PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs

Mauricio Soroco, Jialin Song, Mengzhou Xia, Kye Emond, Weiran Sun, Wuyang Chen
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:56499-56530, 2025.

Abstract

We present PDE-Controller, a framework that enables large language models (LLMs) to control systems governed by partial differential equations (PDEs). Traditional LLMs have excelled in commonsense reasoning but fall short in rigorous logical reasoning. 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. 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 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 promise to release all data, model checkpoints, and code upon acceptance.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-soroco25a, title = {{PDE}-Controller: {LLM}s for Autoformalization and Reasoning of {PDE}s}, author = {Soroco, Mauricio and Song, Jialin and Xia, Mengzhou and Emond, Kye and Sun, Weiran and Chen, Wuyang}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {56499--56530}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/soroco25a/soroco25a.pdf}, url = {https://proceedings.mlr.press/v267/soroco25a.html}, abstract = {We present PDE-Controller, a framework that enables large language models (LLMs) to control systems governed by partial differential equations (PDEs). Traditional LLMs have excelled in commonsense reasoning but fall short in rigorous logical reasoning. 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. 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 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 promise to release all data, model checkpoints, and code upon acceptance.} }
Endnote
%0 Conference Paper %T PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs %A Mauricio Soroco %A Jialin Song %A Mengzhou Xia %A Kye Emond %A Weiran Sun %A Wuyang Chen %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-soroco25a %I PMLR %P 56499--56530 %U https://proceedings.mlr.press/v267/soroco25a.html %V 267 %X We present PDE-Controller, a framework that enables large language models (LLMs) to control systems governed by partial differential equations (PDEs). Traditional LLMs have excelled in commonsense reasoning but fall short in rigorous logical reasoning. 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. 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 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 promise to release all data, model checkpoints, and code upon acceptance.
APA
Soroco, M., Song, J., Xia, M., Emond, K., Sun, W. & Chen, W.. (2025). PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:56499-56530 Available from https://proceedings.mlr.press/v267/soroco25a.html.

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