
Burger, M.; E, W.; Ruthotto, L.; Osher, S. J.
1
2021
Machine learning from a continuous viewpoint. I. Zbl 1472.68136
E, Weinan; Ma, Chao; Wu, Lei
18
2020
Machine learning and computational mathematics. Zbl 1473.00035
E, Weinan
14
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
13
2020
Rademacher complexity and the generalization error of residual networks. Zbl 1467.62158
E, Weinan; Ma, Chao; Wang, Qingcan
2
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
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
93
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
56
2019
A mean-field optimal control formulation of deep learning. Zbl 1421.49021
E, Weinan; Han, Jiequn; Li, Qianxiao
33
2019
A priori estimates of the population risk for two-layer neural networks. Zbl 1427.68277
E, Weinan; Ma, Chao; Wu, Lei
32
2019
Solving many-electron Schrödinger equation using deep neural networks. Zbl 1457.81010
Han, Jiequn; Zhang, Linfeng; E, Weinan
21
2019
Uniformly accurate machine learning-based hydrodynamic models for kinetic equations. Zbl 1431.76100
Han, Jiequn; Ma, Chao; Ma, Zheng; E, Weinan
20
2019
Stochastic modified equations and dynamics of stochastic gradient algorithms. I: Mathematical foundations. Zbl 1484.62106
Li, Qianxiao; Tai, Cheng; E, Weinan
17
2019
Model reduction with memory and the machine learning of dynamical systems. Zbl 1473.35450
Ma, Chao; Wang, Jianchun; E, Weinan
14
2019
Applied stochastic analysis. Zbl 1430.60002
E, Weinan; Li, Tiejun; Vanden-Eijnden, Eric
8
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
338
2018
The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems. Zbl 1392.35306
E, Weinan; Yu, Bing
243
2018
Exponential convergence of the deep neural network approximation for analytic functions. Zbl 1475.65007
E, Weinan; Wang, Qingcan
31
2018
Maximum principle based algorithms for deep learning. Zbl 1467.68156
Li, Qianxiao; Chen, Long; Tai, Cheng; E, Weinan
23
2018
Deep potential: a general representation of a many-body potential energy surface. Zbl 1488.81038
Han, Jiequn; Zhang, Linfeng; Car, Roberto; E, Weinan
14
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
214
2017
A proposal on machine learning via dynamical systems. Zbl 1380.37154
E, Weinan
81
2017
Noisy Hegselmann-Krause systems: phase transition and the \(2R\)-conjecture. Zbl 1393.82013
Wang, Chu; Li, Qianxiao; E, Weinan; Chazelle, Bernard
7
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
11
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
10
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
16
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
5
2013
Mean field limit of a dynamical model of polymer systems. Zbl 1301.60071
E, Weinan; Shen, Hao
4
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; Weinan, E.; Engquist, Björn; Vanden-Eijnden, Eric
166
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
28
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
95
2011
The gentlest ascent dynamics. Zbl 1237.37052
E, Weinan; Zhou, Xiang
41
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
22
2011
Derivation of continuum models for the moving contact line problem based on thermodynamic principles. Zbl 1414.76019
Ren, Weiqing; E, Weinan
19
2011
The electronic structure of smoothly deformed crystals: Wannier functions and the Cauchy-Born Rule. Zbl 1253.74025
E, Weinan; Lu, Jianfeng
19
2011
Contact line dynamics on heterogeneous surfaces. Zbl 1308.76212
Ren, Weiqing; E, Weinan
12
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
11
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
31
2010
Study of noise-induced transitions in the Lorenz system using the minimum action method. Zbl 1202.34104
Zhou, Xiang; E, Weinan
20
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
13
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
10
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
9
2010
On the crystallization of 2D hexagonal lattices. Zbl 1180.82191
E, Weinan; Li, Dong
42
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
28
2009
A general strategy for designing seamless multiscale methods. Zbl 1280.76038
E, Weinan; Ren, Weiqing; Vanden-Eijnden, Eric
17
2009
Pole-based approximation of the Fermi-Dirac function. Zbl 1188.41007
Lin, Lin; Lu, Jianfeng; Ying, Lexing; E, Weinan
11
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
24
2008
Optimal partition and effective dynamics of complex networks. Zbl 1205.68031
E, Weinan; Li, Tiejun; Vanden-Eijnden, Eric
11
2008
The Andersen thermostat in molecular dynamics. Zbl 1148.82019
E, Weinan; Li, Dong
9
2008
Sequential multiscale modeling using sparse representation. Zbl 1364.76156
Garcia-Cervera, Carlos J.; Ren, Weiqing; Lu, Jianfeng; E, Weinan
1
2008
Breakdown of homogenization for the random Hamilton-Jacobi equations. Zbl 1207.70018
E, Weinan; Wehr, Jan; Xin, Jack
1
2008
Heterogeneous multiscale methods: a review. Zbl 1164.65496
E, Weinan; Engquist, Bjorn; Li, Xiantao; Ren, Weiqing; Vanden-Eijnden, Eric
196
2007
Cauchy-Born rule and the stability of crystalline solids: Static problems. Zbl 1106.74019
E, Weinan; Ming, Pingbing
80
2007
Boundary conditions for the moving contact line problem. Zbl 1146.76513
Ren, Weiqing; E, Weinan
45
2007
Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. Zbl 1162.80003
E, Weinan; Liu, Di; Vanden-Eijnden, Eric
38
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
37
2007
Cauchy-Born rule and the stability of crystalline solids: dynamic problems. Zbl 1138.35063
E, Weinan; Ming, Pingbing
22
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
9
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
5
2007
A sub-linear scaling algorithm for computing the electronic structure of materials. Zbl 1133.35429
García-Cervera, Carlos J.; Lu, Jianfeng; E, Weinan
4
2007
Towards a theory of transition paths. Zbl 1101.82016
E, Weinan; Vanden-Eijnden, Eric
49
2006
A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit. Zbl 1131.76010
E, Weinan; Zhang, Pingwen
22
2006
Variational boundary conditions for molecular dynamics simulations of solids at low temperature. Zbl 1114.74326
Li, Xiantao; E, Weinan
13
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
116
2005
Analysis of multiscale methods for stochastic differential equations. Zbl 1080.60060
E, Weinan; Liu, Di; Vanden-Eijnden, Eric
74
2005
Multiscale modeling of the dynamics of solids at finite temperature. Zbl 1120.74313
Li, Xiantao; E, Weinan
37
2005
Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. Zbl 1143.76541
Ren, Weiqing; E, Weinan
36
2005
Self-induced stochastic resonance in excitable systems. Zbl 1109.34040
Muratov, Cyrill B.; Vanden-Eijnden, Eric; E, Weinan
24
2005
The heterogeneous multi-scale method for homogenization problems. Zbl 1086.65521
E, Weinan; Engquist, Björn
23
2005
Analysis of the local quasicontinuum method. Zbl 1188.74019
E, Weinan; Ming, Pingbing
11
2005
The heterogeneous multiscale method based on the discontinuous Galerkin method for hyperbolic and parabolic problems. Zbl 1080.65090
Chen, Shanqin; E, Weinan; Shu, Chi-Wang
10
2005
Numerical methods for multiscale transport equations and application to two-phase porous media flow. Zbl 1089.76049
Yue, Xingye; E, Weinan
9
2005
Minimum action method for the study of rare events. Zbl 1050.60068
E, Weinan; Ren, Weiqing; Vanden-Eijnden, Eric
116
2004
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.
44
2004
...and 77 more Documents
all top 5

Cited by 5,228 Authors

68 E, Weinan
47 Abdulle, Assyr
45 Lu, Jianfeng
44 Chung, Tsz Shun Eric
36 Efendiev, Yalchin R.
33 Jentzen, Arnulf
32 Liu, Jianguo
30 Vanden-Eijnden, Eric
29 Wang, Cheng
24 Ortner, Christoph
22 Leung, Wingtat
22 Lin, Lin
21 Ren, Weiqing
21 Zhang, Pingwen
20 Karniadakis, George Em
20 Zhang, Lei
19 Cui, Junzhi
19 Du, Qiang
19 Ying, Lexing
18 Li, Xiantao
18 Masmoudi, Nader
17 Chen, Jingrun
16 Han, Jiequn
16 Hutzenthaler, Martin
16 Yang, Haizhao
16 Yang, Hanchun
16 Yang, Xiaofeng
16 Zhang, Zhifei
15 Constantin, Peter
15 Engquist, Bjorn E.
15 Hou, Thomas Yizhao
15 Mishra, Siddhartha
15 Schütte, Christof
15 Schwab, Christoph
15 Shen, Jie
15 Vicol, Vlad C.
15 Zhang, Lei
14 Titi, Edriss Saleh
14 Wang, Xiaoming
14 Zhang, Yanyan
13 Duan, Jinqiao
13 Eyink, Gregory L.
13 Gao, Yuan
13 Majda, Andrew J.
13 Samaey, Giovanni
13 Vasilyeva, Maria V.
13 Wu, Jiahong
13 Zhang, Yu
12 Cameron, Maria Kourkina
12 Kevrekidis, Ioannis George
12 Liu, Cheng-Jie
12 Liu, Wing Kam
12 Luskin, Mitchell
12 Osher, Stanley Joel
12 Stuart, Andrew M.
12 Székelyhidi, László jun.
12 Xu, Jie
12 Yang, Tong
11 Bakhtin, Yuri Yu.
11 Bétermin, Laurent
11 De Lellis, Camillo
11 Henning, Patrick
11 Lelièvre, Tony
11 Li, Qianxiao
11 Liu, Wei
11 Ma, Qiang
11 Mattingly, Jonathan Christopher
11 Ming, Pingbing
11 Ohlberger, Mario
11 Stefanelli, Ulisse
11 Tang, Shaoqiang
11 Yamazaki, Kazuo
11 Yang, Zhiqiang
11 Yue, Xingye
11 Zhang, Zhiwen
10 Degond, Pierre
10 Friedrich, Manuel
10 Hartmann, Carsten
10 Isett, Philip
10 Kuksin, Sergeĭ B.
10 Le Bris, Claude
10 Li, Shihu
10 Owhadi, Houman
10 Vilmart, Gilles
10 Wan, Xiaoliang
10 Wang, Wei
10 Yang, Jerry Zhijian
9 Drivas, Theodore Dimitrios
9 Glatt-Holtz, Nathan E.
9 Hairer, Martin
9 Hoang, Viet Ha
9 Hu, Guanghui
9 Lai, Mingchih
9 Legoll, Frédéric
9 Liu, Chun
9 Liu, Di
9 Nersesyan, Vahagn
9 Pavliotis, Grigorios A.
9 Roberts, Anthony John
9 Süli, Endre E.
...and 5,128 more Authors
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Cited in 404 Serials

452 Journal of Computational Physics
162 Computer Methods in Applied Mechanics and Engineering
116 Multiscale Modeling & Simulation
103 Archive for Rational Mechanics and Analysis
103 Journal of Scientific Computing
97 Journal of Differential Equations
91 Physica D
85 Journal of Computational and Applied Mathematics
72 Communications in Computational Physics
69 SIAM Journal on Scientific Computing
65 Communications in Mathematical Physics
55 Journal of Statistical Physics
52 Computers & Mathematics with Applications
50 SIAM Journal on Mathematical Analysis
47 Mathematics of Computation
45 Journal of Fluid Mechanics
45 Journal of Mathematical Analysis and Applications
41 Journal of Mathematical Physics
38 Computers and Fluids
38 Computational Mechanics
37 Applied Numerical Mathematics
37 Journal of Nonlinear Science
34 Applied Mathematics and Computation
34 SIAM Journal on Numerical Analysis
32 Communications on Pure and Applied Mathematics
32 Stochastic Processes and their Applications
31 Nonlinearity
31 Physics of Fluids
31 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis
29 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods
28 International Journal for Numerical Methods in Engineering
28 M\(^3\)AS. Mathematical Models & Methods in Applied Sciences
27 Discrete and Continuous Dynamical Systems. Series B
24 Science China. Mathematics
24 Research in the Mathematical Sciences
23 Journal of Functional Analysis
23 Numerische Mathematik
22 Journal of Mathematical Fluid Mechanics
20 Discrete and Continuous Dynamical Systems
19 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire
19 SIAM Journal on Applied Mathematics
19 Calculus of Variations and Partial Differential Equations
19 Advances in Computational Mathematics
18 Applicable Analysis
17 ZAMP. Zeitschrift für angewandte Mathematik und Physik
17 Applied Mathematics Letters
17 Chaos
17 Communications in Nonlinear Science and Numerical Simulation
17 Nonlinear Analysis. Real World Applications
17 Discrete and Continuous Dynamical Systems. Series S
16 Applied Mathematics and Optimization
16 Journal de Mathématiques Pures et Appliquées. Neuvième Série
16 Stochastic and Partial Differential Equations. Analysis and Computations
15 The Annals of Probability
15 Communications in Partial Differential Equations
15 Comptes Rendus. Mathématique. Académie des Sciences, Paris
14 International Journal for Numerical Methods in Fluids
14 Proceedings of the American Mathematical Society
14 Transactions of the American Mathematical Society
14 Applied Mathematical Modelling
14 Acta Numerica
13 Advances in Mathematics
13 BIT
13 Probability Theory and Related Fields
12 Journal of the Mechanics and Physics of Solids
12 Chinese Annals of Mathematics. Series B
12 Acta Applicandae Mathematicae
12 Mathematical Problems in Engineering
12 Analysis and Applications (Singapore)
12 SN Partial Differential Equations and Applications
11 Mathematical Methods in the Applied Sciences
11 Mathematics and Computers in Simulation
11 Quarterly of Applied Mathematics
11 Numerical Methods for Partial Differential Equations
11 European Journal of Applied Mathematics
11 The Annals of Applied Probability
11 Computational Geosciences
11 Annals of PDE
10 Neural Networks
10 Communications on Pure and Applied Analysis
10 SIAM Journal on Applied Dynamical Systems
10 Journal of Physics A: Mathematical and Theoretical
10 Kinetic and Related Models
10 East Asian Journal on Applied Mathematics
10 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences
9 Bulletin of Mathematical Biology
9 Journal of the European Mathematical Society (JEMS)
9 Journal of Machine Learning Research (JMLR)
9 Journal of Hyperbolic Differential Equations
9 Analysis & PDE
9 Advances in Mathematical Physics
9 SIAM Journal on Mathematics of Data Science
8 Inverse Problems
8 SIAM Journal on Control and Optimization
8 Japan Journal of Industrial and Applied Mathematics
8 SIAM Review
8 Journal of Mathematical Sciences (New York)
8 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations
8 European Journal of Mechanics. A. Solids
8 Oberwolfach Reports
...and 304 more Serials
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Cited in 51 Fields

1,862 Partial differential equations (35-XX)
1,600 Numerical analysis (65-XX)
1,386 Fluid mechanics (76-XX)
677 Probability theory and stochastic processes (60-XX)
483 Computer science (68-XX)
463 Statistical mechanics, structure of matter (82-XX)
437 Mechanics of deformable solids (74-XX)
224 Dynamical systems and ergodic theory (37-XX)
195 Calculus of variations and optimal control; optimization (49-XX)
162 Biology and other natural sciences (92-XX)
139 Ordinary differential equations (34-XX)
103 Systems theory; control (93-XX)
102 Statistics (62-XX)
96 Game theory, economics, finance, and other social and behavioral sciences (91-XX)
93 Optics, electromagnetic theory (78-XX)
92 Quantum theory (81-XX)
73 Mechanics of particles and systems (70-XX)
68 Approximations and expansions (41-XX)
63 Operations research, mathematical programming (90-XX)
62 Classical thermodynamics, heat transfer (80-XX)
58 Geophysics (86-XX)
31 Global analysis, analysis on manifolds (58-XX)
30 Functional analysis (46-XX)
28 Linear and multilinear algebra; matrix theory (15-XX)
26 Real functions (26-XX)
25 Harmonic analysis on Euclidean spaces (42-XX)
25 Operator theory (47-XX)
23 Integral equations (45-XX)
20 Information and communication theory, circuits (94-XX)
14 Measure and integration (28-XX)
13 General and overarching topics; collections (00-XX)
13 Differential geometry (53-XX)
12 Astronomy and astrophysics (85-XX)
11 Combinatorics (05-XX)
8 Difference and functional equations (39-XX)
6 Functions of a complex variable (30-XX)
6 Convex and discrete geometry (52-XX)
6 Relativity and gravitational theory (83-XX)
5 History and biography (01-XX)
5 Potential theory (31-XX)
4 Special functions (33-XX)
3 Number theory (11-XX)
2 Algebraic geometry (14-XX)
2 Several complex variables and analytic spaces (32-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 Abstract harmonic analysis (43-XX)
1 Integral transforms, operational calculus (44-XX)
1 General topology (54-XX)

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