SymPlex: A Structure-Aware Transformer for Symbolic PDE Solving

2026-02-03

AI Summary

This work proposes a reinforcement learning–based approach for discovering symbolic solutions to partial differential equations (PDEs) in the absence of ground-truth symbolic supervision. By formulating the search for symbolic expressions as a tree-structured decision process, the authors introduce SymFormer—a structure-aware architecture that integrates tree-relative self-attention with grammar-constrained autoregressive decoding to directly optimize analytic expressions satisfying the PDE and its boundary conditions in symbolic space. This method overcomes the representational limitations of conventional sequence-based generative models and successfully recovers exact closed-form solutions for several non-smooth and parametric PDEs, thereby demonstrating the effectiveness and promise of deep learning in interpretable, symbolic mathematical reasoning.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Computer Vision: Visual Reasoning & Symbolic Representations
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• symbolic PDE solving
• analytical solutions
• non-smooth behavior
• parametric dependence
• interpretable solutions

Innovation

• Symbolic PDE solving
• Structure-aware Transformer
• Tree-relative self-attention
• Grammar-constrained decoding
• Reinforcement learning for symbolic regression