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  • Open Access

Near-Equilibrium Propagation Training in Nonlinear Wave Systems

Karol Sajnok1,* and Michał Matuszewski1,2,†

  • *Contact author: ksajnok@cft.edu.pl
  • †Contact author: mmatuszewski@cft.edu.pl
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PRX Intelligence 1, 013022 – Published 22 September, 2026

DOI: https://doi.org/10.1103/qd75-znrn

Abstract

Backpropagation learning algorithm, the workhorse of modern artificial intelligence, is notoriously difficult to implement in physical neural networks. Equilibrium propagation (EP) is an alternative with comparable efficiency and strong potential for in situ training. We extend EP learning to both discrete and continuous complex-valued wave systems. In contrast to previous EP implementations, our scheme is valid in the weakly dissipative regime and readily applicable to a wide range of physical settings, even without well-defined nodes, where trainable internode connections can be replaced by trainable local potential. We test the method in driven-dissipative exciton-polariton condensates governed by generalized Gross-Pitaevskii dynamics. Numerical studies on standard benchmarks, including a simple logical task and handwritten-digit recognition, demonstrate stable convergence, establishing a practical route to in situ learning in physical systems in which system control is restricted to local parameters.

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PRX Intelligence 1, 013022– Published 22 September, 2026
  • Received 17 October 2025
  • Revised 20 February 2026
  • Accepted 4 August 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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