Weinan E 
Professor, Department of Mathematics and
Program in Applied and Computational Mathematics
Princeton University
Princeton, NJ 085441000 U.S.A.
Phone: (609)2583683 ~ Fax: (609)2581735
weinan@math.princeton.edu 

Slides of the talk at the 100th anniversary of Professor Feng Kang,
"Machine Learning and Computational Mathematics"
Slides of the talk at MSML2020,
"Towards a Mathematical Understanding of Machine Learning:
what we know and what we don't"
Slides of the talk at IPAM,
"Machine LearningBased Multiscale Modeling"
Slides of the talk at Monterey,
"Deep learning based algorithms for high dimensional PDEs and control"
Slides of the talk at ICIAM 2019,
"Machine Learning: Mathematical Theory and Scientific Applications"
Recent review articles
Weinan E,
"Machiene learning and computational mathematics"
, 2020.
Weinan E, Chao Ma, Stephan Wojtowytsch, and Lei Wu,
"Towards a Mathematical Understanding of
Neural NetworkBased Machine Learning:
what we know and what we don’t"
, 2020.
Weinan E, Jiequn Han and Arnulf Jentzen,
"Algorithms for Solving High Dimensional PDEs: From
Nonlinear Monte Carlo to Machine Learning"
, 2020.
Weinan E, Jiequn Han and Linfeng Zhang,
"Integrating Machine Learning with PhysicsBased Modeling"
, 2020.
Current Research Interests and Highlights:
Continuous formulation of machine learning (papers 10, 17, 34)
Monte Carlolike estimates of the generalization error for
shallow and deep neural network models (papers 28, 29, 30)
Analysis of the stochastic gradient descent algorithm using SDEs (papers 2, 6, 7)
The first machine learningbased algorithms for solving high dimensional
control problems (paper 4).
The first machine learningbased
algorithms for solving high dimensional nonlinear PDEs (papers 8, 9, 15)
Endtoend neural network models for interatomic potentials (the Deep
Potential) and molecular dynamics (Deep Potential based molecular dynamics or DeePMD)
(papers 14, 16, 22).
Concurrent learning algorithm for automatically
generating the data and the machine learningbased model for interatomic potentials
(DPGEN) (papers 24).
Machine learningbased moment closure hydrodynamic model
for kinetic equations (paper 32).
Recent Publications List:
52. Weinan E, Stephan Wojtowytsch,
"A priori estimates for classification problems using neural networks"
, 2020.
51. Yucheng Yang, Yue Pang, Guanhua Huang, and Weinan E,
"The Knowledge Graph for
Macroeconomic Analysis with Alternative Big Data"
, 2020.
50. Yucheng Yang, Zhong Zheng, and Weinan E,
"Interpretable Neural Networks for Panel
Data Analysis in Economics"
, 2020.
49. Chao Ma, Lei Wu and Weinan E,
"A Qualitative Study of the Dynamic Behavior of Adaptive
Gradient Algorithms"
, 2020.
48. Zhong Li, Jiequn Han, Weinan E, and Qianxiao Li,
"On the Curse of Memory in Recurrent Neural Networks:
Approximation and Optimization Analysis"
, 2020.
47. Haijun Yu, Xinyuan Tian, Weinan E, and Qianxiao Li,
"OnsagerNet: Learning Stable and Interpretable Dynamics using a
Generalized Onsager Principle"
, 2020.
46. Chao Ma, Lei Wu and Weinan E,
"The Slow Deterioration of the Generalization Error of the
Random Feature Model"
, 2020.
45. Yixiao Chen, Linfeng Zhang, Han Wang and Weinan E,
"DeePKS: a comprehensive datadriven approach towards chemically
accurate density functional theory"
, 2020.
44. Weinan E, Stephan Wojtowytsch,
"On the Banach spaces associated with multilayer ReLU networks: Function
representation, approximation theory and gradient descent dynamics"
, 2020.
43. Pinchen Xie, Weinan E,
"Coarsegrained spectral projection (CGSP): A scalable and
parallelizable deep learningbased approach to quantum unitary dynamics"
, 2020.
42. Chao Ma, Lei Wu, Weinan E,
"The QuenchingActivation Behavior of the Gradient Descent
Dynamics for Twolayer Neural Network Models"
, 2020.
41. Weinan E, Stephan Wojtowytsch,
"Representation formulas and pointwise properties for Barron functions"
, 2020.
40. Stephan Wojtowytsch, Weinan E,
"Can Shallow Neural Networks Beat the Curse of Dimensionality? A mean field training
perspective"
, 2020.
39. Weinan E, Stephan Wojtowytsch,
"Kolmogorov Width Decay and Poor Approximators in Machine Learning: Shallow Neural Networks,
Random Feature Models and Neural Tangent Kernels"
, 2020.
38. Yixiao Chen, Linfeng Zhang, Han Wang, Weinan E,
"Ground state energy functional with HartreeFock efficiency and chemical accuracy"
, 2020.
37. Weile Jia, Han Wang, Mohan Chen, Denghui Lu, Jiduan Liu, Lin Lin, Roberto Car, Weinan E,
Linfeng Zhang,
"Pushing the limit of molecular dynamics with ab initio accuracy to 100 million atoms with
machine learning"
, 2020.
36. Huan Lei, Lei Wu and Weinan E,
"Machine learning based nonNewtonian fluid model with
molecular fidelity"
, 2020.
35. Weinan E, Chao Ma and Lei Wu,
"On the Generalization Properties of Minimumnorm Solutions for Overparameterized Neural Network Models"
, 2019.
34. Weinan E, Chao Ma and Lei Wu,
"Machine Learning from a Continuous Viewpoint"
, 2019.
33. Weinan E and Yajun Zhou,
"A mathematical model for linguistic universals"
, 2019.
32. Jiequn Han, Chao Ma, Zheng Ma, Weinan E,
"Uniformly Accurate Machine Learning Based Hydrodynamic Models for Kinetic Equations"
, 2019.
31. L.F. Zhang, M.H. Chen, X.F. Wu, H. Wang, W. E and R. Car,
"Deep neural networks for Wannier function centers"
, 2019.
30. W. E, C. Ma and L. Wu
"Barron spaces and
compositional function spaces for neural network models"
, 2019.
29. W. E, C. Ma and Q.C. Wang
"A priori estimates of the
population risk for residual networks'' , 2019.
28. W. E, C. Ma and L. Wu
"A priori estimates
for two layer neural networks" , 2019.
27. W. E, C. Ma, Q.C. Wang and L. Wu
"Analysis of the Gradient
Descent Algorithm for a Deep Neural Network Model with Skipconnections"
, 2019.
26. W. E, C. Ma and L. Wu
"A Comparative Analysis of
the Optimization and Generalization Property of Twolayer Neural Network
and Random Feature Models Under Gradient Descent Dynamics"
, 2019.
25.5. HsinYu Ko, Linfeng Zhang, Biswajit Santra, Han Wang, Weinan E, Robert A.
DiStasio Jr., Roberto Car
" Isotope Effects in Liquid Water
via Deep Potential Molecular Dynamics"
Molecular Physics, to appear, 2019.
25. L. Zhang, D.Y. Lin, H. Wang, R. Car and W. E,
"Active
learning of uniformly accurate interatomic potentials for materials simulation''
Phys. Rev. Materials 3, 023804 – Published 25 February 2019.
24. Jiequn Han, Linfeng Zhang, Weinan E,
"Solving ManyElectron Schrodinger Equation Using
Deep Neural Networks''
Journal of Computational Physics 399, 108929 , 2019.
23. L. Wu, C. Ma and W. E,
"How SGD Selects the Global Minima in Overparameterized Learning: A Stability Perspective"
NIPS, 2018.
22. L. Zhang, J. Han, H. Wang, W. Saidi, R. Car and W. E,
"Endtoend Symmetry Preserving Interatomic Potential Energy Model for Finite and Extended Systems''
NIPS, 2018.
21. L. Zhang, J. Han, H. Wang, R. Car and W. E,
"DeepPCG:Constructing
coarsegrained models via deep neural networks''
J. Chem. Phys. 149, 034101 (2018); https://doi.org/10.1063/1.5027645.
20. L. Zhang, W. E and L. Wang
"MongeAmpere flow for generative
modeling''
arxiv.org/abs/1809.10188 2018.
19. Q. Li, L. Chen, C. Tai and W. E,
"Maximum Principle Based Algorithms for Deep Learning''
JMLR, vol.18, no.165, pp.129, 2018,
https://arxiv.org/pdf/1710.09513v1.pdf.
18. C. Ma. J.C. Wang and W. E,
"Model reduction
with memory and machine learning of dynamical systems''
Comm. Comput. Phys., vol.25, no.4, pp. 947962, 2019,
17. W. E, J. Han an Q. Li,
"A MeanField Optimal Control Formulation of Deep Learning''
Research in Mathematical Sciences, vol. 6, no.10, 2018.
16. L. F. Zhang, J. Han, R. Car, H. Wang and W. E,
"Deep Potential Molecular Dynamics: A scalable model with the accuracy of quantum mechanics''
Phys. Rev. Lett., vol. 120, no. 14, pp.143001, 2018.
15. J. Han, A. Jentzen and W. E,
"Solving highdimensional partial differential equations using deep learning''
Proc. Natl. Acad. Sci., vol. 115, no. 34, pp. 85058510, 2018.
14. J. Han, L. F. Zhang, R. Car and W. E,
"Deep Potential: A
General Representation of a ManyBody Potential Energy Surface''
Comm. Comput. Phys., vol. 23, no. 3, pp. 629639, 2018
13. W. E and Q.C. Wang
"Exponential convergence of the deep neural network approximation for
analytic functions ''
Science China Mathematics, vol. 61, Issue 10, pp 1733–1740, 2018
12. L. F. Zhang, H. Wang and W. E,
"Reinforced dynamics for the enhanced sampling in large atomic and molecular systems. I. Basic Methodology''
J. Chem. Phys., vol. 148, pp.124113, 2018.
11. H. Wang, L. F. Zhang, J. Han and W. E,
"DeePMDkit: A deep learning package for manybody potential energy representation and molecular dynamics''
Comput. Phys. Comm., vol. 228, pp. 178184, 2018.
10. W. E,
"A Proposal on Machine Learning via Dynamical Systems''
Comm. Math. Stat., vol.5, no.1., pp.111, 2017.
9. W. E and B. Yu,
"The Deep Ritz method: A deep learningbased numerical algorithm for solving variational problems''
Comm. Math. Stats., vol. 6, no. 1, pp. 112. 2018.
8. W. E, J. Han and A. Jentzen,
"Deep learningbased numerical methods for highdimensional parabolic partial differential equations and backward stochastic differential equations''
Comm. Math. Stats., vol. 5, no. 4, pp. 349380, 2017.
7. Q. Li, C. Tai and W. E,
"Stochastic modified equations and the dynamics of stochastic gradient algorithms''
JMLR, vol. 20, no. 40, pp. 147, 2019.
6. Q. Li, C. Tai and W. E,
"Stochastic modified equations and adaptive stochastic gradient algorithms''
ICML, 2017.
5. W. E and Y. Wang,
"Optimal convergence rates of the universal approximation error''
Research in Mathematical Sciences,vol. 4, no. 2, 2017.
4. J. Han and W. E,
"Deep learning approximation for stochastic control problems''
arxiv.org/abs/1611.07422, NIPS Workshop on Deep Reinforcement Learning, 2016.
3. C. Tai and W. E,
"Multiscale adaptive representation of signals, I''
JMLR, vol.17, no. 140, pp. 138, 2016.
2. Q. Li, C. Tai and W. E,
"Dynamics of stochastic gradient algorithms''
, 2016
1. C. Tai, T. Xiao, X. Wang and W. E,
"Convolutional neural networks with lowrank
regularization'' ICLR, 2016.
Multilevel Picard method:
Weinan E, Martin Hutzenthaler, Arnulf Jentzen and Thomas Kruse,
"Multilevel Picard iterations for solving smooth semilinear parabolic heat equations''
, 2016.
Research:
My current work focuses on the mathematical theory of machine learning
and integrating machine learning with multiscale modeling.
Research summary: My work draws inspiration from various
disciplines of sciences and has made an impact in fluid dynamics, chemistry, material
sciences, and soft condensed matter physics. I have contributed to the resolution of
some long standing scientific problems such as the Burgers turbulence problem (which
was the original motivation of Burgers for proposing the wellknown Burgers equation),
the CauchyBorn rule for crystalline solids (which indeed dates back to Cauchy, and
provides a microscopic foundation for the elasticity theory), and the moving contact
line problem (which is still largely open). A common theme is to try bringing clarity
to scientific issues through mathematics. A second theme is multiscale and/or
multiphysics problems. I have also worked on building the mathematical framework and
finding effective numerical algorithms for modeling rare events which is a very
difficult class of problems involving multiple time scales (string method, minimum
action methods, transition path theory, etc). I have also worked on multiscale
analysis and algorithms (e.g. the heterogeneous multiscale method)
for stochastic simulation algorithms, homogenization problems,
problems with multiple time scales, complex fluids, etc. My book (Principles of
MultiScale Modeling, Cambridge Univ Press) provides a broad introduction to this subject. A third theme is to develop and analyze algorithms in general. In computational fluid mechanics, I was involved in analyzing and developing vorticitybased methods, the project method and the gauge method. In density functional theory (DFT), my collaborators and I have developed the
PEXSI algorithm, which is so far the most efficient algorithm for DFT.
Microscopic Mechanisms of Equilibrium Melting of a Solid
The microscopic mechanism of the melting process of simple solids has been an outstanding issue for a long time. Lindemann and Max Born each proposed his own version of melting criterion. Classical nucleation theory also gives a prediction about the melting pathway. So far direct detailed theoretical or experimental investigation of this process has not been possible. However, with the advent of advanced simulation algorithms (free energy sampling mehods, string methods for computing transition pathways, etc), it is now possible to study these problems computationally.
PEXSI Webpage
Electronic structure analysis, using for example density functional theory,
is at the core of material science and chemistry.
It is also among the most challenging problems in computational science.
We developed the PEXSI algorithm (pole expansion + selected inversion)
which, for the first time, has brought the
computational complexity of density functional theory from cubic scaling to quadratic
scaling for general three dimensional systems.
This algorithm has been implemented in SIESTA.
Here are some examples of the work I have been involved with (click on the ``+'' sign to read more):
Burgers turbulence
We have analyzed the
statistical properties of solutions to the Burgers equation with random
initial data and random forcing. This series of work provided answers to some
of the questions that Burgers proposed back in the early 20th century, and
resolved some of controversies concerning the asymptotics of the probability distribution
functions for the random forced Burgers equation.
 W. E and E. VandenEijnden. Asymptotic theory for the probability density functions in Burgers turbulence. Phys. Rev. Lett., vol. 83, no. 13, pp. 25722575, 1999.
 W. E, K. Khanin, A. Mazel and Ya. Sinai. Probability distribution functions for the random forced Burgers equation. Phys. Rev. Lett., vol. 78, no. 10, pp. 19041907, 1997.
 M. Avellaneda and W. E. Statistical properties of shocks in Burgers turbulence. Comm. Math. Phys., vol. 172, no. 1, pp. 1338, 1995.
 M. Avellaneda, R. Ryan and W. E. PDFs for velocity and velocity gradients in Burgers' turbulence. Phys. Fluids, vol. 7, no. 12, pp. 30673071, 1995.
From quantum and molecular mechanics to macroscopic theories of solids (CauchyBorn rule and related topics)
The objective here is to understand solids at the level of quantum mechanics or molecular
mechanics. As a byproduct, we give a rigorous derivation of the macroscopic continuum models of solids. A key ingredient in this analysis is to understand the various levels of stability conditions (quantum, classical but at the atomic level and classical but at the macro level).
 W. E and J. Lu, "The KohnSham equation for deformed crystals," Memoire of the American Math Socieety, 2012.
 W. E and J. Lu, "The electronic structure of smoothly deformed crystals: Wannier functions and the CauchyBorn rule," Arch. Ration. Mech. Anal., vol. 199, pp. 407433, 2011.
 W. E and J. Lu, "The electronic structure of smoothly deformed crystals: CauchyBorn rule for the nonlinear tightbinding model," Comm. Pure Appl. Math., vol. 63, pp. 14321468, 2010.
 W. E, J. Lu and X. Yang, "Effective Maxwell equations from timedependent density functional theory," Acta Math. Sinica, vol. 27, pp. 339368, 2011.
 W. E, J. Lu and X. Yang, "Asymptotic analysis of the quantum dynamics: The BlochWigner transform and Bloch dynamics," Acta. Appl. Math. Sinica., (25 July 2011), pp. 112.
 W. E and D. Li, "On the crystallization of 2D hexagonal lattices," Comm. Math. Phys., vol. 286, no. 3, pp. 10991140, 2009.
 W. E and P.B. Ming, "CauchyBorn Rule and the Stability of Crystalline Solids: Dynamic Problems," Acta. Math. Appl. Sin. Engl. Ser., vol. 23, no. 4, pp. 529550, 2007.
 W. E and P.B. Ming, "CauchyBorn Rule and the Stability of Crystalline Solids: Static Problems," Arch. Rat. Mech. Anal., vol. 183, no. 2, pp. 241297, 2007.
Stochastic PDEs
We have developed a new way of studying stochastic PDEs, by
viewing the stationary solutions as functionals of the stochastic forcing.
This has led to a very elegant description of the stationary solutions
of the stochastic Burgers equation and the
stochastic passive scalar equation as well as the ergodicity of the
stochastic NavierStokes equation.
 W. E and D. Liu. Gibbsian dynamics and invariant measures for stochastic dissipative PDEs. J. Stat. Phys., vol. 108, no. 56, pp. 11251156, 2002.
 W. E. "Stochastic PDES in turbulence theory," Proc. 1st Intl. Congress Chinese Math. (Beijing, 1998), pp. 2746. AMS/IP Stud. Adv. Math, vol. 20, Amer. Math. Soc., Providence, RI, 2001.
 W. E and J.C. Mattingly. Ergodicity for the NavierStokes equation with degenerate random forcing: Finitedimensional approximation. Comm. Pure Appl. Math., vol. 54, no. 11, pp. 13861402, 2001.
 W. E, J.C. Mattingly and Ya. Sinai. Gibbsian dynamics and ergodicity for the stochastically forced NavierStokes equation. Comm. Math. Phys., vol. 224, no. 1, pp. 83106, 2001.
 W. E and J.C. Mattingly. Ergodicity for the NavierStokes equation with degenerate random forcing: Finitedimensional approximation. Comm. Pure Appl. Math., vol. 54, no. 11, pp. 13861402, 2001.
 W. E, J.C. Mattingly and Ya. Sinai. Gibbsian dynamics and ergodicity for the stochastically forced NavierStokes equation. Comm. Math. Phys., vol. 224, no. 1, pp. 83106, 2001.
 W. E and E. VandenEijnden. Generalized flows, intrinsic stochasticity and turbulent transport. Proc. Natl. Acad. Sci., vol. 97, no. 15, pp. 82008205, 2000.
Modeling rare events
My work on modeling rare events (joint with Weiqing Ren and Eric VandenEijnden)
has centered around developing the
string method, which is now quite popular in cmputational
chemistry and begins to get popularity
in material science, as well as the transition path theory, which is a general
theoretical framework for analyzing transition events in complex systems.
 W. E and X. Zhou, The gentlest ascent dynamics.Nonlinearity, vol. 24, no. 6, pp. 1831, 2011.
 W. E and E. VandenEijnden, The transition path theory and pathfinding algorithms for the study of rare events.Ann. Rev. Phys. Chem., vol. 61, pp. 391420, 2010.
 X. Cheng, L. Lin, W. E, AC. Shi, and P. Zhang, Nucleation of Ordered Phases in Block Copolymers. Phys. Rev. Lett., vol. 104, pp. 1483011483014, 2010.
 X. Wan, X. Zhou, and W. E, Study of noiseinduced transitions in the KuramotoSivashinsky equation via the minimum action method. Nonlinearity, vol. 23, no. 3, pp. 475494, 2010.
 X. Zhou and W. E, Study of noiseinduced transitions in the Lorenz system using the minimum action method. Comm. Math. Sci., vol. 8, pp. 341355, 2010.
 W. E, W. Ren, E. VandenEijnden. Simplified and improved string method for computing the minimum energy paths in barriercrossing events. J. Chem. Phys., vol. 126, no. 16, 164103, 2007.
 W. E and E. VandenEijnden. Towards a theory of transition paths. J. Stat. Phys., vol. 123, No. 3, 503523, 2006.
 W. Ren, E. VandenEijnden, P. Maragakis and W. E. Transition pathways in complex systems: Application of the finite temperature string method to the alanine dipeptide. J. Chem. Phys., vol. 123, 134109, 2005.
 W. E, W. Ren and E. VandenEijnden. Finite temperature string method for the study of rare events. J. Phys. Chem. B, 109, 66886693, 2005.
 W. E, W. Ren and E. VandenEijnden. String method for the study of rare events. Phys. Rev. B, vol. 66, no. 5, 052301, 2002.
Multiscale methods
We have developed the framework of the heterogeneous
multiscale method (HMM). HMM has led to very promising applications to stochastic
simulation algorithms, ODEs with multiple time scales, and many other areas.
It also provides a very nice framework for analyzing multiscale methods.
 A. Abdulle, W. E, B. Engquist and E. VandenEijnden, The heterogenous multiscale methods. Acta. Numerica, pp. 187, 2012.
 W. E, B. Engquist, X. Li, W. Ren and E. VandenEijnden. Heterogeneous multiscale methods: A review. Comm. Comput. Phys., vol. 2, no. 3, pp. 367450, 2007.
 W. E, P.B. Ming and P.W. Zhang. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc., vol. 18, no. 1, pp. 121156, 2005.
 W. E, D. Liu and E. VandenEijnden. Analysis of multiscale methods for stochastic differential equations. Comm. Pure Appl. Math., vol. 58, No. 11, 15441585, 2005.
 W. E. Analysis of the heterogeneous multiscale method for ordinary differential equations. Comm. Math. Sci., vol. 1, no. 3, pp. 423436, 2003.
 W. E and B. Engquist. The heterogeneous multiscale methods. Comm. Math. Sci., vol. 1, no. 1, pp. 87132, 2003.
 W. E and B. Engquist. Multiscale modeling and computation. Notices Amer. Math. Soc., vol. 50, no. 9, pp. 10621070, 2003.
 W. E, D. Liu and E. VandenEijnden. Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys., vol. 221, no. 1, pp. 158180, 2007.
 W. E, D. Liu and E. VandenEijnden. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys., vo. 123, 194107, 2005.
Soft condensed matter physics
We have developed the first general nonlinear model
for smectic A liquid crystals and used it to study the interesting filamentary
structures arising in isotropicsmectic phase transition. We have also
developed models for the dynamics of membranes and polymer phase separations
that are consistent with thermodynamics.
In addition, we have developed models for general inhomogeneous liquid crystal
polymer systems using
the oneparticle probability distribution function as the order parameter.
 D. Hu, P. Zhang and W. E. Continuum theory of a moving membrane. Phys. Rev. E, vol. 75, no. 4, 041605, 2007.
 Q. Wang, W. E, C. Liu, P.W. Zhang. Kinetic theory for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential. Phys. Rev. E, vol. 65, no. 5, 051504, 2002.
 W. E and P. Zhang. A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit. Methods Appl Anal., vol 13, no. 2, pp. 181198, 2006.
 D. Zhou, P. Zhang and W. E. Modified models of polymer phase separation. Phys. Rev. E, vol. 73, 061801, 2006.
 C.B. Muratov and W. E. Theory of phase separation kinetics in polymerliquid crystal systems. J. Chem. Phys., vol. 116, no. 11, pp. 47234734, 2002.
 W. E and P. PalffyMuhoray. Dynamics of filaments during the isotropicsmectic A phase transition. J. Nonlin. Sci., vol. 9, no. 4, pp. 417437, 1999.
 W. E. Nonlinear continuum theory of smecticA liquid crystals. Arch. Rat. Mech. Anal., vol. 137, no. 2, pp. 159175, 1997.
 W. E and P. PalffyMuhoray. Phase separation in incompressible systems. Phys. Rev. E, vol. 55, no. 4, pp. R3844R3846 , 1997.
 F. Otto and W. E. Thermodynamically driven incompressible fluid mixtures. J. Chem. Phys., vol. 107, no. 23, pp. 1017710184, 1997.
Computational fluid dynamics
JianGuo Liu and I addressed long time
controversies in vorticity boundary conditions and the numerical boundary layers
for the projection method.
A posteriori error estimates
In my master degree thesis completed in 1985 under the supervision of Prof. Huang
Hongci, I established some of the earliest results on a posteriori error estimates
for finite element methods. I introduced the Clement interpolation technique,
and proved upper and lower bounds for local error estimators.
Weak KAM theory
Under the influence of Jurgen Moser, I independently (of Fathi) developed the weak KAM theory.
This was one of the first application of PDE methods to the study of
dynamical systems. The most interesting aspect is to study the
implication of weak solutions of the HamiltonJacobi equation to Hamiltonian systems.
This gives an alternative (and much simplified) viewpoint for the AubryMather theory.
Numerical algorithms for KohnSham density functional theory
 L. Lin, C. Yang, J. Lu, L. Ying and W. E, "A fast parallel algorithm for selected inversion of structured sparse matrix with application to 2D electronic structure calculations," SIAM J. Sci. Computing, vol. 33, 13291351, 2011.
 W. E, T. Li and J. Lu, "Localized basis of eigensubspaces," Proc. Natl. Acad. Sci. USA, vol. 109, pp. 12731278, 2010.
 L. Lin, C. Yang, J. C. Meza, L. Ying and W. E, "SelInv  An algorithms for selected inversion of a sparse symmetric matrix," ACM Transactions on Mathematical Software, vol. 37, no. 4, pp. 40:140:19, 2011.
 L. Lin, C. Yang, J. Lu, L. Ying and W. E, "A fast parallel algorithm for selected inversion of structured sparse matrices with application to 2D electronic structure calculation," Lawrence Berkeley National Laboratory. LBNL Paper LBNL2677E. Retrieved from: http://escholarship.org/uc/item/46q6w084, 2010.
 L. Lin, J. Lu, L. Ying and W. E, "Polebased approximation of the FermiDirac function," Chin. Ann. Math., vol. 30B, pp. 729742, 2009.
 L. Lin, J. Lu, L. Ying, R. Car and W. E, "Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems," Comm. Math. Sci., vol. 7, pp. 755777, 2009.
 L. Lin, J. Lu, R. Car and W. E, "Multipole representation of the Fermi operator with application to electronic structure analysis of metallic systems," Phys. Rev. B, vol. 79, no. 11, pp. 11513311511310, 2009.
 W. Gao and W. E, "Orbital minimization with localization," Discrete and Continuous Dynamical Systems,vol. 23, no. 12, pp. 249264, 2009.
W. E and Jianchun Wang,
"A thermodynamic study of the twodimensional pressuredriven channel flow",
Discrete and Continuous Dynamical Systems,vol.36, no. 8, pp. 43494366, 2016.
Q. Li and W. E,
"The free action for nonequilbirium systems'',
J. Stat. Phys., vol. 161, no. 2, 300325, 2015.
Other topics I have made contributions to include: Onsager's conjecture on
the energy conservation for weak solutions of the 3D Euler's equation,
homogenization and twoscale convergence,
singularity formation in solutions of Prandtl's equation, GinzburgLandau
vortices, micromagnetics and the LandauLifshitz equation, stochastic resonance,
etc.
String Method Webpage
HMM webpage
Mathematical theory of solids at the atomic
and macroscopic scales
The main objective is to develop a rigorous mathematical theory
for solids. This requires understanding models of solids at the
electronic, atomistic and continuum level, as well as the relation
between these models. Problems of interest include: (1). The
crystallization problem: Why solids take the form of crystal lattice at
zero temperature? (2). The CauchyBorn rule, which serves as a
connection between atomistic and continuum models of solids.
 W. E and D. Li. On the
crystallization of 2d hexagonal lattice. Comm. Math. Phys.,
submitted.
 W. E and J.F. Lu. The continuum limit and QMcontinuum approximation of quantum mechanical models of solids. Comm. Math. Sci., vol. 5, no. 3, pp. 679696, 2007.
 W. E and J.F. Lu. The elastic continuum limit of the tight binding model. Chinese Ann. Math. Ser. B, vol. 28, no. 6, pp. 665676, 2007.
 W. E and P.B. Ming. CauchyBorn rule and the stability of crystalline solids: Dynamic problems. Acta Math. Appl. Sin. Engl. Ser., vol. 23, no. 4, pp. 529550, 2007.
 W. E and P. B. Ming. CauchyBorn rule and the stability of crystalline solids: Static problems. Arch. Rat. Mech. Anal., vol. 183, no. 2, pp. 241297, 2007.
Electronic structure, density functional theory
The main objective is to understand the mathematical foundation
of electronic structure analysis, to develop and analysis efficient
algorithms.
General issues in multiscale modeling
 S. Chen, W. E, Y. Liu and C.W. Shu. A discontinuous Galerkin implementation of a domain decomposition method for kinetichydrodynamic coupling multiscale problems in gas dynamics and device simulations. J. Comput. Phys., vol. 225, no. 2, pp. 13141330, 2007.
 W. E, B. Engquist, X. Li, W. Ren and E. VandenEijnden. Heterogeneous multiscale methods: A review. Comm. Comput. Phys., vol. 2, no. 3, pp. 367450, 2007.
 W. E and J.F. Lu. Seamless multiscale modeling via dynamics on fiber bundles. Comm. Math. Sci., vol. 5, no. 3, pp. 649663, 2007.
 X. Yue and W. E. The local microscale problem in the multiscale modelling of strongly heterogeneous media: Effect of boundary conditions and cell size. J. Comput. Phys., vol. 222, no. 2, pp. 556572, 2007.
 S. Chen, W. E and C.W. Shu. The heterogeneous multiscale method based on the discontinuous galerkin method for hyperbolic and parabolic problems. Multiscale Model. Simul., vol. 3, no. 4, pp. 871894, 2005.
 W. E and B. Engquist. The heterogeneous multiscale method for homogenization problems. Multiscale Methods in Sci. and Eng., pp. 89110. Lect. Notes in Comput. Sci. Eng., vol. 44, Springer, Berlin, 2005.
 W. E and P.B. Ming. Analysis of the local quasicontinuum method. Frontiers and Prospects of Contemp. Appl. Math., pp. 1832. Contemporary Appl. Math., vol. 6, Higher Education Press, Beijing, 2005.
 W. E, P.B. Ming and P.W. Zhang. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc., vol. 18, no. 1, pp. 121156, 2005.
 W. E, D. Liu and E. VandenEijnden. Analysis of multiscale methods for stochastic differential equations. Comm. Pure Appl. Math., vol. 58, No. 11, 15441585, 2005.
 W. E and B. Engquist. The heterogeneous multiscale method. Second Intl. Congress of Chinese Mathematicians. Proc. of ICCM2001, Taipei, pp. 5774, New Studies in Advanced Mathematics, vol. 4, Intl. Press, 2004.
 W. E, X. Li, E. VandenEijnden. Some recent progress in multiscale modeling. Multiscale Modelling and Simulation, pp. 322. Lect. Notes Comput. Sci. Eng., vol. 39, Springer, Berlin, 2004.
 W. E and X.T. Li. Analysis of the heterogeneous multiscale method for gas dynamics. Methods Appl. Anal., vol. 11, no. 4, pp. 557572, 2004.
 W. E and P.B. Ming. Analysis of multiscale methods. J. Comput. Math., vol. 22, no. 2, pp. 210219, 2004.
 W. E nd X. Yue. Heterogeneous multiscale method for locally selfsimilar problems. Comm. Math. Sci., vol. 2, no. 1, pp. 137144, 2004.
 W. E. Analysis of the heterogeneous multiscale method for ordinary differential equations. Comm. Math. Sci., vol. 1, no. 3, pp. 423436, 2003.
 A. Abdulle and W. E. Finite difference heterogeneous multiscale method for homogenization problems. J. Comput. Phys., vol. 191, no. 1 pp. 1839, 2003.
 L.T. Cheng and W. E. The heterogeneous multiscale method for interface dynamics. Recent advances in scientific computing and partial differential equations (Hong Kong, 2002), pp. 4353, Contemp. Math., vol. 330, Amer. Math. Soc., Providence, RI, 2003.
 W. E and B. Engquist. The heterogeneous multiscale methods. Comm. Math. Sci., vol. 1, no. 1, pp. 87132, 2003.
 W. E and B. Engquist. Multiscale modeling and computation. Notices Amer. Math. Soc., vol. 50, no. 9, pp. 10621070, 2003.
 W. E, B. Engquist and Z. Huang. Heterogeneous multiscale method: A general methodology for multiscale modeling. Phys. Rev. B, vol. 67, no. 9, 092101, 2003.
Problems with multiple time scales
 T. Li, A. Abdulle and W. E. Effectiveness of implicit methods for stiff stochastic differential equations. Comm. Comput. Phys., vol. 3, no. 2, pp. 295307, 2008.
 W. E, D. Liu and E. VandenEijnden. Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys., vol. 221, no. 1, pp. 158180, 2007.
 W. E, D. Liu and E. VandenEijnden. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys., vo. 123, 194107, 2005.
 W. E, D. Liu and E. VandenEijnden. Analysis of multiscale methods for stochastic differential equations. Comm. Pure Appl. Math., vol. 58, No. 11, 15441585, 2005.
 W. E and X.T. Li. Analysis of the heterogeneous multiscale method for gas dynamics. Methods Appl. Anal., vol. 11, no. 4, pp. 557572, 2004.
Stochastic chemical kinetic systems
Multiscale modeling of solids
 W. Guo, T. P. Schulze and W. E. Simulation of impurity diffusion in a strained nanowire using offlattice KMC. Comm. Comput. Phys., vol. 2, no. 1, pp. 164176, 2007.
 X. Li and W. E. Variational boundary conditions for molecular dynamics simulations of crystalline solids at finite temperature: Treatment of the thermal bath. Phys. Rev. B, vol 76, no. 10, 104107, 2007.
 J.Z. Yang and W. E. Generalized CauchyBorn rules for elastic deformation of sheets, plates, and rods: Derivation of continuum models from atomistic models. Phys. Rev. B, vol. 74, no 18, 184110, 2006.
 Y. Xiang, H. Wei, P.B. Ming and W. E. A generalized Peierls�Nabarro model for curved dislocations and core structures of dislocation loops in Al and Cu. Acta Materialia, in press, available online 14 January 2008.
 W. E, J.F. Lu, J.Z. Yang. Uniform accuracy of the quasicontinuum method. Phys. Rev. B, vol. 74, 214115, 2006.
 X.T. Li and W. E. Variational boundary conditions for molecular dynamics simulation of solids at low temperature. Comm. Comput. Phys., vol. 1, No. 1, 135175, 2006.
 N. Choly, G. Lu, W. E and E. Kaxiras. Multiscale simulations in simple metals: A densityfunctional based methodology. Phys. Rev. B, vol. 71, 094101, 2005.
 X.T. Li and W. E. Multiscale modeling of the dynamics of solids at finite temperature. J. Mech. Phys. Solids, vol. 53, 16501685, 2005.
 W. E and X.T. Li. Multiscale modeling of crystalline solids. Handbook of Materials Modeling, Part A, edited by S. Yip., pp. 14911506, Springer Netherlands, 2005.
 Y. Xiang and W. E. Misfit elastic energy and a continuum model for epitaxial growth with elasticity on vicinal surfaces. Phys. Rev. B, vol. 69, no. 3, 035409, 2004.
 Y. Xiang, D.J. Srolovitz, L.T. Cheng and W. E. Level set simulations of dislocationparticle bypass mechanisms. Acta Materialia, vol. 52, no. 7, pp. 17451760, 2004.
 Y. Xiang, L.T. Cheng, D.J. Srolovitz and W. E. A level set method for dislocation dynamics. Acta Materialia, vol. 51, no. 18, pp. 54995518, 2003.
 T. Schulze, P. Smereka and W. E. Coupling kinetic MonteCarlo and continuum models with application to epitaxial growth. J. Comput. Phys., vol 189, no. 1, pp. 197211, 2003.
 Y. Xiang and W. E. Nonlinear evolution equation for the stressdriven morphological instability. J. Appl. Phys., vol. 91, no. 11, pp. 94149422, 2002.
 W. E and Z. Huang. A dynamic atomisticcontinuum method for the simulation of crystalline materials. J. Comput. Phys., vol. 182, no. 1, pp. 234261, 2002.
 W. E and Z. Huang. Matching conditions in atomisticcontinuum modeling of materials. Phys. Rev. Lett., vol. 87, no. 13, 135501, 2001.
 W. E and N.K. Yip. Continnum theory of epitaxial crystal growth. I. J. Stat. Phys., vol. 104, no. 12, pp. 221253, 2001.
 M.I. Mendelev, D.J. Srolovitz and W. E. Grainboundary migration in the presence of diffusing impurities: simulations and analytical models. Philos. Mag. A, vol. 81, no. 9, pp. 22432269, 2001.
 T. Schulze and W. E. A continuum model for the growth of epitaxial films. J. Crystal Growth, vol. 222, no. 12, pp. 414425, 2001.
 W. E and N.K. Yip. Continuum limits of step flow models. EQUADIFF 99 Proc. Intl. Conf. Differential Equations, vol. 1, 2 (Berlin, 1999), pp. 448453, World Sci. Publishing, River Edge, NJ, 2000.
 R. Caflisch, W. E, M. Gyure, B. Merriman and C. Ratsch. Kinetic model for a step edge in epitaxial growth. Phys. Rev. E, vol. 59, no. 6, pp. 68796887, 1999.
Multiscale modeling of complex fluids
 D. Hu, P. Zhang and W. E. Continuum theory of a moving membrane. Phys. Rev. E, vol. 75, no. 4, 041605, 2007.
 W. E and P. Zhang. A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit. Methods Appl Anal., vol 13, no. 2, pp. 181198, 2006.
 D. Zhou, P. Zhang and W. E. Modified models of polymer phase separation. Phys. Rev. E, vol. 73, 061801, 2006.
 W. Ren and W. E. Heterogeneous multiscale method for the modeling of complex fluids and microfluidics. J. Comput. Phys., vol. 204, no. 1, pp. 126, 2005.
 S. Succi, W. E and E. Kaxiras. Lattice boltzmann methods for multiscale fluid problems. Handbook of Materials Modeling, Part B, pp. 24752486, Springer Netherlands, 2005.
 X. Nie, S. Chen, W. E and M. Robbins. Hybrid continuumatomistic simulation of singular corner flow. Phys. Fluids, vol. 16, no. 10, pp. 35793591, 2004.
 T.J. Li, E. VandenEijnden, P.W. Zhang and W. E. Stochastic models of polymeric fluids at small Deborah number. J. NonNewtonian Fluid Mechanics, vol. 121, no. 23, pp. 117125, 2004.
 X. Nie, S. Chen, W. E and M.O. Robbins. A continuum and molecular dynamics hybrid method for micro and nanofluid flow. J. Fluid Mech., vol. 500, pp. 5564, 2004.
 W. E, T.J. Li and P.W. Zhang. Wellposedness for the dumbbell model of polymeric fluids. Comm. Math. Phys., vol. 248, no. 2, pp. 409427, 2004.
 T.J. Li, E. VandenEijnden, P.W. Zhang and W. E. Stochastic models of polymeric fluids at small Deborah number. J. NonNewtonian Fluid Mechanics, vol. 121, 117125, 2004.
 W. E, T.J. Li, P.W. Zhang. Convergence of a stochastic method for the modeling of polymeric fluids. Acta Math. Appl. Sin., vol. 18, no. 4, pp. 529536, 2002.
 C.B. Muratov and W. E. Theory of phase separation kinetics in polymerliquid crystal systems. J. Chem. Phys., vol. 116, no. 11, pp. 47234734, 2002.
 P. PalffyMuhoray, T. Kosa and W. E. Brownian motors in the photoalignment of liquid crystals. Appl. Phys. A, vol. 75, no. 2, pp. 293300, 2002.
 Q. Wang, W. E, C. Liu, P.W. Zhang. Kinetic theory for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential. Phys. Rev. E, vol. 65, no. 5, 051504, 2002.
 W. E and P. PalffyMuhoray. Dynamics of filaments during the isotropicsmectic A phase transition. J. Nonlin. Sci., vol. 9, no. 4, pp. 417437, 1999.
 W. E and P. PalffyMuhoray. Orientational ratchets and angular momentum balance in the Janossy effect. Mol. Cryst. Liq. Cryst., vol. 320, no. 1, pp. 193206, 1998.
 W. E. Nonlinear continuum theory of smecticA liquid crystals. Arch. Rat. Mech. Anal., vol. 137, no. 2, pp. 159175, 1997.
 W. E and P. PalffyMuhoray. Phase separation in incompressible systems. Phys. Rev. E, vol. 55, no. 4, pp. R3844R3846 , 1997.
 F. Otto and W. E. Thermodynamically driven incompressible fluid mixtures. J. Chem. Phys., vol. 107, no. 23, pp. 1017710184, 1997.
Multiscale methods for multiscale PDEs
 X. Yue and W. E. The local microscale problem in the multiscale modelling of strongly heterogeneous media: Effect of boundary conditions and cell size. J. Comput. Phys., vol. 222, no. 2, pp. 556572, 2007.
 W. E and B. Engquist. The heterogeneous multiscale method for homogenization problems. Multiscale Methods in Sci. and Eng., pp. 89110. Lect. Notes in Comput. Sci. Eng., vol. 44, Springer, Berlin, 2005.
 W. E, P.B. Ming and P.W. Zhang. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc., vol. 18, no. 1, pp. 121156, 2005.
 X. Yue and W. E. Numerical methods for multiscale transport equations and application to twophase porous media flow. J. Comput. Phys., vol. 210, no. 2, pp. 656675, 2005.
 A. Abdulle and W. E. Finite difference heterogeneous multiscale method for homogenization problems. J. Comput. Phys., vol. 191, no. 1 pp. 1839, 2003.
The moving contact line problem and microfluidics
Homogenization theory
 B. Engquist and W. E. Large time behavior and homogenization of solutions of twodimensional conservation laws. Comm. Pure Appl. Math., vol. 46, no. 1, pp. 126, 1993.
 W. E and C.W. Shu. Effective equations and the inverse cascade theory for Kolmogorov flows. Phys. Fluids A, vol. 5, no. 4, pp. 9981010, 1993.
 W. E. Propagation of oscillations in the solutions of 1D compressible fluid equations. Comm. Partial Differential Equations, vol. 17, no. 34, pp. 545552, 1992.
 W. E. Homogenization of linear and nonlinear transport equations. Comm. Pure Appl. Math., vol. 45, no. 3, pp. 301326, 1992.
 W. E. Homogenization of scalar conservation laws with oscillatory forcing terms. SIAM J. Appl. Math., vol. 52, no. 4, pp. 959972, 1992.
 W. E and D. Serre. Correctors for the homogenization of conservation laws with oscillatory forcing terms. Asymptotic Anal., vol. 5, no. 4, pp. 311316, 1992.
 W. E. A class of homogenization problems in the calculus of variations. Comm. Pure Appl. Math., vol. 44, no. 7, pp. 733759, 1991.
 W. E and R.V. Kohn. The initial value problem for measurevalued solutions of a canonical 2x2 system with linearly degenerate fields. Comm. Pure Appl. Math., vol. 44, no. 89, pp. 9811000, 1991.
 W. E and H. Yang. Numerical study of oscillatory solutions of the gasdynamic equations. Stud. Appl. Math., vol. 85, no. 1, pp. 2952, 1991.
 W. E and T.Y. Hou. Homogenization and convergence of the vortex method for 2D Euler equations with oscillatory vorticity fields. Comm. Pure Appl. Math., vol. 43, no. 7, pp. 821855, 1990.
Analysis of stochastic partial differential equations
 W. E and D. Liu. Gibbsian dynamics and invariant measures for stochastic dissipative PDEs. J. Stat. Phys., vol. 108, no. 56, pp. 11251156, 2002.
 W. E. Stochastic PDES in turbulence theory. Proc. 1st Intl. Congress Chinese Math. (Beijing, 1998), pp. 2746. AMS/IP Stud. Adv. Math, vol. 20, Amer. Math. Soc., Providence, RI, 2001.
 W. E and J.C. Mattingly. Ergodicity for the NavierStokes equation with degenerate random forcing: Finitedimensional approximation. Comm. Pure Appl. Math., vol. 54, no. 11, pp. 13861402, 2001.
 W. E, J.C. Mattingly and Ya. Sinai. Gibbsian dynamics and ergodicity for the stochastically forced NavierStokes equation. Comm. Math. Phys., vol. 224, no. 1, pp. 83106, 2001.
 W. E. Stochastic hydrodynamics. Current Developments in Mathematics, 2000, pp. 109147, Intl. Press, Somerville, MA, 2000.
 W. E, K. Khanin, A. Mazel and Ya. Sinai. Invariant measures for Burgers equation with stochastic forcing. Ann. of Math., vol. 151, no. 3, pp. 877960, 2000.
 W. E and Ya. Sinai. Recent results on mathematical and statistical hydrodynamics. Russ. Math. Survey, vol. 55, no. 4, 635666, 2000.
 W. E, Yu. Rykov and Ya. Sinai. Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Comm. Math. Phys., vol. 177, no. 2, pp. 349380, 1996.
Rare events: String method, minimum action method and transition path theory
 W. E, W. Ren, E. VandenEijnden. Simplified and improved string method for computing the minimum energy paths in barriercrossing events. J. Chem. Phys., vol. 126, no. 16, 164103, 2007.
 T. Qian, W. Ren, J. Shi, W. E and P. Sheng. Numerical study of metastability due to tunneling: The quantum string method. Phys. A, vol. 379, no. 2, pp. 491502, 2007.
 W. E and E. VandenEijnden. Towards a theory of transition paths. J. Stat. Phys., vol. 123, No. 3, 503523, 2006.
 W. E, W. Ren and E. VandenEijnden. Transition pathways in complex systems: Reaction coordinates, isocommittor surfaces and transition tubes. Chem. Phys. Lett., vol. 413, no. 13, 242247, 2005.
 W. Ren, E. VandenEijnden, P. Maragakis and W. E. Transition pathways in complex systems: Application of the finite temperature string method to the alanine dipeptide. J. Chem. Phys., vol. 123, 134109, 2005.
 W. E, W. Ren and E. VandenEijnden. Finite temperature string method for the study of rare events. J. Phys. Chem. B, 109, 66886693, 2005.
 W. E, W. Ren, E. VandenEijnden. Minimum action method for the study of rare events. Comm. Pure Appl. Math., vol. 57, no. 5, pp. 637656, 2004.
 W. E and E. VandenEijnden. Metastability, conformation dynamics, and transition pathways in complex systems. Multiscale Modelling and Simulation, pp. 3568, Lect. Notes Comput. Sci. Eng., vol. 39, Springer, Berlin, 2004.
 W. E, W. Ren and E. VandenEijnden. Energy landscape and thermally activated switching of submicronsized ferromagnetic elements. J. Appl. Phys., vol. 93, no. 4, pp. 22752282, 2003.
 W. E, W. Ren and E. VandenEijnden. String method for the study of rare events. Phys. Rev. B, vol. 66, no. 5, 052301, 2002.
 W. E, W. Ren and E. VandenEijnden. Energy landscapes and rare events. ICM Report, vol. 1, pp. 621630, Higher Ed. Press, Beijing, 2002.
Stochastic chemical kinetic systems
``Burgers turbulence'' and passive scalar
turbulence
 W. E and E. VandenEijnden. A note on generalized flows. Phys. D, vol. 183, no. 34, pp. 159174, 2003.
 W. E. Stochastic PDES in turbulence theory. Proc. 1st Intl. Congress Chinese Math. (Beijing, 1998), pp. 2746. AMS/IP Stud. Adv. Math, vol. 20, Amer. Math. Soc., Providence, RI, 2001.
 W. E and E. VandenEijnden. Turbulent Prandtl number effect on passive scalar advection. Phys. D, vol. 152153, pp. 636645, 2001.
 W. E and E. VandenEijnden. Statistical theory for the stochastic Burgers equation in the inviscid limit. Comm. Pure Appl. Math., vol. 53, no. 7, pp. 852901, 2000.
 W. E and E. VandenEijnden. Another note on forced Burgers turbulence. Phys. Fluids, vol. 12, no. 1, pp. 149154, 2000.
 W. E and E. VandenEijnden. Generalized flows, intrinsic stochasticity and turbulent transport. Proc. Natl. Acad. Sci., vol. 97, no. 15, pp. 82008205, 2000.
 W. E and E. VandenEijnden. On the statistical solution of the Riemann equation and its implications for Burgers turbulence. Phys. Fluids, vol. 11, no. 8, pp. 21492153, 1999.
 W. E and E. VandenEijnden. Asymptotic theory for the probability density functions in Burgers turbulence. Phys. Rev. Lett., vol. 83, no. 13, pp. 25722575, 1999.
 W. E, K. Khanin, A. Mazel and Ya. Sinai. Probability distribution functions for the random forced Burgers equation. Phys. Rev. Lett., vol. 78, no. 10, pp. 19041907, 1997.
 M. Avellaneda and W. E. Statistical properties of shocks in Burgers turbulence. Comm. Math. Phys., vol. 172, no. 1, pp. 1338, 1995.
 M. Avellaneda, R. Ryan and W. E. PDFs for velocity and velocity gradients in Burgers' turbulence. Phys. Fluids, vol. 7, no. 12, pp. 30673071, 1995.
General issues in stochastic modeling
 C.B. Muratov, E. VandenEijnden, W. E. Noise can play an organizing role for the recurrent dynamics in excitable media. Proc. Natl. Acad. Sci., vol. 104, no. 3, pp. 702707, 2007.
 C.B. Muratov, E. VandenEijnden and W. E. Selfinduced stochastic resonance in excitable systems. Phys. D, vol. 210, no. 34, pp. 227240, 2005.
 P. PalffyMuhoray, T. Kosa, W. E. Brownian ratchets and the photoalignment of liquid crystals. Braz. J. Phys., vol.32 no.2b, pp. 552563, Sao Paulo, 2002.
 P. PalffyMuhoray, T. Kosa and W. E. Dynamics of a Light Driven Molecular Motor. Mol. Cryst. Liq. Cryst., vol. 375, no. 1, pp. 577592, 2002.
 T. Kosa, W. E and P. PalffyMuhoray. Brownian motors in the photoalignment of liquid crystals. Intl J. Eng. Sci., vol. 38, no. 910, pp. 10771084, 2000.
 W. E and P. PalffyMuhoray. Domain size in the presence of random fields. Phys. Rev. E, vol. 57, no. 1, pp. 135137, 1998.
Incompressible flow: Projection methods, vorticitybased methods and gauge methods
 W. E and J.G. Liu. Gauge method for viscous incompressible flows. Comm. Math. Sci., vol. 1, no. 2, pp. 317332, 2003.
 W. E and J.G. Liu. Projection method III: Spatial discretization on the staggered grid. Math. Comp., vol. 71, no. 237, pp. 2747, 2002.
 W. E. Numerical methods for viscous incompressible flows: some recent advances. Advances in scientific computing, p. 29, Science Press, 2001.
 J.G. Liu and W. E. Simple finite element method in vorticity formulation for incompressible flows. Math. Comp., vol. 70, no. 234, pp. 579593, 2001.
 W. E and J.G. Liu. Gauge finite element method for incompressible flows. Intl. J. Numer. Methods in Fluids, vol. 34, no. 8, pp. 701710, 2000.
 W. E and J.G. Liu. Finite difference schemes for incompressible flows in the velocityimpulse density formulation. J. Comput. Phys., vol. 130, no. 1, 6776, 1997.
 W. E and J.G. Liu. Finite difference methods for 3D viscous incompressible flows in the vorticityvector potential formulation on nonstaggered grids. J. Comput. Phys., vol. 138, no. 1, 5782, 1997.
 W. E and J.G. Liu. Vorticity boundary condition and related issues for finite difference schemes. J. Comput. Phys., vol. 124, no. 2, pp. 368382, 1996.
 W. E and J.G. Liu. Essentially compact schemes for unsteady viscous incompressible flows. J. Comput. Phys., vol. 126, no. 1, pp. 122138, 1996.
 W. E and J.G. Liu. Projection method II: GodunovRyabenki analysis. SIAM J. Numer. Anal., vol. 33, no. 4, pp. 15971621, 1996.
 W. E and J.G. Liu. Finite difference schemes for incompressible flows in vorticity formulations. Vortex flows and related numerical methods, II (Montreal, PQ, 1995), pp. 181195, ESAIM Proc., vol. 1, Soc. Math. Appl. Indust., Paris, 1996.
 W. E and J.G. Liu. Projection method I: Convergence and numerical boundary layers. SIAM J. Numer. Anal., vol. 32, no. 4, pp. 10171057, August, 1995.
 Z.T. Chen and W. E. Convergence of Legendre methods for NavierStokes equations. J. Comput. Math., vol. 12, no. 4, pp. 298311, 1994.
 W. E and C.W. Shu. A numerical resolution study of high order essentially nonoscillatory schemes applied to incompressible flow. J. Comput. Phys., vol. 110, no. 1, pp. 3946, 1994.
 W. E. Convergence of Fourier methods for the NavierStokes equations. SIAM J. Numer. Anal., vol. 30, no. 3, pp. 650674, 1993.
 W. E. Convergence of spectral methods for Burgers' equation. SIAM J. Numer. Anal., vol. 29, no. 6, pp. 15201541, 1992.
A posterior error estimates
Work done in Master degree thesis, under the guidance of
Professor Hongci Huang at the Chinese Academy of Sciences. The main
focus is on finite element for problems with corner singularities.
Issues discussed include: A posterior error estimates, direct and
inverseerror estimates on locally refined domains, convergence of
multigrid methods on such domains, etc.
 W. E, M. Mu and H.C. Huang. A posteriori error estimates in finite element methods. Chinese Quart. J. Math., (Chinese) vol. 3, no. 1, pp. 97107, 1988.
 W. E, H.C. Huang and W. Han. Error analysis of local refinements of polygonal domains. J. Comput. Math., vol. 5, no. 1, pp. 8994, 1987.
 H.C. Huang and W. E. A posteriori error estimates for finite element methods for onedimensional boundary value problems. Chinese Quart. J. Math., (Chinese) vol. 2, no. 1, pp. 4347, 1987.
 H.C. Huang, W. E and M. Mu. Extrapolation combined with multigrid method for solving finite element equations. J. Comput. Math., vol. 4, no. 4, pp. 362367, 1986.
Miscellaneous topics
Euler equations, boundary layer problem, AubryMather theory, micromagnetics and the LandauLifshitz equation, vortex dynamcis in GinzburgLandau theory
 W. E, D. Li. The Andersen thermostat in molecular dynamics. Comm. Pure Appl Math., vol. 61, no. 1, pp. 96136, 2008.
 W. E. Boundary layer theory and the zeroviscosity limit of the NavierStokes equation. Acta Math. Sin., vol. 16, no. 2, pp. 207218, 2000.
 W. E. AubryMather theory and periodic solutions of the forced Burgers equation. Comm. Pure Appl. Math., vol. 52, no. 7, pp. 811828, 1999.
 W. E and B. Engquist. Blowup of solutions of the unsteady Prandtl's equation. Comm. Pure Appl. Math., vol. 50, no. 12, pp. 12871293, 1997.
 P. Constantin, W. E and E.S. Titi. Onsager's conjecture on the energy conservation for solutions of Euler's equation. Comm. Math. Phys., vol. 165, no. 1, pp. 207209, 1994.
 W. E and C.W. Shu. Smallscale structures in Boussinesq convection. Phys. Fluids, vol. 6, no. 1, pp. 4958, 1994.
 Z. Cai and W. E. Hierarchical method for elliptic problems using wavelet. Comm. Appl. Numer. Methods, vol. 8, no 11, pp. 819825, 1992.
 T.F. Chan, W. E and J. Sun. Domain decomposition interface preconditioners for fourthorder elliptic problems. Appl. Numer. Math., vol. 8, no 45, pp. 317331, 1991.
 W. E. The optimal parameters of the AOR method and their effect. Math. Numer. Sin., (Chinese) vol. 6, no. 3, 329333, 1984.
Micromagnetics and LandauLifshitz equation
 X.P. Wang, K. Wang and W. E. Simulations of 3D domain wall structures in thin films. Discrete Contin. Dyn. Syst. Ser. B, vol. 6, no. 2, pp. 373389, 2006.
 C.J. GarciaCervera and W. E. Improved GaussSeidel projection method for micromagnetics simulations. IEEE Trans. Magnetics, vol. 39, no. 3, pp. 17661770, 2003.
 C.J. GarciaCervera, Z. Gimbutas and W. E. Accurate numerical methods for micromagnetics simulations with general geometries. J. Comput. Phys., vol. 184, no. 1, pp. 3752, 2003.
 C.J. GarciaCervera and W. E. Effective dynamics for ferromagnetic thin films. J. Appl. Phys., vol. 90, no. 1, pp. 370374, 2001.
 X.P. Wang, C.J. GarciaCervera and W. E. A GaussSeidel projection method for micromagnetics simulations. J. Comput. Phys., vol. 171, no. 1, pp. 357372, 2001.
 W. E and X.P. Wang. Numerical methods for the LandauLifshitz equation. SIAM J. Numer. Anal., vol. 38, no. 5, pp. 16471665, 2000.
GinzburgLandau vortices
 W. E. Dynamics of vortices in superconductors. World Congress of Nonlinear Analysts '92, vol. 4 (Tampa, FL, 1992), pp. 38113821, de Gruyter, Berlin, 1996.
 W. E. Dynamics of vortices in GinzburgLandau theories with applications to superconductivity. Phys. D, vol. 77, no. 4, pp. 383404, 1994.
 W. E. Dynamics of vortex liquids in GinzburgLandau theories with applications to superconductivity. Phys. Rev. B, vol. 50, no. 2, pp. 11261135, 1994.
 Stochastic Hydrodynamics (PS), in "Current Development in Mathematics", 2001.
 Mathematics and Sciences (PS), written for the Beijing Intelligencer, ICM 2002.
 Multiscale Modeling and Computation (PS), appeared in Notice of AMS.
 The Heterogeneous Multiscale Method: A Review (PDF),
appeared in Comm. Comput. Phys.
 Modeling rare events (PPT), talk at American Conference of Theoretical Chemistry.
 HMM papers