The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems
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Summary
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.
- Type
- article
- Published
- 2017-09-30
- Cited by
- 1,938
- References
- 15
- Access
- Open access
- OpenAlex
- https://openalex.org/W2760972773
- Semantic Scholar
- https://api.semanticscholar.org/CorpusID:2988078
Keywords
Ritz method, Deep learning, Eigenvalues and eigenvectors, Nonlinear system, Descent (aeronautics)
References
- Analysis Of The Finite Element Method
- G. Strang / G. J. Fix, An Analysis of the Finite Element Method. (Series in Automatic Computation. XIV + 306 S. m. Fig. Englewood Clifs, N. J. 1973. Prentice‐Hall, Inc.
- Deep Residual Learning for Image Recognition
- A Proposal on Machine Learning via Dynamical Systems
- Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations
- Overcoming the curse of dimensionality: Solving high-dimensional partial differential equations using deep learning
- Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics
- Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations
- Solving high-dimensional partial differential equations using deep learning
- Densely Connected Convolutional Networks
- Deep Learning
- Partial Differential Equations
- Deep Learning
- Adam: A Method for Stochastic Optimization
- Partial Differential Equations
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- A unified deep artificial neural network approach to partial differential equations in complex geometries
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- Solving for high-dimensional committor functions using artificial neural networks
- Deep Multiscale Model Learning
- Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations
- Topological Properties of the Set of Functions Generated by Neural Networks of Fixed Size
- Relu Deep Neural Networks and Linear Finite Elements
- Deep Global Model Reduction Learning
- A machine learning framework for data driven acceleration of computations of differential equations
- DNN Expression Rate Analysis of High-Dimensional PDEs: Application to Option Pricing
- A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
- A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients
- Analysis of the generalization error: Empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
- Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations
- Frequency Principle in Deep Learning with General Loss Functions and Its Potential Application
- A mesh-free method for interface problems using the deep learning approach
- Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
- Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks
- Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks
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