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.

翻译：我们提出SymPlex，一种用于发现偏微分方程解析符号解的强化学习框架，无需依赖真实表达式。SymPlex将符号偏微分方程求解建模为树结构决策过程，并仅利用偏微分方程及其边界条件优化候选解。其核心是SymFormer，一种结构感知Transformer，它通过树相对自注意力建模层次化符号依赖关系，并通过语法约束的自回归解码确保句法有效性，克服了基于序列的生成器表达能力有限的缺陷。与在离散化或隐函数空间中近似求解的数值和神经网络方法不同，SymPlex直接在符号表达式空间中操作，从而产生可解释且人类可读的解，这些解天然能表示非光滑行为和显式参数依赖性。实证结果表明，基于深度学习的符号方法能够精确恢复非光滑及含参数的偏微分方程解。