The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems
A deep learning-based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations, which is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions.
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.org
- 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…
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.…
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…
A Proposal on Machine Learning via Dynamical Systems
- E. Weinan
- Computer Science
- 22 March 2017
The idea of using continuous dynamical systems to model general high-dimensional nonlinear functions used in machine learning and the connection with deep learning is discussed.
Convolutional neural networks with low-rank regularization
- Cheng Tai, Tong Xiao, Xiaogang Wang, E. Weinan
- Computer ScienceInternational Conference on Learning…
- 19 November 2015
A new algorithm for computing the low-rank tensor decomposition for removing the redundancy in the convolution kernels and is more effective than iterative methods for speeding up large CNNs.
Stochastic Modified Equations and Adaptive Stochastic Gradient Algorithms
- Qianxiao Li, Cheng Tai, E. Weinan
- Computer ScienceInternational Conference on Machine Learning
- 19 November 2015
The method of stochastic modified equations (SME) is developed, in which stochastics gradient algorithms are approximated in the weak sense by continuous-time stochastically differential equations, which provides a general methodology for the analysis and design of Stochastic gradient algorithms.
Invariant measures for Burgers equation with stochastic forcing
In this paper we study the following Burgers equation
du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t)
where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise…
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