Variational Monte Carlo - Bridging concepts of machine learning and high dimensional partial differential equations

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.

Keywords

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

URI

https://doi.org/10.34657/3468
https://oa.tib.eu/renate/handle/123456789/1987

Citation

Eigel, M., Trunschke, P., Schneider, R., & Wolf, S. (2018). Variational Monte Carlo - Bridging concepts of machine learning and high dimensional partial differential equations (Vol. 2544). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik. https://doi.org//10.20347/WIAS.PREPRINT.2544

Collections

Mathematik

License

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