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  • Open Access

Beyond gradient descent: Adam for analog Ising machines

Stijn Van Vooren*, Guy Van der Sande, and Guy Verschaffelt

  • *Contact author: Stijn.Van.Vooren@vub.be
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Phys. Rev. E 114, 045305 – Published 13 October, 2026

DOI: https://doi.org/10.1103/qg6g-nkf3

Abstract

As Moore's law reaches its limits, Ising machines offer a promising alternative computing approach for difficult optimization problems. However, many analog, time-continuous Ising machines rely on gradient-descent-like dynamics to find solutions, which can limit speed and robustness. We investigate whether momentum and Adam optimization can improve these systems. Since these optimizers are traditionally formulated in discrete time, we derive continuous-time versions suitable for analog, time-continuous Ising-machine dynamics. On Max-Cut benchmarks, we find that Adam-based dynamics substantially reduce time to target and improve solution quality compared with gradient-descent- and momentum-based dynamics. We further introduce a first-order continuous-time approximation of Adam that is intended as a simpler starting point for future physical implementations and while performing better than the full Adam formulation in a continuous-time setting. We also study a purely algorithmic discrete-time setting, where the performance gap is reduced on easier problem instances, while the Adam-based update rule performs best on harder weighted problem instances. These results identify continuous-time Adam dynamics as a powerful design principle for analog Ising machines.

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Phys. Rev. E 114, 045305– Published 13 October, 2026
  • Received 29 May 2026
  • Accepted 27 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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