Tensor network approaches for learning non-linear dynamical laws

Abstract

Given observations of a physical system, identifying the underlying non-linear governing equation is a fundamental task, necessary both for gaining understanding and generating deterministic future predictions. Of most practical relevance are automated approaches to theory building that scale efficiently for complex systems with many degrees of freedom. To date, available scalable methods aim at a data-driven interpolation, without exploiting or offering insight into fundamental underlying physical principles, such as locality of interactions. In this work, we show that various physical constraints can be captured via tensor network based parameterizations for the governing equation, which naturally ensures scalability. In addition to providing analytic results motivating the use of such models for realistic physical systems, we demonstrate that efficient rank-adaptive optimization algorithms can be used to learn optimal tensor network models without requiring a~priori knowledge of the exact tensor ranks. As such, we provide a physics-informed approach to recovering structured dynamical laws from data, which adaptively balances the need for expressivity and scalability.

17 pages, 8 figures

References in corpus (11)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Area laws for the entanglement entropy - a review
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Entanglement renormalization
• Entropy scaling and simulability by Matrix Product States
• On the Expressive Power of Deep Learning: A Tensor Analysis
• Deep Learning and Quantum Entanglement: Fundamental Connections with Implications to Network Design
• Tensor-based algorithms for image classification
• Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning
• Neural-Guided Symbolic Regression with Asymptotic Constraints
• Recovery guarantees for polynomial approximation from dependent data with outliers

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• Asymptotic Log-Det Sum-of-Ranks Minimization via Tensor (Alternating) Iteratively Reweighted Least Squares
• Convergence bounds for nonlinear least squares and applications to tensor recovery
• WaveTrain: A Python Package for Numerical Quantum Mechanics of Chain-Like Systems Based on Tensor Trains