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.
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Acknowledgements
We are grateful to Professor Ruo Li and Dr. Zhanxing Zhu for very helpful discussions. The work of E and Yu is supported in part by the National Key Basic Research Program of China 2015CB856000, Major Program of NNSFC under Grant 91130005, DOE Grant DE-SC0009248, and ONR Grant N00014-13-1-0338.
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E, W., Yu, B. The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems. Commun. Math. Stat. 6, 1–12 (2018). https://doi.org/10.1007/s40304-018-0127-z
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DOI: https://doi.org/10.1007/s40304-018-0127-z