Variational Monte Carlo - Bridging Concepts of Machine Learning and High Dimensional Partial Differential Equations
For researchers in uncertainty quantification and scientific computing, this work offers a theoretically grounded framework for solving high-dimensional parametric PDEs.
The paper develops a statistical learning approach for parametric PDEs in uncertainty quantification, minimizing empirical risk over a model class. It provides a unified convergence analysis combining numerical analysis and statistics, with numerical experiments using hierarchical tensors.
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.