神经预测-校正器：利用强化学习解决同伦问题
Neural Predictor-Corrector: Solving Homotopy Problems with Reinforcement Learning
AI总结：
本文提出神经预测-校正器，通过强化学习统一解决同伦问题，实现高效稳定的方法。
摘要：
同伦范式是一种解决复杂问题的一般原则，广泛应用于鲁棒优化、全局优化、多项式根寻找和采样等多个领域。实际求解器通常采用预测-校正（PC）结构，但依赖于手工设计的启发式方法来确定步长和迭代终止，这些方法往往不理想且任务特定。为此，我们将这些问题统一到一个框架下，从而设计出通用的神经求解器。基于这种统一的观点，我们提出了神经预测-校正器（NPC），用自动学习的策略取代手工设计的启发式方法。NPC将策略选择视为一个顺序决策问题，并利用强化学习自动发现高效的策略。为进一步增强泛化能力，我们引入了消融训练机制，使一类问题能够通过一次离线训练进行处理，并在新实例上实现高效的在线推理。在四个代表性同伦问题上的实验表明，我们的方法能够有效泛化到未见过的实例。它在效率上始终优于经典和专用基线方法，同时在不同任务上表现出更优越的稳定性，突显了将同伦方法统一到单一神经框架中的价值。
英文摘要：
The Homotopy paradigm, a general principle for solving challenging problems, appears across diverse domains such as robust optimization, global optimization, polynomial root-finding, and sampling. Practical solvers for these problems typically follow a predictor-corrector (PC) structure, but rely on hand-crafted heuristics for step sizes and iteration termination, which are often suboptimal and task-specific. To address this, we unify these problems under a single framework, which enables the design of a general neural solver. Building on this unified view, we propose Neural Predictor-Corrector (NPC), which replaces hand-crafted heuristics with automatically learned policies. NPC formulates policy selection as a sequential decision-making problem and leverages reinforcement learning to automatically discover efficient strategies. To further enhance generalization, we introduce an amortized training mechanism, enabling one-time offline training for a class of problems and efficient online inference on new instances. Experiments on four representative homotopy problems demonstrate that our method generalizes effectively to unseen instances. It consistently outperforms classical and specialized baselines in efficiency while demonstrating superior stability across tasks, highlighting the value of unifying homotopy methods into a single neural framework.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
