用于偏微分方程工作流的大语言模型
Large language models for partial differential equation workflows
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中文总结 AI 辅助
本文探讨大语言模型（LLMs）辅助偏微分方程（PDE）工作流在模型构建、求解器生成等阶段的进展，指出其受限于数据集稀缺与仿真-现实差距，是科学AI系统开发的关键试验台。
中文摘要 AI 辅助
偏微分方程（PDEs）在科学与工程领域要具备可操作性，并非作为孤立的公式，而是作为可执行的工作流存在，这些工作流连接着建模假设、控制方程、数值求解器、诊断工具与决策环节。大语言模型（LLMs）正开始通过关联自然语言、符号数学、代码、求解器输出及反馈来支持此类工作流。本文考察了LLM辅助PDE研究在三个阶段的最新进展：控制模型的发现与构建、可执行数值求解器的生成与修订，以及利用仿真反馈支持控制、设计与优化。在这些阶段中，当前系统主要充当工作流级别的接口。尽管取得了这些进展，该领域仍受限于高质量数据集与基准的稀缺，尤其在知识发现与实际应用场景中，专家标注、可执行问题构建及任务级反馈需要大量领域投入。另一项挑战是基于仿真的结果与现实世界科学及工程系统之间持续存在的差距，这限制了数值仿真、控制策略及优化设计向实际场景的直接迁移。这些挑战使得LLM辅助的PDE工作流成为开发科学AI系统的关键试验台，这类系统能够关联语言、计算、物理约束与现实世界决策。
英文摘要
Numerical methods and deep learning have advanced the modelling and simulation of systems governed by partial differential equations (PDEs), while formulating equations, configuring solvers, and translating simulations into design and control decisions continue to demand substantial domain expertise and computational resources. Large language models (LLMs) offer opportunities to automate and coordinate these tasks by combining scientific knowledge, mathematical reasoning, code generation, and tool use. Here we review LLM-assisted PDE research across three stages, including discovery, solving, and optimization. We examine how LLMs interact with physical knowledge, numerical methods, and computational feedback to formulate models, implement solution procedures, and guide design and control. We also review existing benchmarks and evaluation practices for assessing scientific validity, computational performance, and workflow reliability. Finally, we discuss three interconnected challenges and opportunities: developing mathematical and physical reasoning tailored to specific PDE problems; building transferable domain expertise through domain-specific and multimodal post-training; and integrating active experimental design with scientific validation for open-ended problems. These directions outline a path towards a more substantive role for LLMs in developing and testing scientific ideas and computational methods across PDE research.
