用强化学习找偏微分方程的解析解，直接输出可读公式。
SymPlex: A Structure-Aware Transformer for Symbolic PDE Solving
- 将符号求解建模为树结构决策，用语法约束生成合法表达式。
- 在无真值情况下成功恢复非光滑和含参的精确解析解。
- 适合需要可解释性解的物理建模与理论推导场景。
我们提出SymPlex，一种无需真值表达式的强化学习框架，用于发现偏微分方程（PDE）的解析符号解。SymPlex将符号求解建模为树结构决策过程，仅依赖PDE及其边界条件优化候选解。核心是SymFormer，一种结构感知Transformer，通过树相对自注意力建模符号层级依赖，并利用语法约束自回归解码保证表达式语法正确性，突破了序列生成器的表达能力局限。与数值或神经方法在离散或隐函数空间近似解不同，SymPlex直接在符号表达式空间操作，生成可解释且人类可读的解，能自然表征非光滑行为和显式参数依赖。实验证明该方法可基于深度学习符号方法精确恢复非光滑及含参的PDE解。
原文摘要 · Abstract (English)
We propose SymPlex, a reinforcement learning framework for discovering analytical symbolic solutions to partial differential equations (PDEs) without access to ground-truth expressions. SymPlex formulates symbolic PDE solving as tree-structured decision-making and optimizes candidate solutions using only the PDE and its boundary conditions. At its core is SymFormer, a structure-aware Transformer that models hierarchical symbolic dependencies via tree-relative self-attention and enforces syntactic validity through grammar-constrained autoregressive decoding, overcoming the limited expressivity of sequence-based generators. Unlike numerical and neural approaches that approximate solutions in discretized or implicit function spaces, SymPlex operates directly in symbolic expression space, enabling interpretable and human-readable solutions that naturally represent non-smooth behavior and explicit parametric dependence. Empirical results demonstrate exact recovery of non-smooth and parametric PDE solutions using deep learning-based symbolic methods.
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