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The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems
TLDR
The heterogeneous multiscale method*
- A. Abdulle, E. Weinan, B. Engquist, E. Vanden-Eijnden
- PhysicsActa Numerica
- 19 April 2012
The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework.…
Onsager's conjecture on the energy conservation for solutions of Euler's equation
- P. Constantin, E. Weinan, E. Titi
- Mathematics
- 1 October 1994
We give a simple proof of a result conjectured by Onsager [1] on energy conservation for weak solutions of Euler's equation.
Heterogeneous multiscale methods: A review
- E. Weinan, B. Engquist, Xiantao Li, W. Ren, E. Vanden-Eijnden
- Mathematics
- 2007
This paper gives a systematic introduction to HMM, the heterogeneous multiscale methods, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be…
Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations
- E. Weinan, Jiequn Han, Arnulf Jentzen
- Computer ScienceArXiv
- 15 June 2017
We study a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, which is based on an analogy between the…
Analysis of the heterogeneous multiscale method for elliptic homogenization problems
- E. Weinan, P. Ming, Pingwen Zhang
- Mathematics
- 16 September 2004
The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the…
Convolutional neural networks with low-rank regularization
- Cheng Tai, Tong Xiao, Xiaogang Wang, E. Weinan
- Computer ScienceICLR
- 19 November 2015
TLDR
A Proposal on Machine Learning via Dynamical Systems
- E. Weinan
- Computer Science
- 22 March 2017
TLDR
Stochastic Modified Equations and Adaptive Stochastic Gradient Algorithms
- Qianxiao Li, Cheng Tai, E. Weinan
- Computer ScienceICML
- 19 November 2015
TLDR
String method for the study of rare events
- E. Weinan, W. Ren, E. Vanden-Eijnden
- Physics
- 25 May 2002
We present an efficient method for computing the transition pathways, free energy barriers, and transition rates in complex systems with relatively smooth energy landscapes. The method proceeds by…
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