用少量数据让大模型精准求解各类偏微分方程
OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inference
- 基于算子推断，用少量解数据训练模型
- 在未见参数和边界条件下仍保持高精度
- 支持自然语言描述任务，适合工程与科研场景
求解多样化的偏微分方程（PDE）是科学与工程的基础。大型语言模型（LLMs）在代码生成、符号推理和工具使用方面表现出色，但在异构环境下可靠求解PDE仍具挑战。已有基于LLM的代码生成和Transformer基础模型在PDE学习中取得进展，但泛化到未见参数和边界条件时，执行成功率与数值精度之间存在持续权衡。本文提出OpInf-LLM，一种通过算子推断实现参数化PDE求解的LLM框架。该框架仅需少量解数据，即可准确预测多种PDE实例，包括未见参数与配置，并实现与LLMs的无缝集成，支持自然语言任务描述及物理特征合理参数化。其低计算开销与统一求解流程，在异构设置下实现高执行成功率，为基于LLM的可泛化降阶建模开辟新可能。
原文摘要 · Abstract (English)
Solving diverse partial differential equations (PDEs) is fundamental in science and engineering. Large language models (LLMs) have demonstrated strong capabilities in code generation, symbolic reasoning, and tool use, but reliably solving PDEs across heterogeneous settings remains challenging. Prior work on LLM-based code generation and transformer-based foundation models for PDE learning has shown promising advances. However, a persistent trade-off between execution success rate and numerical accuracy arises, particularly when generalization to unseen parameters and boundary conditions is required. In this work, we propose OpInf-LLM, an LLM parametric PDE solving framework via operator inference. The proposed framework leverages small amounts of solution data to enable accurate prediction of diverse PDE instances, including unseen parameters and configurations, and provides seamless integration with LLMs for natural language task specification and physics-based reasoning of proper feature parameterization. Its low computational demands and unified solution pipeline further enable a high execution success rate across heterogeneous settings, opening new possibilities for generalizable reduced-order modeling in LLM-based PDE solving.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。