# Beyond Gradient Descent: Adam for Analog Ising Machines

arXiv: https://arxiv.org/abs/2606.03917 · HTML: https://mlsanity.com/paper/2606.03917/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 24 references that are in the map.
- First posted on arXiv 2 Jun 2026.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Convergence and Dynamical Behavior of the ADAM Algorithm for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1810.02263/) (2018)
- [A general system of differential equations to model first order adaptive algorithms](https://mlsanity.com/paper/1810.13108/) (2018)
- [BayesOpt: A Bayesian Optimization Library for Nonlinear Optimization, Experimental Design and Bandits](https://mlsanity.com/paper/1405.7430/) (2014)
- [Noise-injected analog Ising machines enable ultrafast statistical sampling and machine learning](https://mlsanity.com/paper/2112.11534/) (2021)
- [Accelerating Continuous Variable Coherent Ising Machines via Momentum](https://mlsanity.com/paper/2401.12135/) (2024)

## Abstract (arXiv)

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.