The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems

Weinan E , Bing Yu

Communications in Mathematics and Statistics ›› 2018, Vol. 6 ›› Issue (1) : 1 -12.

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Communications in Mathematics and Statistics ›› 2018, Vol. 6 ›› Issue (1) : 1 -12. DOI: 10.1007/s40304-018-0127-z
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The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems

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Abstract

We propose a deep learning-based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz Method is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions. The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning. We illustrate the method on several problems including some eigenvalue problems.

Keywords

Deep Ritz Method / Variational problems / PDE / Eigenvalue problems

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Weinan E, Bing Yu. The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems. Communications in Mathematics and Statistics, 2018, 6(1): 1-12 DOI:10.1007/s40304-018-0127-z

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Funding

National Natural Science Foundation of China(91130005)

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