PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs

TL;DR

PDE-Controller leverages large language models to formalize, reason about, and control systems governed by PDEs, significantly advancing AI applications in applied mathematics and engineering.

Contribution

The paper introduces PDE-Controller, a comprehensive framework combining datasets, models, and metrics to enable LLMs to handle PDE formalization and control tasks.

Findings

Outperforms existing models in PDE reasoning and autoformalization.

Achieves up to 62% utility gain in PDE control tasks.

Provides publicly available datasets, models, and code.

Abstract

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/.