用大模型生成微分方程求解器，提升科学计算自动化水平
CodePDE: An Inference Framework for LLM-driven PDE Solver Generation
- 将求解偏微分方程转化为代码生成任务，利用大模型推理生成代码
- 在多种典型方程上实现强性能，验证了推理时算法的有效性
- 揭示了模型可靠性与复杂度的权衡，适合科研自动化场景
偏微分方程（PDE）是建模物理系统的基础，但求解仍具挑战。传统数值求解器依赖专家知识且计算成本高，而基于神经网络的求解器需要大量训练数据且可解释性差。本文将PDE求解视为代码生成任务，提出首个基于大语言模型（LLM）的推理框架CodePDE，用于生成PDE求解器。通过全面评估LLM在推理阶段的推理、调试、自优化和测试时扩展能力，结果表明：结合先进推理算法与缩放策略，LLM可在多种代表性PDE问题上实现优异表现。研究还发现新见解，如求解器可靠性与复杂度之间的权衡、基于LLM的求解代理设计原则，以及模型在困难任务中的失败模式。这些发现为构建更强大可靠的科学计算型大模型系统提供指导。
原文摘要 · Abstract (English)
Partial differential equations (PDEs) are fundamental to modeling physical systems, yet solving them remains a complex challenge. Traditional numerical solvers rely on expert knowledge to implement and are computationally expensive, while neural-network-based solvers require large training datasets and often lack interpretability. In this work, we frame PDE solving as a code generation task and introduce CodePDE, the first inference framework for generating PDE solvers using large language models (LLMs). With CodePDE, we present a thorough evaluation on critical capacities of LLM for PDE solving: reasoning, debugging, self-refinement, and test-time scaling. CodePDE shows that, with advanced inference-time algorithms and scaling strategies, LLMs can achieve strong performance across a range of representative PDE problems. We also identify novel insights into LLM-driven solver generation, such as trade-offs between solver reliability and sophistication, design principles for LLM-powered PDE solving agents, and failure modes for LLM on hard tasks. These insights offer guidance for building more capable and reliable LLM-based scientific engines.
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