PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
This work addresses the underexplored area of PDEs in AI-for-math, bridging language generation and PDE systems for complex scientific and engineering applications, though it appears incremental as it builds on existing LLM capabilities.
The paper tackles the problem of enabling large language models (LLMs) to control systems governed by partial differential equations (PDEs) by transforming informal natural language instructions into formal specifications and executing reasoning steps, achieving up to a 62% improvement in utility gain for PDE control compared to existing models.
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/.