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

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 atthis https URL.

@article{li2025_2505.08783,
  title={ CodePDE: An Inference Framework for LLM-driven PDE Solver Generation },
  author={ Shanda Li and Tanya Marwah and Junhong Shen and Weiwei Sun and Andrej Risteski and Yiming Yang and Ameet Talwalkar },
  journal={arXiv preprint arXiv:2505.08783},
  year={ 2025 }
}