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

A statistical learning approach for high-dimensional 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.