
The ExpectationMaximization Algorithm for Continuoustime Hidden Markov Models
We propose a unified framework that extends the inference methods for cl...
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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 ...
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Some observations on partial differential equations in Barron and multilayer spaces
We use explicit representation formulas to show that solutions to certai...
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Generalization and Memorization: The Bias Potential Model
Models for learning probability distributions such as generative models ...
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Towards Theoretically Understanding Why SGD Generalizes Better Than ADAM in Deep Learning
It is not clear yet why ADAMalike adaptive gradient algorithms suffer f...
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Interpretable Neural Networks for Panel Data Analysis in Economics
The lack of interpretability and transparency are preventing economists ...
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The Knowledge Graph for Macroeconomic Analysis with Alternative Big Data
The current knowledge system of macroeconomics is built on interactions ...
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A priori estimates for classification problems using neural networks
We consider binary and multiclass classification problems using hypothe...
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Machine Learning and Computational Mathematics
Neural networkbased machine learning is capable of approximating functi...
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Towards a Mathematical Understanding of Neural NetworkBased Machine Learning: what we know and what we don't
The purpose of this article is to review the achievements made in the la...
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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...
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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...
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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...
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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...
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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 ...
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DeePKS: a comprehensive datadriven approach towards chemically accurate density functional theory
We propose a general machine learningbased framework for building an ac...
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On the Banach spaces associated with multilayer ReLU networks: Function representation, approximation theory and gradient descent dynamics
We develop Banach spaces for ReLU neural networks of finite depth L and ...
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Coarsegrained spectral projection (CGSP): A scalable and parallelizable deep learningbased approach to quantum unitary dynamics
We propose the coarsegrained spectral projection method (CGSP), a deep ...
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The QuenchingActivation Behavior of the Gradient Descent Dynamics for Twolayer Neural Network Models
A numerical and phenomenological study of the gradient descent (GD) algo...
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Representation formulas and pointwise properties for Barron functions
We study the natural function space for infinitely wide twolayer neural...
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Integrating Machine Learning with PhysicsBased Modeling
Machine learning is poised as a very powerful tool that can drastically ...
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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 twolayer neural networ...
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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...
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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 DeePMDkit, which, upon training a deep ne...
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Machine learning based nonNewtonian fluid model with molecular fidelity
We introduce a machinelearningbased framework for constructing continu...
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Machine Learning from a Continuous Viewpoint
We present a continuous formulation of machine learning, as a problem in...
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On the Generalization Properties of Minimumnorm Solutions for Overparameterized Neural Network Models
We study the generalization properties of minimumnorm solutions for thr...
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A Mathematical Model for Linguistic Universals
Inspired by chemical kinetics and neurobiology, we propose a mathematica...
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Deep neural network for Wannier function centers
We introduce a deep neural network (DNN) model that assigns the position...
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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...
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Analysis of the Gradient Descent Algorithm for a Deep Neural Network Model with Skipconnections
The behavior of the gradient descent (GD) algorithm is analyzed for a de...
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A Comparative Analysis of the Optimization and Generalization Property of Twolayer Neural Network and Random Feature Models Under Gradient Descent Dynamics
A fairly comprehensive analysis is presented for the gradient descent dy...
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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...
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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...
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Active Learning of Uniformly Accurate Interatomic Potentials for Materials Simulation
An active learning procedure called Deep Potential Generator (DPGEN) is...
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A Priori Estimates of the Generalization Error for Twolayer Neural Networks
New estimates for the generalization error are established for the twol...
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MongeAmpère Flow for Generative Modeling
We present a deep generative model, named MongeAmpère flow, which build...
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Model Reduction with Memory and the Machine Learning of Dynamical Systems
The wellknown MoriZwanzig theory tells us that model reduction leads t...
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A MeanField Optimal Control Formulation of Deep Learning
Recent work linking deep neural networks and dynamical systems opened up...
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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...
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Understanding and Enhancing the Transferability of Adversarial Examples
Stateoftheart deep neural networks are known to be vulnerable to adve...
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DeePMDkit: A deep learning package for manybody potential energy representation and molecular dynamics
Recent developments in manybody potential energy representation via dee...
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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...
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Maximum Principle Based Algorithms for Deep Learning
The continuous dynamical system approach to deep learning is explored in...
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The Deep Ritz method: A deep learningbased numerical algorithm for solving variational problems
We propose a deep learning based method, the Deep Ritz Method, for numer...
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Machine learning approximation algorithms for highdimensional fully nonlinear partial differential equations and secondorder backward stochastic differential equations
Highdimensional partial differential equations (PDE) appear in a number...
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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 ...
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Deep learningbased numerical methods for highdimensional parabolic partial differential equations and backward stochastic differential equations
We propose a new algorithm for solving parabolic partial differential eq...
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Deep Learning Approximation for Stochastic Control Problems
Many real world stochastic control problems suffer from the "curse of di...
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Stochastic modified equations and adaptive stochastic gradient algorithms
We develop the method of stochastic modified equations (SME), in which s...
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Weinan E
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Professor, Department of Mathematics and Program in Applied and Computational Mathematics at Princeton University