The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems
This provides a novel computational approach for researchers in numerical analysis and machine learning, though it appears incremental as it adapts existing deep learning techniques to variational problems.
The authors tackled solving variational problems from partial differential equations by proposing the Deep Ritz Method, a deep learning-based numerical algorithm that is nonlinear, adaptive, and scalable to high dimensions, demonstrating its application on eigenvalue problems.
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.