PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
Authors: name: Mauricio Soroco, name: Jialin Song, name: Mengzhou Xia, name: Kye Emond, name: Weiran Sun, name: Wuyang Chen

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/.