SymPlex: A Structure-Aware Transformer for Symbolic PDE Solving
This addresses the challenge of obtaining interpretable, human-readable solutions for PDEs, which is important for scientists and engineers, though it appears incremental as it builds on existing symbolic and neural methods.
The authors tackled the problem of discovering analytical symbolic solutions to partial differential equations (PDEs) without ground-truth expressions, proposing SymPlex, a reinforcement learning framework that formulates this as tree-structured decision-making and uses a structure-aware Transformer for grammar-constrained decoding, resulting in exact recovery of non-smooth and parametric PDE solutions.
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.