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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.
The heterogeneous multiscale method*
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
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
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
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
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
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.
A Proposal on Machine Learning via Dynamical Systems
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.
Stochastic Modified Equations and Adaptive Stochastic Gradient Algorithms
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.
String method for the study of rare events
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