神经预测校正器：基于强化学习的同伦问题求解方法

摘要

同倫範式作為解決複雜問題的通用原理，廣泛存在於魯棒優化、全局優化、多項式求根和抽樣等領域。針對這些問題的實際求解器通常採用預測-校正架構，但依賴人工設計的步長與迭代終止啓發式規則，這些規則往往存在次優性且僅適用於特定任務。為此，我們將上述問題統一於單一框架，據此設計出通用神經求解器。基於此統一視角，我們提出神經預測-校正器，用自動學習的策略替代人工啓發規則。該方法將策略選擇建模為序列決策問題，通過強化學習自動發現高效求解策略。為提升泛化能力，我們引入攤銷訓練機制，實現對問題類別的一次性離線訓練及新實例的高效在線推理。在四個典型同倫問題上的實驗表明，本方法能有效泛化至未見實例，在效率上持續超越經典方法與專用基線，並展現出跨任務的卓越穩定性，彰顯了將同倫方法統一至神經框架的價值。

English

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.

神经预测校正器：基于强化学习的同伦问题求解方法

Neural Predictor-Corrector: Solving Homotopy Problems with Reinforcement Learning

摘要

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