Beyond Gradient Descent: Adam for Analog Ising Machines

2026-06-02

AI Summary

This work addresses the limitations of conventional analog Ising machines, which suffer from slow convergence and insufficient robustness due to gradient-descent-like dynamics. The authors propose, for the first time, a continuous-time Adam optimization dynamics tailored for analog Ising machines and derive its first-order approximation to facilitate physical implementation. By introducing an adaptive momentum mechanism into the continuous-time domain, the proposed dynamics significantly accelerates convergence and enhances solution quality on Max-Cut benchmark problems. Notably, it achieves state-of-the-art performance on weighted, hard instances, thereby demonstrating its effectiveness and superiority in solving combinatorial optimization tasks.

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.

Problem

• Ising machines
• optimization problems
• gradient descent
• analog computing
• continuous-time dynamics

Innovation

• continuous-time Adam
• analog Ising machines
• optimization dynamics
• momentum methods
• Max-Cut