Tensor networks- structure learning, uncertainty quantification, and PDE solutions
Abstract
This proposal seeks to develop automated computational techniques with quantified uncertainty for using tensor networks to exploit low-dimensional structure in solutions for high dimensional forward and inverse problems. We investigate tensor-network approaches to discover this structure and seek to answer some fundamental computational questions. For example, we seek to understand how do we choose an optimal tensor network structure. How do we adapt structures to specific data and problem settings. How do we leverage the unique multi-linear nature of tensor networks to perform rapid inference. How do we embed these methods to enable fast forward and inverse uncertainty quantification. To address these questions, we develop probabilistic computational mathematical tools. Specifically, we will develop fast computation routines for tensor networks with arbitrary structure through randomized numerical linear algebra; investigate the performance of probabilistic ensembles of tensor network structures - as opposed to deterministic or probabilistic representations of fixed structures; discover and investigate randomized graph construction strategies to progressively build a network that balances edge growth and rank growth; and develop probabilistic tensor network approaches to the solution of high-dimensional PDEs and high-dimensional data problems (tensors with more than billions of elements) with quantified uncertainty for both forward and inverse problems.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 06, 2025
- Source ID
- FA95502410246
Entities
People
- Alex Gorodetsky
Organizations
- Air Force Office of Scientific Research
- Board of Regents of the University of Michigan
- United States Air Force