PDE-Controller：LLMs 用于偏微分方程的自动形式化和推理

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

尽管最近的数学人工智能取得了纯数学方面的进展，但应用数学领域，特别是偏微分方程（PDEs），尽管在现实世界中具有重要应用，仍然未被充分探索。我们提出了PDE-Controller，这是一个框架，使得大型语言模型（LLMs）能够控制由偏微分方程（PDEs）管理的系统。我们的方法使LLMs能够将非正式的自然语言指令转换为正式规范，然后执行推理和规划步骤，以提高PDE控制的效用。我们构建了一个全面的解决方案，包括数据集（人工编写的案例和200万个合成样本）、数学推理模型和新颖的评估指标，所有这些都需要大量的工作。我们的PDE-Controller在推理、自动形式化和程序合成方面明显优于提示最新的开源和GPT模型，为PDE控制实现了高达62%的效用增益。通过弥合语言生成和PDE系统之间的差距，我们展示了LLMs在解决复杂科学和工程挑战方面的潜力。我们将在https://pde-controller.github.io/发布所有数据、模型检查点和代码。

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/.