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Interpretable tensor-neural networks as topological invariants

D. O. Oriekhov*, Stan Bergkamp, Guliuxin Jin, Juan Daniel Torres Luna, Badr Zouggari, Sibren van der Meer, Naoual El Yazidi, and Eliska Greplova†

  • *Contact author: d.oriekhov@tudelft.nl
  • †Contact author: e.greplova@tudelft.nl
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Phys. Rev. Research 8, 033211 – Published 20 August, 2026

DOI: https://doi.org/10.1103/grxl-jcbs

Abstract

Much attention has been devoted to the use of machine learning to approximate physical concepts. Yet, due to challenges in interpretability of machine learning techniques, the question of what physics machine learning models are able to learn remains open. Here, we bridge the concept of a physical quantity and its machine learning approximation in the context of the original application of neural networks in physics: topological phase classification. We construct a hybrid tensor-neural network object that exactly expresses the real-space topological invariant and rigorously assess its trainability and generalization. Specifically, we benchmark the accuracy and trainability of a tensor-neural network to multiple types of neural networks, thus exemplifying the differences in trainability and representational power. Our work highlights the challenges in learning topological invariants and constitutes a stepping stone toward more accurate and better generalizable machine learning representations in condensed matter physics.

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Phys. Rev. Research 8, 033211– Published 20 August, 2026
  • Received 4 December 2025
  • Accepted 10 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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