PDE-Controller：LLM 用於偏微分方程的自動形式化和推理

PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs

February 3, 2025

作者: Mauricio Soroco, Jialin Song, Mengzhou Xia, Kye Emond, Weiran Sun, Wuyang Chen

cs.AI

摘要

最近在數學人工智慧領域取得了一些進展，尤其是在純數學方面，但應用數學領域，特別是偏微分方程（PDEs），儘管具有重要的現實應用，仍然未被充分探索。我們提出了PDE-Controller，這是一個框架，使得大型語言模型（LLMs）能夠控制由偏微分方程（PDEs）控制的系統。我們的方法使得LLMs能夠將非正式的自然語言指令轉換為正式規範，然後執行推理和規劃步驟，以提高PDE控制的效用。我們構建了一個全面的解決方案，包括數據集（人工編寫案例和200萬個合成樣本）、數學推理模型和新穎的評估指標，所有這些都需要大量的努力。我們的PDE-Controller在推理、自動形式化和程序合成方面明顯優於最新的開源和GPT模型，PDE控制的效用增益可達62％。通過彌合語言生成和PDE系統之間的差距，我們展示了LLMs在應對複雜科學和工程挑戰方面的潛力。我們將在https://pde-controller.github.io/上發布所有數據、模型檢查點和代碼。

English

While recent AI-for-math has made strides in pure mathematics, areas of applied mathematics, particularly PDEs, remain underexplored despite their significant real-world applications. We present PDE-Controller, a framework that enables large language models (LLMs) to control systems governed by partial differential equations (PDEs). Our approach enables LLMs to transform informal natural language instructions into formal specifications, and then execute reasoning and planning steps to improve the utility of PDE control. We build a holistic solution comprising datasets (both human-written cases and 2 million synthetic samples), math-reasoning models, and novel evaluation metrics, all of which require significant effort. Our PDE-Controller significantly outperforms prompting the latest open-source and GPT models in reasoning, autoformalization, and program synthesis, achieving up to a 62% improvement in utility gain for PDE control. By bridging the gap between language generation and PDE systems, we demonstrate the potential of LLMs in addressing complex scientific and engineering challenges. We will release all data, model checkpoints, and code at https://pde-controller.github.io/.