SymPlex: A Structure-Aware Transformer for Symbolic PDE Solving

TL;DR

SymPlex introduces a structure-aware Transformer framework that discovers symbolic solutions to PDEs directly in expression space, enabling interpretable, exact, and non-smooth solutions without ground-truth expressions.

Contribution

It presents SymPlex, a novel reinforcement learning approach with SymFormer, a Transformer that models hierarchical symbolic dependencies and enforces syntax, advancing symbolic PDE solving.

Findings

Exact recovery of non-smooth PDE solutions

Handles parametric PDEs effectively

Operates directly in symbolic expression space

Abstract

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.