用强化学习逐步求解非线性代数方程，支持换元技巧。
Reinforcement Learning for Symbolic Equation Solving
- 将代数题建模为动态动作空间的马尔可夫决策过程，用树形策略网络求解
- 在封闭方程上准确率达93%，开放族方程达79%（束搜索）
- 仅靠奖励学习换元时机，尤其对指数类问题提升显著
我们提出一个强化学习代理，分步求解符号方程，涵盖非线性封闭方程（如根式、指数、三角函数）以及需换元（CoV）的受控开放类问题（如配方法）。将代数问题建模为具有动态动作空间和树状策略（TreeMLP）的马尔可夫决策过程，主策略仅通过奖励学习，无需监督解迹；换元操作由监督生成器提供，可替换为符号计算系统调用。在封闭方程上，该代理在CommonCore数据集上以0.93的贪心准确率超越此前最优（ConPoLe的0.925）；在四个手设计的受限开放族（二次、三次、四次、指数）上，束搜索准确率达0.79，贪心为0.67，超过最强非学习搜索方法（A*，0.64）。学习到的换元时机仅在指数族中有效，该族需嵌套换元，自然规则在保留测试集上零解，而策略仅靠奖励即解决75%。在10倍规模下出现尖锐种子级双峰现象；采用UCB学习进度课程训练显示缓解趋势但不显著。不声称具备通用开放方程求解能力：所有开放方程结果均限于这四类可控家族。
原文摘要 · Abstract (English)
We present a reinforcement-learning agent that solves symbolic equations step by step, covering both nonlinear closed equations (radicals, exponentials, trigonometric) and a controlled class of restricted-open families requiring a change of variables (CoV) such as completing the square. We cast algebra as an MDP with a dynamic action space and a tree-structured policy (TreeMLP). The main policy learns from reward alone with no supervised solution traces; the CoV substitution comes from a supervised generator interchangeable with a CAS call. On closed equations the agent matches the prior best on CommonCore (0.93 greedy vs. ConPoLe's 0.925) under a single policy. On four hand-designed restricted-open families (quadratic, cubic, quartic, exponential) it reaches 0.79 beam / 0.67 greedy, exceeding the strongest non-learned search (A-star, 0.64). Learned CoV timing has content only on the exponential family, the one requiring a nested CoV, where a natural rule solves none of the held-out equations while the policy solves 75% from reward alone. At 10x scale a sharp seed-level bimodality emerges; a UCB learning-progress curriculum shows a non-significant positive trend toward mitigating it. We do not claim general open-equation solving: every open-equation result is confined to these four controlled families.
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