A quantum inspired approach to learning dynamical laws from
data---block-sparsity and gauge-mediated weight sharing
- URL: http://arxiv.org/abs/2208.01591v3
- Date: Tue, 20 Feb 2024 11:19:47 GMT
- Title: A quantum inspired approach to learning dynamical laws from
data---block-sparsity and gauge-mediated weight sharing
- Authors: J. Fuksa, M. G\"otte, I. Roth, J. Eisert
- Abstract summary: We propose a scalable and numerically robust method for recovering dynamical laws of complex systems.
We use block-sparse tensor train representations of dynamical laws, inspired by similar approaches in quantum many-body systems.
We demonstrate the performance of the method numerically on three one-dimensional systems.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Recent years have witnessed an increased interest in recovering dynamical
laws of complex systems in a largely data-driven fashion under meaningful
hypotheses. In this work, we propose a scalable and numerically robust method
for this task, utilizing efficient block-sparse tensor train representations of
dynamical laws, inspired by similar approaches in quantum many-body systems.
Low-rank tensor train representations have been previously derived for
dynamical laws of one-dimensional systems. We extend this result to efficient
representations of systems with $K$-mode interactions and controlled
approximations of systems with decaying interactions. We further argue that
natural structure assumptions on dynamical laws, such as bounded polynomial
degrees, can be exploited in the form of block-sparse support patterns of
tensor-train cores. Additional structural similarities between interactions of
certain modes can be accounted for by weight sharing within the ansatz. To make
use of these structure assumptions, we propose a novel optimization algorithm,
block-sparsity restricted alternating least squares with gauge-mediated weight
sharing. The algorithm is inspired by similar notions in machine learning and
achieves a significant improvement in performance over previous approaches. We
demonstrate the performance of the method numerically on three one-dimensional
systems -- the Fermi-Pasta-Ulam-Tsingou system, rotating magnetic dipoles and
point particles interacting via modified Lennard-Jones potentials, observing a
highly accurate and noise-robust recovery.
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