CodePDE: An Inference Framework for LLM-driven PDE Solver Generation

• URL: http://arxiv.org/abs/2505.08783v1
• Date: Tue, 13 May 2025 17:58:08 GMT
• Title: CodePDE: An Inference Framework for LLM-driven PDE Solver Generation
• Authors: Shanda Li, Tanya Marwah, Junhong Shen, Weiwei Sun, Andrej Risteski, Yiming Yang, Ameet Talwalkar,
• Abstract summary: Partial differential equations (PDEs) are fundamental to modeling physical systems.<n>Traditional numerical solvers rely on expert knowledge to implement and are computationally expensive.<n>We introduce CodePDE, the first inference framework for generating PDE solvers using large language models.
• Score: 57.15474515982337
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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). Leveraging advanced inference-time algorithms and scaling strategies, CodePDE unlocks critical capacities of LLM for PDE solving: reasoning, debugging, selfrefinement, and test-time scaling -- all without task-specific tuning. CodePDE achieves superhuman performance across a range of representative PDE problems. We also present a systematic empirical analysis of LLM generated solvers, analyzing their accuracy, efficiency, and numerical scheme choices. Our findings highlight the promise and the current limitations of LLMs in PDE solving, offering a new perspective on solver design and opportunities for future model development. Our code is available at https://github.com/LithiumDA/CodePDE.

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