# Tensor network approaches for learning non-linear dynamical laws

arXiv: https://arxiv.org/abs/2002.12388 · HTML: https://mlsanity.com/paper/2002.12388/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Feb 2020.

## Built on by (most cited first, 7 of 7)

- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [AI Descartes: Combining Data and Theory for Derivable Scientific Discovery](https://mlsanity.com/paper/2109.01634/) (2021)
- [Scalably learning quantum many-body Hamiltonians from dynamical data](https://mlsanity.com/paper/2209.14328/) (2022)
- [Prospects for quantum advantage in machine learning from the representability of functions](https://mlsanity.com/paper/2512.15661/) (2025)
- [Convergence bounds for nonlinear least squares and applications to tensor recovery](https://mlsanity.com/paper/2108.05237/) (2021)
- [A quatum inspired neural network for geometric modeling](https://mlsanity.com/paper/2401.01801/) (2024)

## Abstract (arXiv)

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.