Weinan E

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Princeton University ∙ berkeley college

Professor, Department of Mathematics and Program in Applied and Computational Mathematics at Princeton University

The Expectation-Maximization Algorithm for Continuous-time Hidden Markov Models

We propose a unified framework that extends the inference methods for cl...
03/31/2021 ∙ by Qingcan Wang, et al. ∙ 0 ∙ share

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On the emergence of tetrahedral symmetry in the final and penultimate layers of neural network classifiers

A recent numerical study observed that neural network classifiers enjoy ...
12/10/2020 ∙ by Weinan E, et al. ∙ 5 ∙ share

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Some observations on partial differential equations in Barron and multi-layer spaces

We use explicit representation formulas to show that solutions to certai...
12/02/2020 ∙ by Weinan E, et al. ∙ 0 ∙ share

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Generalization and Memorization: The Bias Potential Model

Models for learning probability distributions such as generative models ...
11/29/2020 ∙ by Hongkang Yang, et al. ∙ 0 ∙ share

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Towards Theoretically Understanding Why SGD Generalizes Better Than ADAM in Deep Learning

It is not clear yet why ADAM-alike adaptive gradient algorithms suffer f...
10/12/2020 ∙ by Pan Zhou, et al. ∙ 46 ∙ share

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Interpretable Neural Networks for Panel Data Analysis in Economics

The lack of interpretability and transparency are preventing economists ...
10/11/2020 ∙ by Yucheng Yang, et al. ∙ 0 ∙ share

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The Knowledge Graph for Macroeconomic Analysis with Alternative Big Data

The current knowledge system of macroeconomics is built on interactions ...
10/11/2020 ∙ by Yucheng Yang, et al. ∙ 0 ∙ share

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A priori estimates for classification problems using neural networks

We consider binary and multi-class classification problems using hypothe...
09/28/2020 ∙ by Weinan E, et al. ∙ 0 ∙ share

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Machine Learning and Computational Mathematics

Neural network-based machine learning is capable of approximating functi...
09/23/2020 ∙ by Weinan E, et al. ∙ 0 ∙ share

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Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't

The purpose of this article is to review the achievements made in the la...
09/22/2020 ∙ by Weinan E, et al. ∙ 270 ∙ share

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On the Curse of Memory in Recurrent Neural Networks: Approximation and Optimization Analysis

We study the approximation properties and optimization dynamics of recur...
09/16/2020 ∙ by Zhong Li, et al. ∙ 1 ∙ share

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A Qualitative Study of the Dynamic Behavior of Adaptive Gradient Algorithms

The dynamic behavior of RMSprop and Adam algorithms is studied through a...
09/14/2020 ∙ by Chao Ma, et al. ∙ 27 ∙ share

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OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle

We propose a systematic method for learning stable and interpretable dyn...
09/06/2020 ∙ by Haijun Yu, et al. ∙ 0 ∙ share

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Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning

In recent years, tremendous progress has been made on numerical algorith...
08/31/2020 ∙ by Weinan E, et al. ∙ 0 ∙ share

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The Slow Deterioration of the Generalization Error of the Random Feature Model

The random feature model exhibits a kind of resonance behavior when the ...
08/13/2020 ∙ by Chao Ma, et al. ∙ 45 ∙ share

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DeePKS: a comprehensive data-driven approach towards chemically accurate density functional theory

We propose a general machine learning-based framework for building an ac...
08/01/2020 ∙ by Yixiao Chen, et al. ∙ 0 ∙ share

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On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics

We develop Banach spaces for ReLU neural networks of finite depth L and ...
07/30/2020 ∙ by Weinan E, et al. ∙ 11 ∙ share

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Coarse-grained spectral projection (CGSP): A scalable and parallelizable deep learning-based approach to quantum unitary dynamics

We propose the coarse-grained spectral projection method (CGSP), a deep ...
07/19/2020 ∙ by Pinchen Xie, et al. ∙ 0 ∙ share

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The Quenching-Activation Behavior of the Gradient Descent Dynamics for Two-layer Neural Network Models

A numerical and phenomenological study of the gradient descent (GD) algo...
06/25/2020 ∙ by Chao Ma, et al. ∙ 6 ∙ share

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Representation formulas and pointwise properties for Barron functions

We study the natural function space for infinitely wide two-layer neural...
06/10/2020 ∙ by Weinan E, et al. ∙ 0 ∙ share

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Integrating Machine Learning with Physics-Based Modeling

Machine learning is poised as a very powerful tool that can drastically ...
06/04/2020 ∙ by Weinan E, et al. ∙ 0 ∙ share

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Can Shallow Neural Networks Beat the Curse of Dimensionality? A mean field training perspective

We prove that the gradient descent training of a two-layer neural networ...
05/21/2020 ∙ by Stephan Wojtowytsch, et al. ∙ 51 ∙ share

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Kolmogorov Width Decay and Poor Approximators in Machine Learning: Shallow Neural Networks, Random Feature Models and Neural Tangent Kernels

We establish a scale separation of Kolmogorov width type between subspac...
05/21/2020 ∙ by Weinan E, et al. ∙ 22 ∙ share

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86 PFLOPS Deep Potential Molecular Dynamics simulation of 100 million atoms with ab initio accuracy

We present the GPU version of DeePMD-kit, which, upon training a deep ne...
04/24/2020 ∙ by Denghui Lu, et al. ∙ 0 ∙ share

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Machine learning based non-Newtonian fluid model with molecular fidelity

We introduce a machine-learning-based framework for constructing continu...
03/07/2020 ∙ by Huan Lei, et al. ∙ 12 ∙ share

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Machine Learning from a Continuous Viewpoint

We present a continuous formulation of machine learning, as a problem in...
12/30/2019 ∙ by Weinan E, et al. ∙ 33 ∙ share

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On the Generalization Properties of Minimum-norm Solutions for Over-parameterized Neural Network Models

We study the generalization properties of minimum-norm solutions for thr...
12/15/2019 ∙ by Weinan E, et al. ∙ 10 ∙ share

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A Mathematical Model for Linguistic Universals

Inspired by chemical kinetics and neurobiology, we propose a mathematica...
07/29/2019 ∙ by Weinan E, et al. ∙ 12 ∙ share

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Deep neural network for Wannier function centers

We introduce a deep neural network (DNN) model that assigns the position...
06/27/2019 ∙ by Linfeng Zhang, et al. ∙ 0 ∙ share

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Barron Spaces and the Compositional Function Spaces for Neural Network Models

One of the key issues in the analysis of machine learning models is to i...
06/18/2019 ∙ by Weinan E, et al. ∙ 1 ∙ share

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Analysis of the Gradient Descent Algorithm for a Deep Neural Network Model with Skip-connections

The behavior of the gradient descent (GD) algorithm is analyzed for a de...
04/10/2019 ∙ by Weinan E, et al. ∙ 15 ∙ share

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A Comparative Analysis of the Optimization and Generalization Property of Two-layer Neural Network and Random Feature Models Under Gradient Descent Dynamics

A fairly comprehensive analysis is presented for the gradient descent dy...
04/08/2019 ∙ by Weinan E, et al. ∙ 25 ∙ share

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A Priori Estimates of the Population Risk for Residual Networks

Optimal a priori estimates are derived for the population risk of a regu...
03/06/2019 ∙ by Weinan E, et al. ∙ 6 ∙ share

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Stochastic Modified Equations and Dynamics of Stochastic Gradient Algorithms I: Mathematical Foundations

We develop the mathematical foundations of the stochastic modified equat...
11/05/2018 ∙ by Qianxiao Li, et al. ∙ 0 ∙ share

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Active Learning of Uniformly Accurate Inter-atomic Potentials for Materials Simulation

An active learning procedure called Deep Potential Generator (DP-GEN) is...
10/28/2018 ∙ by Linfeng Zhang, et al. ∙ 0 ∙ share

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A Priori Estimates of the Generalization Error for Two-layer Neural Networks

New estimates for the generalization error are established for the two-l...
10/15/2018 ∙ by Weinan E, et al. ∙ 0 ∙ share

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Monge-Ampère Flow for Generative Modeling

We present a deep generative model, named Monge-Ampère flow, which build...
09/26/2018 ∙ by Linfeng Zhang, et al. ∙ 0 ∙ share

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Model Reduction with Memory and the Machine Learning of Dynamical Systems

The well-known Mori-Zwanzig theory tells us that model reduction leads t...
08/10/2018 ∙ by Chao Ma, et al. ∙ 0 ∙ share

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A Mean-Field Optimal Control Formulation of Deep Learning

Recent work linking deep neural networks and dynamical systems opened up...
07/03/2018 ∙ by Weinan E, et al. ∙ 0 ∙ share

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Exponential Convergence of the Deep Neural Network Approximation for Analytic Functions

We prove that for analytic functions in low dimension, the convergence r...
07/01/2018 ∙ by Weinan E, et al. ∙ 0 ∙ share

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Understanding and Enhancing the Transferability of Adversarial Examples

State-of-the-art deep neural networks are known to be vulnerable to adve...
02/27/2018 ∙ by Lei Wu, et al. ∙ 0 ∙ share

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DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics

Recent developments in many-body potential energy representation via dee...
12/11/2017 ∙ by Han Wang, et al. ∙ 0 ∙ share

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Reinforced dynamics for enhanced sampling in large atomic and molecular systems. I. Basic Methodology

A new approach for efficiently exploring the configuration space and com...
12/10/2017 ∙ by Linfeng Zhang, et al. ∙ 0 ∙ share

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Maximum Principle Based Algorithms for Deep Learning

The continuous dynamical system approach to deep learning is explored in...
10/26/2017 ∙ by Qianxiao Li, et al. ∙ 1 ∙ share

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The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems

We propose a deep learning based method, the Deep Ritz Method, for numer...
09/30/2017 ∙ by Weinan E, et al. ∙ 1 ∙ share

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Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations

High-dimensional partial differential equations (PDE) appear in a number...
09/18/2017 ∙ by Christian Beck, et al. ∙ 0 ∙ share

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Towards Understanding Generalization of Deep Learning: Perspective of Loss Landscapes

It is widely observed that deep learning models with learned parameters ...
06/30/2017 ∙ by Lei Wu, et al. ∙ 0 ∙ share

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Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations

We propose a new algorithm for solving parabolic partial differential eq...
06/15/2017 ∙ by Weinan E, et al. ∙ 0 ∙ share

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Deep Learning Approximation for Stochastic Control Problems

Many real world stochastic control problems suffer from the "curse of di...
11/02/2016 ∙ by Jiequn Han, et al. ∙ 0 ∙ share

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Stochastic modified equations and adaptive stochastic gradient algorithms

We develop the method of stochastic modified equations (SME), in which s...
11/19/2015 ∙ by Qianxiao Li, et al. ∙ 0 ∙ share

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Weinan E

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