Weinan E | Academic Influence

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Weinan E

Mathematics

#744

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#1350

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Measure Theory

#506

World Rank

#714

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Mathematics

Weinan E's Degrees

PhD Mathematics Princeton University
Bachelors Mathematics Peking University

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According to Wikipedia, Weinan E is a Chinese mathematician. He is known for his pathbreaking work in applied mathematics and machine learning. His academic contributions include novel mathematical and computational results in stochastic differential equations; design of efficient algorithms to compute multiscale and multiphysics problems, particularly those arising in fluid dynamics and chemistry; and pioneering work on the application of deep learning techniques to scientific computing. In addition, he has worked on multiscale modeling and the study of rare events.

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Weinan E's Published Works

Number of citations in a given year to any of this author's works

Total number of citations to an author for the works they published in a given year. This highlights publication of the most important work(s) by the author

Published Works

Finite temperature string method for the study of rare events. (2002) (1197)
Solving high-dimensional partial differential equations using deep learning (2017) (1057)
Heterogeneous multiscale methods: A review (2007) (755)
The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems (2017) (690)
The Heterognous Multiscale Methods (2003) (678)
Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics (2017) (642)
The heterogeneous multiscale method* (2012) (575)
Simplified and improved string method for computing the minimum energy paths in barrier-crossing events. (2007) (573)
Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations (2017) (565)
Onsager's conjecture on the energy conservation for solutions of Euler's equation (1994) (516)
A Proposal on Machine Learning via Dynamical Systems (2017) (489)
Transition-path theory and path-finding algorithms for the study of rare events. (2010) (482)
DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics (2017) (443)
Towards a Theory of Transition Paths (2006) (391)
Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics (1996) (367)
Convolutional neural networks with low-rank regularization (2015) (363)
Invariant measures for Burgers equation with stochastic forcing (2000) (338)
Analysis of the heterogeneous multiscale method for elliptic homogenization problems (2004) (309)
Heterogeneous multiscale method: A general methodology for multiscale modeling (2003) (272)
Minimum action method for the study of rare events (2004) (243)
Vorticity Boundary Condition and Related Issues for Finite Difference Schemes (1996) (241)
Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations (2017) (232)
Boundary Conditions for the Moving Contact Line Problem (2007) (227)
Principles of Multiscale Modeling (2011) (226)
Stochastic Modified Equations and Adaptive Stochastic Gradient Algorithms (2015) (225)
Gibbsian Dynamics and Ergodicity¶for the Stochastically Forced Navier–Stokes Equation (2001) (224)
Projection method I: convergence and numerical boundary layers (1995) (217)
Analysis of multiscale methods for stochastic differential equations (2005) (208)
Cauchy–Born Rule and the Stability of Crystalline Solids: Static Problems (2007) (204)
BLOWUP OF SOLUTIONS OF THE UNSTEADY PRANDTL'S EQUATION (1997) (203)
Active Learning of Uniformly Accurate Inter-atomic Potentials for Materials Simulation (2018) (201)
Transition pathways in complex systems: Reaction coordinates, isocommittor surfaces, and transition tubes (2005) (195)
End-to-end Symmetry Preserving Inter-atomic Potential Energy Model for Finite and Extended Systems (2018) (194)
Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. (2005) (190)
Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics (2005) (189)
Boundary Layer Theory and the Zero-Viscosity Limit of the Navier-Stokes Equation (2000) (187)
Maximum Principle Based Algorithms for Deep Learning (2017) (184)
Matching conditions in atomistic-continuum modeling of materials. (2001) (179)
Transition pathways in complex systems: Application of the finite-temperature string method to the alanine dipeptide. (2005) (179)
Small‐scale structures in Boussinesq convection (1998) (163)
Essentially Compact Schemes for Unsteady Viscous Incompressible Flows (1996) (158)
Towards Understanding Generalization of Deep Learning: Perspective of Loss Landscapes (2017) (158)
Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation (2011) (157)
DP-GEN: A concurrent learning platform for the generation of reliable deep learning based potential energy models (2019) (150)
Deep Learning Approximation for Stochastic Control Problems (2016) (145)
A mean-field optimal control formulation of deep learning (2018) (144)
Multi-scale Modeling and Computation (2003) (141)
GAUGE METHOD FOR VISCOUS INCOMPRESSIBLE FLOWS (2003) (140)
Uniform accuracy of the quasicontinuum method (2006) (138)
Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales (2007) (136)
Deep Potential: a general representation of a many-body potential energy surface (2017) (134)
Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity (1994) (133)
The Heterogeneous Multiscale Method: A Review (2007) (129)
Homogenization of linear and nonlinear transport equations (1992) (127)
A Gauss-Seidel projection method for micromagnetics simulations (2001) (126)
Ergodicity for the Navier‐Stokes equation with degenerate random forcing: Finite‐dimensional approximation (2001) (126)
Multiscale modeling of the dynamics of solids at finite temperature (2005) (125)
Solving many-electron Schrödinger equation using deep neural networks (2018) (122)
Probability Distribution Functions for the Random Forced Burgers Equation (1997) (122)
Finite difference heterogeneous multi-scale method for homogenization problems (2003) (120)
DeePCG: Constructing coarse-grained models via deep neural networks. (2018) (120)
Pushing the Limit of Molecular Dynamics with Ab Initio Accuracy to 100 Million Atoms with Machine Learning (2020) (118)
A dynamic atomistic-continuum method for the simulation of crystalline materials (2001) (118)
Well-Posedness for the Dumbbell Model of Polymeric Fluids (2004) (116)
Analysis of the heterogeneous multiscale method for ordinary differential equations (2003) (113)
Optimal partition and effective dynamics of complex networks (2008) (112)
SelInv---An Algorithm for Selected Inversion of a Sparse Symmetric Matrix (2010) (111)
AUBRY-MATHER THEORY AND PERIODIC SOLUTIONS OF THE FORCED BURGERS EQUATION (1999) (107)
Towards Theoretically Understanding Why SGD Generalizes Better Than ADAM in Deep Learning (2020) (106)
On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations (2017) (104)
Numerical Methods for the Landau-Lifshitz Equation (2000) (104)
A comparative analysis of optimization and generalization properties of two-layer neural network and random feature models under gradient descent dynamics (2019) (103)
The local microscale problem in the multiscale modeling of strongly heterogeneous media: Effects of boundary conditions and cell size (2007) (103)
Microscopic mechanisms of equilibrium melting of a solid (2014) (101)
Nucleation of ordered phases in block copolymers. (2010) (99)
A level set method for dislocation dynamics (2003) (99)
Adaptive minimum action method for the study of rare events. (2008) (97)
Continuum models for the contact line problem (2010) (97)
Multiscale simulations in simple metals: A density-functional-based methodology (2004) (97)
Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems (2009) (94)
Exponential convergence of the deep neural network approximation for analytic functions (2018) (92)
Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't (2020) (91)
Self-induced stochastic resonance in excitable systems (2005) (91)
Stochastic Modified Equations and Dynamics of Stochastic Gradient Algorithms I: Mathematical Foundations (2018) (90)
A general strategy for designing seamless multiscale methods (2009) (90)
Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning (2020) (90)
Barron Spaces and the Compositional Function Spaces for Neural Network Models (2019) (89)
The gentlest ascent dynamics (2010) (88)
Statistical properties of shocks in Burgers turbulence (1995) (87)
Energy landscape and thermally activated switching of submicron-sized ferromagnetic elements (2003) (87)
Level set simulations of dislocation-particle bypass mechanisms (2004) (87)
A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow. Final Report (1992) (84)
86 PFLOPS Deep Potential Molecular Dynamics simulation of 100 million atoms with ab initio accuracy (2020) (84)
DeFine: deep convolutional neural networks accurately quantify intensities of transcription factor-DNA binding and facilitate evaluation of functional non-coding variants (2018) (84)
The heterogeneous multi-scale method for homogenization problems (2005) (80)
Overcoming the curse of dimensionality: Solving high-dimensional partial differential equations using deep learning (2017) (79)
Model Reduction with Memory and the Machine Learning of Dynamical Systems (2018) (78)
Machine learning from a continuous viewpoint, I (2019) (76)
Statistical theory for the stochastic Burgers equation in the inviscid limit (1999) (73)
A Generalized Peierls-Nabarro Model for Curved Dislocations and Core Structures of Dislocation Loops in Al and Cu (2008) (73)
A Proposal on Machine Learning via Dynamical Systems (2017) (72)
On the Crystallization of 2D Hexagonal Lattices (2009) (70)
Asymptotic theory for the probability density functions in burgers turbulence (1999) (69)
Dynamics of vortex liquids in Ginzburg-Landau theories with applications to superconductivity. (1994) (68)
Understanding and Enhancing the Transferability of Adversarial Examples (2018) (68)
Generalized flows, intrinsic stochasticity, and turbulent transport (2000) (68)
Multiscale modeling (2019) (67)
Effectiveness of implicit methods for stiff stochastic differential equations (2008) (66)
The Heterogeneous Multi-Scale Method (2002) (65)
Accurate numerical methods for micromagnetics simulations with general geometries (2003) (64)
The Barron Space and the Flow-Induced Function Spaces for Neural Network Models (2021) (64)
Kinetic model for a step edge in epitaxial growth. (1999) (62)
Coupling kinetic Monte-Carlo and continuum models with application to epitaxial growth (2003) (62)
Nonlinear Continuum Theory¶of Smectic-A Liquid Crystals (1997) (60)
Uniformly accurate machine learning-based hydrodynamic models for kinetic equations (2019) (60)
A Priori Estimates of the Population Risk for Residual Networks (2019) (56)
Localized bases of eigensubspaces and operator compression (2010) (56)
Modeling subgrid-scale forces by spatial artificial neural networks in large eddy simulation of turbulence (2020) (55)
Monge-Ampère Flow for Generative Modeling (2018) (55)
Some Recent Progress in Multiscale Modeling (2004) (54)
Multilevel Picard iterations for solving smooth semilinear parabolic heat equations (2016) (53)
Variational boundary conditions for molecular dynamics simulations of crystalline solids at finite temperature: Treatment of the thermal bath (2007) (52)
Metastability, conformation dynamics, and transition pathways in complex systems (2004) (50)
Finite Difference Methods for 3D Viscous Incompressible Flows in the Vorticity-Vector Potential Formulation on Nonstaggered Grids (1997) (49)
A Fast Parallel Algorithm for Selected Inversion of Structured Sparse Matrices with Application to 2D Electronic Structure Calculations (2010) (49)
Reinforced dynamics for enhanced sampling in large atomic and molecular systems. I. Basic Methodology (2017) (49)
Phase Diagram of a Deep Potential Water Model. (2021) (49)
Joint Learning of Response Ranking and Next Utterance Suggestion in Human-Computer Conversation System (2017) (47)
Projection Method II: Godunov--Ryabenki Analysis (1996) (47)
Pole-Based approximation of the Fermi-Dirac function (2009) (46)
A Priori Estimates of the Generalization Error for Two-layer Neural Networks (2018) (45)
Nonlinear evolution equation for the stress-driven morphological instability (2002) (45)
Thermodynamically driven incompressible fluid mixtures (1997) (43)
Projection method III: Spatial discretization on the staggered grid (2002) (43)
Finite Difference Schemes for Incompressible Flows in the Velocity-Impulse Density Formulation (1997) (42)
Isotope effects in liquid water via deep potential molecular dynamics (2019) (42)
Effective dynamics for ferromagnetic thin films (2001) (41)
Deep neural network for the dielectric response of insulators (2020) (41)
Linear-scaling subspace-iteration algorithm with optimally localized nonorthogonal wave functions for Kohn-Sham density functional theory (2009) (40)
Brownian motors in the photoalignment of liquid crystals (2002) (40)
Representation formulas and pointwise properties for Barron functions (2020) (40)
Derivation of continuum models for the moving contact line problem based on thermodynamic principles (2011) (39)
STUDY OF NOISE-INDUCED TRANSITIONS IN THE LORENZ SYSTEM USING THE MINIMUM ACTION METHOD (2010) (38)
Noisy Hegselmann-Krause Systems: Phase Transition and the 2R-Conjecture (2015) (38)
The Andersen thermostat in molecular dynamics (2008) (38)
Large time behavior and homogenization of solutions of two-dimensional conservation laws (1993) (38)
Analysis of multiscale methods (2004) (38)
Atomistic simulations of rare events using gentlest ascent dynamics. (2011) (38)
The Kohn-Sham Equation for Deformed Crystals (2012) (38)
Machine Learning and Computational Mathematics (2020) (38)
A Molecular Kinetic Theory of Inhomogeneous Liquid Crystal Flow and the Small Deborah Number Limit (2006) (37)
Misfit elastic energy and a continuum model for epitaxial growth with elasticity on vicinal surfaces (2004) (37)
Ground State Energy Functional with Hartree-Fock Efficiency and Chemical Accuracy. (2020) (36)
Continuum theory of a moving membrane. (2007) (36)
Analysis of the Local Quasicontinuum Method (2006) (35)
The Heterogeneous Multiscale Method Based on the Discontinuous Galerkin Method for Hyperbolic and Parabolic Problems (2005) (35)
Contact line dynamics on heterogeneous surfaces (2011) (35)
On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics (2020) (35)
Continuum Theory of Epitaxial Crystal Growth. I (2001) (34)
Phase separation in incompressible systems (1997) (33)
The Electronic Structure of Smoothly Deformed Crystals: Wannier Functions and the Cauchy–Born Rule (2011) (33)
Modified models of polymer phase separation. (2006) (33)
Noise can play an organizing role for the recurrent dynamics in excitable media (2007) (33)
Generalized Cauchy-Born rules for elastic deformation of sheets, plates, and rods: Derivation of continuum models from atomistic models (2006) (33)
Variational Boundary Conditions for Molecular Dynamics Simulations of Solids at Low Temperature (2005) (33)
DeePKS: a comprehensive data-driven approach towards chemically accurate density functional theory (2020) (32)
Can Shallow Neural Networks Beat the Curse of Dimensionality? A Mean Field Training Perspective (2020) (32)
A class of homogenization problems in the calculus of variations (1991) (32)
Turbulent Prandtl number effect on passive scalar advection (2001) (31)
Dynamics of Stochastic Gradient Algorithms (2015) (31)
Simple finite element method in vorticity formulation for incompressible flows (2001) (30)
Electronic structure of smoothly deformed crystals: Cauchy‐born rule for the nonlinear tight‐binding model (2010) (29)
Kinetic theory for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential. (2002) (29)
Numerical methods for multiscale transport equations and application to two-phase porous media flow (2005) (29)
Convergence of spectral methods for Burgers' equation (1992) (29)
Probabilistic framework for network partition. (2009) (28)
Homogenization of scalar conservation laws with oscillatory forcing terms (1992) (28)
Grain-boundary migration in the presence of diffusing impurities: Simulations and analytical models (2001) (28)
Deep potential generation scheme and simulation protocol for the Li10GeP2S12-type superionic conductors. (2021) (27)
A multiscale coupling method for the modeling of dynamics of solids with application to brittle cracks (2010) (27)
A deep potential model with long-range electrostatic interactions. (2021) (27)
A numerical method for the study of nucleation of ordered phases (2010) (27)
Method for defect stability diagram from ab initio calculations: A case study of SrTiO3 (2012) (25)
Multipole representation of the Fermi operator with application to the electronic structure analysis of metallic systems (2008) (25)
Kolmogorov width decay and poor approximators in machine learning: shallow neural networks, random feature models and neural tangent kernels (2020) (24)
Dynamics of Filaments during the Isotropic—Smectic A Phase Transition (1999) (24)
Analysis of the Gradient Descent Algorithm for a Deep Neural Network Model with Skip-connections (2019) (24)
Deep Potentials for Materials Science (2022) (22)
Orientational Ratchets and Angular Momentum Balance in the Janossy Effect (1998) (21)
OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle (2020) (21)
Correctors for the homogenization of conservation laws with oscillatory forcing terms (1992) (21)
Spatial feedbacks and the dynamics of savanna and forest (2019) (21)
Structure and Dynamics of Complex Networks (2012) (21)
Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations (2019) (21)
Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations (2013) (21)
On the distinguished limits of the Navier slip model of the moving contact line problem (2014) (20)
The Dawning of a New Era in Applied Mathematics (2021) (20)
PDFs for velocity and velocity gradients in Burgers' turbulence (1995) (20)
Analysis of the heterogeneous multiscale method for gas dynamics (2004) (20)
Gibbsian Dynamics and Invariant Measures for Stochastic Dissipative PDEs (2002) (19)
The initial‐value problem for measure‐valued solutions of a canonical 2 × 2 system with linearly degenerate fields (1991) (19)
Optimization-based string method for finding minimum energy path (2013) (19)
Gauge finite element method for incompressible flows (2000) (18)
On the emergence of tetrahedral symmetry in the final and penultimate layers of neural network classifiers (2020) (18)
Integrating Machine Learning with Physics-Based Modeling (2020) (18)
Sampling saddle points on a free energy surface. (2014) (18)
Recent results on mathematical and statistical hydrodynamics (2000) (18)
A note on generalized flows (2003) (17)
Subcritical bifurcation in spatially extended systems (2012) (17)
Convergence of Fourier methods for the Navier-Stokes equations (1993) (16)
Convergence of a Stochastic Method for the Modeling of Polymeric Fluids (2002) (16)
A discontinuous Galerkin implementation of a domain decomposition method for kinetic-hydrodynamic coupling multiscale problems in gas dynamics and device simulations (2007) (16)
Seamless multiscale modeling via dynamics on fiber bundles (2007) (16)
A sub-linear scaling algorithm for computing the electronic structure of materials (2007) (16)
Functional Frank-Wolfe Boosting for General Loss Functions (2015) (16)
On the Curse of Memory in Recurrent Neural Networks: Approximation and Optimization Analysis (2020) (16)
Multiscale Adaptive Representation of Signals: I. The Basic Framework (2015) (16)
Propagation of oscillations in the solutions of 1-D compressible fluid equations (1992) (16)
Study of the instability of the Poiseuille flow using a thermodynamic formalism (2015) (15)
Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations (2016) (15)
Model the nonlinear instability of wall-bounded shear flows as a rare event: a study on two-dimensional Poiseuille flow (2015) (14)
On the Generalization Properties of Minimum-norm Solutions for Over-parameterized Neural Network Models (2019) (14)
Sequential multiscale modeling using sparse representation (2008) (14)
Effective equations and the inverse cascade theory for Kolmogorov flows (1993) (14)
Theory of phase separation kinetics in polymer–liquid crystal systems (2002) (13)
Some observations on partial differential equations in Barron and multi-layer spaces (2020) (13)
Optimized local basis set for Kohn-Sham density functional theory (2011) (13)
Numerical study of metastability due to tunneling: The quantum string method (2005) (13)
A continuum model for the growth of epitaxial films (2001) (13)
Machine-learning-assisted modeling (2021) (13)
Another note on forced burgers turbulence (1999) (13)
The Slow Deterioration of the Generalization Error of the Random Feature Model (2020) (12)
Efficient sampling of high-dimensional free energy landscapes using adaptive reinforced dynamics (2021) (12)
Response to "Comment on 'Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates' †J. Chem. Phys. 123, 194107 "2005…‡" (2007) (12)
Probing Multi-Scale Energy Landscapes Using the String Method (2002) (12)
Machine learning based non-Newtonian fluid model with molecular fidelity (2020) (11)
Asymptotic analysis of quantum dynamics in crystals: the Bloch-Wigner transform, Bloch dynamics and Berry phase (2013) (11)
Domain decomposition interface preconditioners for fourth-order elliptic problems (1991) (11)
An L2 Analysis of Reinforcement Learning in High Dimensions with Kernel and Neural Network Approximation (2021) (11)
Hyperbolic equations and frequency interactions (1998) (11)
Dynamics of a Light Driven Molecular Motor (2002) (10)
On the emergence of simplex symmetry in the final and penultimate layers of neural network classifiers (2020) (10)
How SGD Selects the Global Minima in Over-parameterized Learning: A Dynamical Stability Perspective (2018) (10)
The Quenching-Activation Behavior of the Gradient Descent Dynamics for Two-layer Neural Network Models (2020) (10)
The Elastic Continuum Limit of the Tight Binding Model* (2007) (10)
Machine Learning: Mathematical Theory and Scientific Applications (2019) (9)
On full history recursive multilevel Picard approximations and numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations (2016) (9)
Multiscale Modeling Of Crystalline Solids (2005) (9)
A Qualitative Study of the Dynamic Behavior of Adaptive Gradient Algorithms (2020) (9)
Effective Maxwell equations from time-dependent density functional theory (2010) (9)
Emergence of spatio-temporal dynamics from exact coherent solutions in pipe flow (2016) (9)
The Landscape of Complex Networks (2012) (9)
Energy landscapes and rare events (2002) (9)
A Priori Estimates for Two-layer Neural Networks (2018) (9)
Heterogenous multiscale method for locally self-similar problems (2004) (8)
On the statistical solution of the Riemann equation and its implications for Burgers turbulence (1999) (8)
Hierarchical method for elliptic problems using wavelet (1992) (8)
Generalized flows, intrinsic stochasticity, and turbulent transport. (2000) (8)
Orbital minimization with localization (2008) (8)
Adaptive coupling of a deep neural network potential to a classical force field. (2018) (7)
Brownian ratchets and the photoalignment of liquid crystals (2002) (7)
Finite difference schemes for incompressible flows in vorticity formulations (1996) (7)
Some observations on high-dimensional partial differential equations with Barron data (2020) (7)
Density-gradient-corrected embedded atom method (2009) (7)
Efficient iterative method for solving the Dirac-Kohn-Sham density functional theory (2012) (6)
The Heterogeneous Multiscale Method and the “ Equation-free ” Approach to Multiscale Modeling (2008) (6)
Lattice Boltzmann Methods for Multiscale Fluid Problems (2005) (6)
Numerical study of oscillatory solutions of the gas-dynamic equations (1991) (6)
Simulations of 3-D domain wall structures in thin films (2005) (6)
Erratum: Kinetic theory for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential [Phys. Rev. E 65, 051504 (2002)] (2005) (6)
A priori estimates for classification problems using neural networks (2020) (6)
Generalization error of GAN from the discriminator’s perspective (2021) (6)
Simulation of Impurity Diffusion in a Strained Nanowire Using Off-Lattice KMC (2006) (6)
Enhancing the Transferability of Adversarial Examples with Noise Reduced Gradient (2018) (5)
Mean field limit of a dynamical model of polymer systems (2013) (5)
A multi-scale sampling method for accurate and robust deep neural network to predict combustion chemical kinetics (2022) (5)
Numerical study of the small scale structures in Boussinesq convection (1992) (5)
Connections between deep learning and partial differential equations (2021) (5)
Approximation and Optimization Theory for Linear Continuous-Time Recurrent Neural Networks (2022) (5)
Optimized local basis function for Kohn-Sham density functional theory (2011) (5)
The continuum limit and QM-continuum approximation of quantum mechanical models of solids (2007) (5)
Some Critical Issues for the "Equation-Free" Approach to Multiscale Modeling (2008) (5)
DLODE: a deep learning-based ODE solver for chemistry kinetics (2020) (5)
Selected Problems in Materials Science (2001) (5)
Stochastic PDEs in Turbulence Theory (2000) (5)
Interpretable Neural Networks for Panel Data Analysis in Economics (2020) (4)
Analytic 4×4 Propagation Matrices for Homogeneous Media (1998) (4)
Applied Stochastic Analysis (2019) (4)
On full history recursive multilevel Picard approximations and numerical approximations of high-dimensional nonlinear parabolic partial differential equations (2016) (4)
Generalization and Memorization: The Bias Potential Model (2020) (4)
Rademacher complexity and the generalization error of residual networks (2020) (4)
Topics in the analysis and computation of stochastic differential equations (2003) (4)
A Machine Learning Enhanced Algorithm for the Optimal Landing Problem (2022) (4)
The Free Action of Nonequilibrium Dynamics (2015) (4)
The Knowledge Graph for Macroeconomic Analysis with Alternative Big Data (2020) (4)
Optimal Paths for Spatially Extended Metastable Systems Driven by Noise (2002) (4)
The landscape of complex networks: Critical nodes and a hierarchical decomposition (2013) (3)
Ab initio multi-scale modeling of ferroelectrics: The case of PbTiO3 (2022) (3)
シート,プレートおよびロッドの弾性変形に対する一般化したCauchy‐Born則:原子レベル模型から連続体模型の導出 (2006) (3)
The Heterogeneous Multiscale Methods with Application to Combustion (2011) (3)
Solving optimal control of rigid-body dynamics with collisions using the hybrid minimum principle (2022) (3)
DeepHAM: A Global Solution Method for Heterogeneous Agent Models with Aggregate Shocks (2021) (3)
Kolmogorov width decay and poor approximators in machine learning: shallow neural networks, random feature models and neural tangent kernels (2021) (3)
Interfacial diffusion aided deformation during nanoindentation (2016) (3)
A Mathematical Model for Linguistic Universals (2019) (3)
Stability and the continuum limit of the spin-polarized Thomas-Fermi-Dirac-von Weizsäcker model (2012) (3)
Hyperbolic Equations and Frequency Interactions, Volume 5 (1999) (3)
Breakdown of homogenization for the random Hamilton-Jacobi equations (2008) (3)
Failure of random materials: A large deviation and computational study (2011) (3)
Analytical Solution of Maxwell's Equations in Lossy and Optically Active Crystals (2000) (3)
DeePKS-kit: A package for developing machine learning-based chemically accurate energy and density functional models (2020) (3)
Domain size in the presence of random fields (1997) (3)
The heterogeneous multiscale method: A ten-year review (2012) (3)
Bridging Traditional and Machine Learning-based Algorithms for Solving PDEs: The Random Feature Method (2022) (2)
Optimal convergence rate of the universal estimation error (2017) (2)
A comparative analysis of optimization and generalization properties of two-layer neural network and random feature models under gradient descent dynamics (2020) (2)
Noisy Hegselmann-Krause system: phase transition and 2-R conjecture (2015) (2)
A deep learning-based ODE solver for chemical kinetics (2020) (2)
Reinforced dynamics of large atomic and molecular systems (2017) (2)
A Mathematical Model for Universal Semantics (2019) (2)
A Priori Estimates of the Generalization Error for Autoencoders (2020) (1)
Cauchy-Born rule and spin density wave for the spin-polarized Thomas-Fermi-Dirac-von Weizsacker model (2011) (1)
Mathematical theory of solids: From quantum mechanics to continuum models (2014) (1)
Dynamics of vortices in superconductors (1995) (1)
Empowering Optimal Control with Machine Learning: A Perspective from Model Predictive Control (2022) (1)
A deep learning-based model reduction (DeePMR) method for simplifying chemical kinetics (2022) (1)
Modified string method for finding minimum energy path (2010) (1)
Homogenization and numerical methods for hyperbolic conservation laws with oscillatory data (1989) (1)
The heterogeneous multi-scale method: A mathematical framework for multi-scale modeling (2005) (1)
Coarse-grained spectral projection: A deep learning assisted approach to quantum unitary dynamics (2021) (1)
A thermodynamic study of the two-dimensional pressure-driven channel flow (2016) (1)
Exact Renormalization Group Analysis of Turbulent Transport by the Shear Flow (2013) (1)
Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations (2017) (1)
Deep neural network for Wannier function centers (2019) (1)
A mean-field optimal control formulation of deep learning (2018) (1)
Initial Value Problem Enhanced Sampling for Closed-Loop Optimal Control Design with Deep Neural Networks (2022) (1)
AMP: A unified framework boosting low resource automatic speech recognition (2023) (0)
Noise-driven transition to quasi-deterministic limit cycle dynamics in excitable systems (2003) (0)
Deep Learning in Finance (2023) (0)
Automatic machine-learning potential generation scheme and simulation protocol for the LiGePS-type superionic conductors (2020) (0)
Deep Learning for Multi-Scale Molecular Modelling (2019) (0)
Addendum to "Gauge Method for Viscous Incompressible Flows" (2003) (0)
Microscopic Mechanisms of Equilibrium Melting of a Solid (2014) (0)
C L ] 2 9 Ju l 2 01 9 A Mathematical Model for Linguistic Universals (2019) (0)
Mean field limit of a dynamical model of polymer systems (2013) (0)
Polymerization Induced Phase Separation in Polymer - Liquid Crystal Systems (1996) (0)
Atomic Structure of Supported Metal Nano Clusters on MoS 2 Monolayer Using Deep Potentials (2019) (0)
Order-order Nucleation in Copolymers: A String Method Approach (2012) (0)
A Proposal on Machine Learning via Dynamical Systems (2017) (0)
Fire sale in financial networks (2014) (0)
Theoretical Studies of Water by Climbing Jacob’s Ladder with Deep Learning (2019) (0)
DeePN2: A deep learning-based non-Newtonian hydrodynamic model (2021) (0)
Multiscale analysis of density functional theory (2009) (0)
Asymptotic analysis of quantum dynamics in crystals: the Bloch-Wigner transform, Bloch dynamics and Berry phase (2011) (0)
Ab Initio Generalized Langevin Equations (2022) (0)
Integrated Multiscale Modeling of Molecular Computing Devices (2012) (0)
Simple finite element methods for unsteady incompressible viscous flow in vorticity formulation (1996) (0)
WEINAN E AND BJORN ENGQUIST (2002) (0)
Stochastic Models of Polymer Systems (2016) (0)
Simple and Efficient Difference Methods for 3D Incompressible Flows Using Vorticity Formulation (1996) (0)
Exact Renormalization Group Analysis of Turbulent Transport by the Shear Flow (2013) (0)
Optimal convergence rate of the universal estimation error (2017) (0)
Linguistic Universals: Language-independent semantic fingerprints. (2019) (0)
Introduction to Data Science (2022) (0)
Analysis of Fourier Methods for Navier-Stokes Equation (1987) (0)
Interface Conditions for Coupled Atomistic and Continuum Models of Solids for Dynamics Problems at Finite Temperature (2006) (0)
Modeling Mesoscale Processes of Scalable Synthesis (Final Report) (2019) (0)
Atomistic simulations of melting and solidification using temperature accelerated molecular dynamics (2013) (0)
Overcoming the Curse of Dimension: Methods Based on Sparse Representation and Adaptive Sampling (2011) (0)
Active Learning of Uniformly Accurate Deep Potential Models for Multicomponent Systems (2019) (0)
Coarse-grained spectral projection (CGSP): A scalable and parallelizable deep learning-based approach to quantum unitary dynamics (2020) (0)
The pole expansion and selected inversion technique for solving Kohn-Sham density functional theory at large scale (2013) (0)
- ' NATURAL CONSTRAINTS ~ IN CRITICAL P ' ) TNT THEORY TO PERIODIC SOLUTIONS OF NATURAL HAMILTONIAN SYSTEMS (0)
Hybrid Auxiliary Field Quantum Monte Carlo for Molecular Systems. (2022) (0)
Density functional theory: analysis and algorithms (2009) (0)
MAC: A unified framework boosting low resource automatic speech recognition (2023) (0)
Noisy Hegselmann-Krause Systems: Phase Transition and the 2R-Conjecture (2017) (0)
DeePMD-kit v2: A software package for Deep Potential models (2023) (0)
SEAMLESS MULTISCALE MODELING VIA DYNAMICS ON FIBER (2007) (0)
Exponential convergence of the deep neural network approximation for analytic functions (2018) (0)
TWO TRAN 51 ENT 5 FOR A RADIAL NEARLY-HORIZONTAL FLOW (2010) (0)
Numerical Study of Nucleation in Diblock Copolymer Melts (2003) (0)
The Barron Space and the Flow-Induced Function Spaces for Neural Network Models (2021) (0)
Spatial feedbacks and the dynamics of savanna and forest (2019) (0)
Continuum limits of step flow models (2000) (0)
Atomistic Models of Materials: Mathematical Challenges (2008) (0)
Solving Boltzmann and Fokker-Planck Equations Using Sparse Representation (2011) (0)
Finding saddle points using Gentlest ascent dynamics (2011) (0)
The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems (2018) (0)
The heterogeneous multiscale method based on the discontinuous Galerkin and the finite volume methods for hyperbolic problems (2005) (0)
The Free Action of Nonequilibrium Dynamics (2015) (0)
Temperature and rate sensitivity of melting in Cu (2013) (0)
Machine-Learned Invertible Coarse Graining for Multiscale Molecular Modeling (2023) (0)
N A ] 1 0 A ug 2 01 7 On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations (2018) (0)
An Expectation-Maximization Algorithm for Continuous-time Hidden Markov Models (2021) (0)
The Random Feature Method for Time-dependent Problems (2023) (0)
CONVERGENCE OF LEGENDRE METHODS FOR NAVIER-STOKES EQUATIONS (1994) (0)
Analysis of Spectral Methods for Burgers' Equation (1987) (0)
Magnetic ordering schemes in R 3 T 4 X 4 compounds (2006) (0)
PRESSURE AND TEMPERATURE DEPENDENCE OF THE PROTON AND DEUTERON SPIN-LAT?iICE RELAXATION TKViES IN LKQUID HZ!5 AND I&S (2011) (0)
On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations (2019) (0)
Machine Learning and Computational Mathematics (2020) (0)

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