# Free-Energy Machine for Combinatorial Optimization

arXiv: https://arxiv.org/abs/2412.09285 · HTML: https://mlsanity.com/paper/2412.09285/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Dec 2024.

## Built on by (most cited first, 7 of 7)

- [An Energy-Based Mechanism for Compositional Behavior](https://mlsanity.com/paper/2512.04745/) (2025)
- [Neural QAOA2: Differentiable Joint Graph Partitioning and Parameter Initialization for Quantum Combinatorial Optimization](https://mlsanity.com/paper/2605.13072/) (2026)
- [Learning efficient representations of complex constraints for scalable optimization](https://mlsanity.com/paper/2603.08283/) (2026)
- [A general optimization solver based on OP-to-MaxSAT reduction](https://mlsanity.com/paper/2604.21961/) (2026)
- [Crystal structure prediction using graph neural combinatorial optimization](https://mlsanity.com/paper/2604.23921/) (2026)
- [Local-Minima-Preserving Continuous Relaxation of Ising Problems](https://mlsanity.com/paper/2606.30333/) (2026)
- [Iterative tensor network transformations for element-wise evaluation of elementary and filtering functions](https://mlsanity.com/paper/2608.17135/) (2026)

## Abstract (arXiv)

Finding optimal solutions to combinatorial optimization problems is pivotal in both scientific and technological domains, within academic research and industrial applications. A considerable amount of effort has been invested in the development of accelerated methods that leverage sophisticated models and harness the power of advanced computational hardware. Despite the advancements, a critical challenge persists, the dual demand for both high efficiency and broad generality in solving problems. In this work, we propose a general method, Free-Energy Machine (FEM), based on the ideas of free-energy minimization in statistical physics, combined with automatic differentiation and gradient-based optimization in machine learning. The algorithm is flexible, solving various combinatorial optimization problems using a unified framework, and is efficient, naturally utilizing massive parallel computational devices such as graph processing units (GPUs) and field-programmable gate arrays (FPGAs). We benchmark our algorithm on various problems including the maximum cut problems, balanced minimum cut problems, and maximum k-satisfiability problems, scaled to millions of variables, across both synthetic, real-world, and competition problem instances. The findings indicate that our algorithm not only exhibits exceptional speed but also surpasses the performance of state-of-the-art algorithms tailored for individual problems. This highlights that the interdisciplinary fusion of statistical physics and machine learning opens the door to delivering cutting-edge methodologies that will have broad implications across various scientific and industrial landscapes.