From Equations to Insights: Unraveling Symbolic Structures in PDEs with LLMs

2025-03-13

AI Summary

Discovering symbolic structures in partial differential equations (PDEs) remains challenging due to the implicit and analytically intractable nature of their underlying symbolic relationships. Method: This work introduces, for the first time, large language models (LLMs) to PDE symbolic structure discovery, proposing an interpretable framework that integrates LLM-based reasoning, symbolic expression parsing, and the finite element method (FEM). The framework enables PDE operator identification, symbolic dependency modeling, and approximate analytical solution generation. Contribution/Results: By bridging symbolic reasoning and numerical approximation while preserving mathematical interpretability, the method significantly improves both the efficiency and accuracy of FEM-based solvers. It demonstrates high accuracy in symbolic relation prediction and strong cross-equation generalization on canonical PDEs—including the heat equation, wave equation, and Poisson equation—thereby establishing a novel paradigm for AI-driven scientific discovery.

Technology Category

• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Machine Learning: Neuro-Symbolic Learning
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation

Abstract

Motivated by the remarkable success of artificial intelligence (AI) across diverse fields, the application of AI to solve scientific problems-often formulated as partial differential equations (PDEs)-has garnered increasing attention. While most existing research concentrates on theoretical properties (such as well-posedness, regularity, and continuity) of the solutions, alongside direct AI-driven methods for solving PDEs, the challenge of uncovering symbolic relationships within these equations remains largely unexplored. In this paper, we propose leveraging large language models (LLMs) to learn such symbolic relationships. Our results demonstrate that LLMs can effectively predict the operators involved in PDE solutions by utilizing the symbolic information in the PDEs. Furthermore, we show that discovering these symbolic relationships can substantially improve both the efficiency and accuracy of the finite expression method for finding analytical approximation of PDE solutions, delivering a fully interpretable solution pipeline. This work opens new avenues for understanding the symbolic structure of scientific problems and advancing their solution processes.

Problem

• Uncovering symbolic relationships in PDEs using LLMs
• Improving efficiency and accuracy of PDE solutions
• Advancing interpretable solution pipelines for scientific problems

Innovation

• Leveraging LLMs to uncover symbolic PDE relationships
• Predicting PDE operators using symbolic information
• Enhancing analytical PDE solutions with interpretable methods