这篇论文用严格的对照实验告诉你：扩散模型在约束求解里什么时候有用、什么时候没用，结果很反直觉——真实系统里随机多起点反而全赢了。做AI for science和优化的人值得看看。
该论文研究了基于扩散模型的提议在连续代数约束求解中的有效性。在候选条件修复排序中，K个增强选项的排名器以少量调用达到穷举搜索上限：平衡非线性菜单准确率0.997 vs 0.236（p<10^-70），多种子平均0.982±0.006。提出的MARC系统将约束转化为因子图，用图神经扩散去噪器提出赋值，再通过精确计算机代数能量下降优化并由符号检查器验证。与随机多起点相比，学习式提议只在高维空间明显胜出（低维持平，变量耦合时优势消失）。在八个真实系统（机器人、定位、优化、代数）中，经典随机多起点全部求解，无一处于学习有利区间。
When Do Learned Diffusion Proposals Help Constraint Solving? A Controlled Study on Continuous Algebraic Systems
Solving a continuous algebraic constraint system requires two decisions: which values satisfy the constraints, and which structural augmentation renders an unsolvable system solvable. Classical solvers answer the first well and the second only by enumeration. On that discrete decision, a candidate-conditioned repair ranker choosing among K augmentations reaches the exhaustive-search ceiling at a fraction of the calls, outperforming random (0.997 vs 0.236 balanced nonlinear menu accuracy; p < 10^-70; 0.982 +/- 0.006 across seeds) and beating a budget-matched per-candidate probe on accuracy and cost. MARC turns such a system into a factor graph, over which a graph-neural diffusion denoiser proposes assignments, descent on an exact computer-algebra energy polishes them, and an exact symbolic checker certifies solutions. Evaluations of diffusion-based proposals rarely include one control: random multi-start under the same refinement budget. Applied to our system, it sharply curtails what the learned proposal contributes on the value decision. Does it beat random multi-start at choosing satisfying assignments? Only narrowly, in a predictable regime. Across trapped low-dimensional families it ties with random restart, but dominates in high dimension, where random search fails. Once variables couple, the advantage is gone. Since all methods share one polish and one checker, best-of-K random multi-start succeeds with probability exactly 1 - (1 - q(n))^K, where q(n) is single-start reachability; one measured constant, with no free parameters, reproduces the entire curve (mean absolute error 0.012). The favorable regime is not specific to our synthetic families: across eight real-world systems in robotics, positioning, optimization, and algebra, classical multi-start solved all eight, none in the learning-favorable regime. We map the regimes in which learned proposals improve solvers.