# PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs

arXiv: https://arxiv.org/abs/2502.00963 · HTML: https://mlsanity.com/paper/2502.00963/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 37 of its 68 references that are in the map.
- First posted on arXiv 3 Feb 2025.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 30 of 37)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models](https://mlsanity.com/paper/2410.05229/) (2024)
- [Robots That Ask For Help: Uncertainty Alignment for Large Language Model Planners](https://mlsanity.com/paper/2307.01928/) (2023)
- [Large Language Models for Mathematical Reasoning: Progresses and Challenges](https://mlsanity.com/paper/2402.00157/) (2024)
- [Iterative Reasoning Preference Optimization](https://mlsanity.com/paper/2404.19733/) (2024)
- [FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI](https://mlsanity.com/paper/2411.04872/) (2024)
- [DeepSeek-Prover: Advancing Theorem Proving in LLMs through Large-Scale Synthetic Data](https://mlsanity.com/paper/2405.14333/) (2024)
- [RedPajama: an Open Dataset for Training Large Language Models](https://mlsanity.com/paper/2411.12372/) (2024)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [A Survey of Deep Learning for Mathematical Reasoning](https://mlsanity.com/paper/2212.10535/) (2022)
- [Large Language Models for Robotics: Opportunities, Challenges, and Perspectives](https://mlsanity.com/paper/2401.04334/) (2024)
- [A Survey on Deep Learning for Theorem Proving](https://mlsanity.com/paper/2404.09939/) (2024)
- [Don't Trust: Verify -- Grounding LLM Quantitative Reasoning with Autoformalization](https://mlsanity.com/paper/2403.18120/) (2024)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Decision-Making Behavior Evaluation Framework for LLMs under Uncertain Context](https://mlsanity.com/paper/2406.05972/) (2024)
- [Multilingual Mathematical Autoformalization](https://mlsanity.com/paper/2311.03755/) (2023)
- [MathCoder2: Better Math Reasoning from Continued Pretraining on Model-translated Mathematical Code](https://mlsanity.com/paper/2410.08196/) (2024)
- [Data-Efficient Learning of Natural Language to Linear Temporal Logic Translators for Robot Task Specification](https://mlsanity.com/paper/2303.08006/) (2023)
- [Decomposing the Enigma: Subgoal-based Demonstration Learning for Formal Theorem Proving](https://mlsanity.com/paper/2305.16366/) (2023)
- [Easy Problems That LLMs Get Wrong](https://mlsanity.com/paper/2405.19616/) (2024)
- [Autoformalizing Euclidean Geometry](https://mlsanity.com/paper/2405.17216/) (2024)
- [TRIGO: Benchmarking Formal Mathematical Proof Reduction for Generative Language Models](https://mlsanity.com/paper/2310.10180/) (2023)
- [How Numerical Precision Affects Arithmetical Reasoning Capabilities of LLMs](https://mlsanity.com/paper/2410.13857/) (2024)
- [Physics-informed neural networks for PDE-constrained optimization and control](https://mlsanity.com/paper/2205.03377/) (2022)
- [DiffPhyCon: A Generative Approach to Control Complex Physical Systems](https://mlsanity.com/paper/2407.06494/) (2024)

## Built on by (most cited first, 23 of 23)

- [CodePDE: An Inference Framework for LLM-driven PDE Solver Generation](https://mlsanity.com/paper/2505.08783/) (2025)
- [Autoformalization in the Era of Large Language Models: A Survey](https://mlsanity.com/paper/2505.23486/) (2025)
- [Enhancing Physics-Informed Neural Networks Through Feature Engineering](https://mlsanity.com/paper/2502.07209/) (2025)
- [Lean Finder: Semantic Search for Mathlib That Understands User Intents](https://mlsanity.com/paper/2510.15940/) (2025)
- [AutoNumerics: An Autonomous, PDE-Agnostic Multi-Agent Pipeline for Scientific Computing](https://mlsanity.com/paper/2602.17607/) (2026)
- [PDEAgent-Bench: A Multi-Metric, Multi-Library Benchmark for PDE Solver Generation](https://mlsanity.com/paper/2605.09636/) (2026)
- [Text-Trained LLMs Can Zero-Shot Extrapolate PDE Dynamics, Revealing a Three-Stage In-Context Learning Mechanism](https://mlsanity.com/paper/2509.06322/) (2025)
- [Towards a Common Framework for Autoformalization](https://mlsanity.com/paper/2509.09810/) (2025)
- [SciML Agents: Write the Solver, Not the Solution](https://mlsanity.com/paper/2509.09936/) (2025)
- [OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inference](https://mlsanity.com/paper/2602.01493/) (2026)
- [Self-Evolving Scientific Agent Designs Physically Reasoned White-Box Fluid Control](https://mlsanity.com/paper/2606.08405/) (2026)
- [Integrating Expert Knowledge into Logical Programs via LLMs](https://mlsanity.com/paper/2502.12275/) (2025)
- [PDE-SHARP: PDE Solver Hybrids through Analysis and Refinement Passes](https://mlsanity.com/paper/2511.00183/) (2025)
- [AutoPDE: Reliable Agentic PDE Solving via Explicitly Represented Solver Strategies](https://mlsanity.com/paper/2606.10752/) (2026)
- [Reinforcement Learning with Verifiable Physics: Post-training LLMs with Continuous Rewards](https://mlsanity.com/paper/2607.10474/) (2026)
- [TopoAlign: A Framework for Aligning Code to Math via Topological Decomposition](https://mlsanity.com/paper/2510.11944/) (2025)
- [Towards Autoformalization of LLM-generated Outputs for Requirement Verification](https://mlsanity.com/paper/2511.11829/) (2025)
- [Evaluating the Formal Reasoning Capabilities of Large Language Models through Chomsky Hierarchy](https://mlsanity.com/paper/2604.02709/) (2026)
- [Discovering Ordinary Differential Equations with LLM-Based Qualitative and Quantitative Evaluation](https://mlsanity.com/paper/2605.07323/) (2026)
- [MioFFAn: an Annotation Software for Formula Formalization with LLM Automation Capabilities](https://mlsanity.com/paper/2607.22552/) (2026)
- [Large language models for partial differential equation workflows](https://mlsanity.com/paper/2608.03600/) (2026)
- [DiffPDE: Masked Diffusion Language Models as PDE Solver](https://mlsanity.com/paper/2608.30532/) (2026)
- [GPUPhysBench: Benchmarking Coding Agents for Correct and Efficient GPU Physics Simulation](https://mlsanity.com/paper/2609.35639/) (2026)

## Abstract (arXiv)

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/.