Variational Monte Carlo - Bridging Concepts of Machine Learning and High Dimensional Partial Differential Equations

TL;DR

This paper introduces a statistical learning method for solving parametric PDEs related to Uncertainty Quantification, combining theoretical analysis and numerical experiments to demonstrate its effectiveness.

Contribution

It develops a unified convergence analysis for a broad class of problems, integrating numerical analysis and statistical theory, and applies it to hierarchical tensor models.

Findings

Effective in high-dimensional PDEs

Convergence guarantees established

Numerical experiments confirm performance

Abstract

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.