让大模型自动理解自然语言并控制偏微分方程系统
PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
- 将自然语言指令转化为偏微分方程的正式规范
- 在控制任务中实现最高62%的性能提升
- 适合科学计算与工程自动化领域的研究者
尽管人工智能在纯数学领域取得进展，但应用数学中的偏微分方程（PDE）仍被严重低估。我们提出PDE-Controller框架，使大语言模型（LLMs）能够控制由偏微分方程描述的系统。该方法将非正式自然语言指令转化为形式化规范，并执行推理与规划以增强PDE控制效果。我们构建了包含人工撰写案例和200万条合成样本的数据集、数学推理模型及新型评估指标，整体工作量巨大。PDE-Controller在推理、自动形式化与程序生成方面显著优于最新开源模型及GPT系列，使PDE控制的效用提升最高达62%。本研究打通语言生成与PDE系统之间的壁垒，展示了大模型解决复杂科学与工程问题的潜力。所有数据、模型检查点与代码已公开于https://pde-controller.github.io/。
原文摘要 · Abstract (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 release all data, model checkpoints, and code at https://pde-controller.github.io/.
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