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The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems
Open accessAbstract
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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Cites
Deep Residual Learning for Image Recognition
He, K., Zhang, X., Ren, S., & Sun, J. (2016) · 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) · doi:10.1109/cvpr.2016.90 · Cited by 235,474
Densely Connected Convolutional Networks
Huang, G., Liu, Z., Van Der Maaten, L., & Weinberger, K. Q. (2017) · 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) · doi:10.1109/cvpr.2017.243 · Cited by 48,566
Cited by
Weak adversarial networks for high-dimensional partial differential equations
Zang, Y., Bao, G., Ye, X., & Zhou, H. (2020) · Journal of Computational Physics · doi:10.1016/j.jcp.2020.109409 · Cited by 444
Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks
Geist, M., Petersen, P., Raslan, M., Schneider, R., & Kutyniok, G. (2021) · Journal of Scientific Computing · doi:10.1007/s10915-021-01532-w · Cited by 57
Du, Z., & Lu, R. (2025) · Batteries · doi:10.3390/batteries11060204 · Cited by 58
Metadata
- Authors
- E, W., & Yu, B.
- Venue
- Communications in Mathematics and Statistics
- Published
- 2018-02-14
- Volume
- 6
- Issue
- 1
- Pages
- 1-12
- ISSN
- 2194-6701 (ISSN-L)
- 2194-671X
- OpenAlex
- W2760972773
- Cited by
- 1,418
- References
- 2
- FWCI
- 32.90
- Type
- journal article
- Language
- en
- Open access
- green
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