Neural and tensor networks for high-dimensional parametric PDEs and sampling
Authors
- Eigel, Martin
ORCID: 0000-0003-2687-4497 - Grasedyck, Lars
- Le, Thong
- Schütte, Janina
ORCID: 0009-0000-9924-3229
2020 Mathematics Subject Classification
- 65C30 35R60 60H35 65N22 65J10 62H12 68T07
Keywords
- Neural networks, tensor networks, parametric PDEs, parametric obstacle problems, high dimensions, sampling
DOI
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.
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