Neural and tensor networks for high-dimensional parametric PDEs and sampling

Abstract

In this chapter, we present various methods based on neural network architectures and tensor decompositions for addressing parametric partial differential equations (PDEs) and sampling from unnormalized densities in high-dimensional settings. Parametric PDEs arise in many areas of science and engineering, where one seeks to solve a PDE efficiently for a wide range of parameter values or where the data is inherently uncertain. Such problems have been analysed extensively in Uncertainty Quantification in recent years. Directly related and also highly relevant for practical applications is a class of statistical inverse problems. It can be approached in a Bayesian framework through sampling techniques. par Two main strategies are central to this chapter: deconstructing complex problems into manageable, simpler subproblems in a sensible way and designing problem-specific model architectures that suit the specific properties and requirements of the problem at hand. The developed methods are supported by theoretical convergence guarantees and demonstrate state-of-the-art performance in numerical experiments.