用大模型破解偏微分方程中的符号规律，提升求解效率与可解释性。
From Equations to Insights: Unraveling Symbolic Structures in PDEs with LLMs
- 用大语言模型挖掘偏微分方程中的符号算子关系。
- 显著提升符号机器学习求解精度与速度。
- 适合对科学计算可解释性有需求的研究者。
受人工智能在多领域取得的显著成功启发，其在解决以偏微分方程（PDEs）形式表达的科学问题方面日益受到关注。尽管现有研究多聚焦于解的理论性质（如适定性、正则性、连续性）及直接基于AI的求解方法，但对揭示方程内部符号关系的探索仍不充分。本文提出利用大语言模型（LLMs）学习此类符号关系。实验表明，LLMs能有效利用方程中的符号信息，理论上和数值上均准确预测解所涉及的算子。进一步证明，发现这些符号关系可显著提升符号机器学习在寻找解析近似解时的效率与准确性，构建出完全可解释的求解流程。该工作为理解科学问题的符号结构并推动求解过程提供了新路径。
原文摘要 · Abstract (English)
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 both theoretically and numerically. Furthermore, we show that discovering these symbolic relationships can substantially improve both the efficiency and accuracy of symbolic machine learning 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.
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