用逆向生成数据+强化学习，让AI高效发现动力系统守恒律。
Learning First Integrals via Backward-Generated Data and Guided Reinforcement Learning
- 逆向生成微分方程与首次积分对，解决高质量训练数据稀缺问题。
- 在挑战性基准上超越Mathematica等大型模型，且计算成本更低。
- 适合需要自动化发现物理守恒律的研究者或工具开发者。
首次积分的发现对理解动力系统的守恒定律具有根本意义。然而，现有符号计算工具和大语言模型在此任务上仍受限于高质量训练数据稀缺，且成功解法常依赖数学直觉。本文提出FISolver，一种基于LLM的求解器。首先，设计“逆向生成”算法，通过从采样的首次积分推导微分方程，系统构建大规模（微分方程，首次积分）数据对，缓解数据瓶颈。其次，对紧凑数学模型进行监督微调，并通过基于编辑距离的奖励函数引导强化学习进一步提升性能。此外，设计数据合成与融合策略，实现从稀疏样本中有效适应复杂问题族。实验表明，FISolver在计算成本显著更低的前提下，显著优于更大规模数学LLM及商用求解器如Mathematica，在挑战性基准上表现优异，揭示了一条自动发现首次积分的新数据驱动路径。
原文摘要 · Abstract (English)
The discovery of first integrals is of fundamental scientific importance for understanding conservation laws in dynamical systems. However, existing symbolic computation tools and Large Language Models (LLMs) remain limited on this task because high-quality training data are scarce and successful solutions often depend on mathematical intuition. This paper presents FISolver, an LLM-based solver developed to address this challenge. First, we introduce a "Backward Generation" algorithm that systematically builds large-scale datasets of (differential equation, first integral) pairs by deriving differential equations from sampled integrals, thereby alleviating the data scarcity bottleneck. Second, we apply supervised fine-tuning to a compact mathematical model and further improve its performance through reinforcement learning with a Levenshtein Distance-based shaped reward. In addition, we design data synthesis and blending strategies that support effective adaptation to difficult problem families from sparse examples. Experiments show that FISolver, while requiring substantially lower computational cost, significantly outperforms larger mathematical LLMs and commercial solvers such as Mathematica on challenging benchmarks, indicating a new data-driven route for automated discovery of first integrals.
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