Combinatorial optimization problems near algorithmic phase transitions represent a fundamental challenge for both classical algorithms and machine learning approaches. Among them, graph coloring stands as a prototypical constraint satisfaction problem exhibiting sharp dynamical and satisfiability thresholds. Here we introduce a physics-inspired neural framework that learns to solve large-scale graph coloring instances by combining graph neural networks with statistical-mechanics principles. Our approach integrates a planting-based supervised signal, symmetry-breaking regularization, and iterative noise-annealed neural dynamics to navigate clustered solution landscapes. When the number of iterations scales quadratically with graph size, the learned solver reaches algorithmic thresholds close to the theoretical dynamical transition in random graphs and achieves near-optimal detection performance in the planted inference regime. The model generalizes from small training graphs to instances orders of magnitude larger, demonstrating that neural architectures can learn scalable algorithmic strategies that remain effective in hard connectivity regions. These results establish a general paradigm for learning neural solvers that operate near fundamental phase boundaries in combinatorial optimization and inference.

翻译：组合优化问题在算法相变附近对经典算法和机器学习方法均构成根本性挑战。其中，图着色作为典型的约束满足问题，展现出尖锐的动力学与可满足性阈值。本文提出一种物理启发的神经计算框架，通过将图神经网络与统计力学原理相结合，学习解决大规模图着色实例。该方法融合了基于植入的监督信号、对称破缺正则化以及迭代噪声退火神经动力学，以导航聚类化的解空间景观。当迭代次数与图规模呈二次方关系时，学习得到的求解器在随机图中可达到接近理论动力学转变的算法阈值，并在植入推断机制中实现接近最优的检测性能。该模型能够从训练阶段的小规模图泛化至规模大数个数量级的实例，证明神经架构能够学习可扩展的算法策略，并在困难连通区域保持有效性。这些结果为学习在组合优化与推断基本相边界附近运行的神经求解器建立了通用范式。