Variational Monte Carlo -- Bridging concepts of machine learning and high dimensional partial differential equations
- Eigel, Martin
- Trunschke, Philipp
- Schneider, Reinhold
- Wolf, Sebastian
2010 Mathematics Subject Classification
- 35R60 47B80 60H35 65C20 65N12 65N22 65J10
- Partial differential equations with random coefficients, tensor representation, tensor train, uncertainty quantification, stochastic finite element methods, log-normal, adaptive methods, ALS, low-rank, reduced basis methods
A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range of problems. A general unified convergence analysis is derived, which takes into account the approximation and the statistical errors. By this, a combination of theoretical results from numerical analysis and statistics is obtained. Numerical experiments illustrate the performance of the method with the model class of hierarchical tensors.
- Adv. Comput. Math., 45 (2019), pp. 2503--2532, DOI 10.1007/s10444-019-09723-8 .