mlsanityData to 14 Oct 2026

PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs

arXiv 2502.00963 ↗First posted 3 Feb 2025cs.LGcited by 23

On the map: LLM reasoning → Mathematical reasoning

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/.

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.

Builds on

37 of 68 references.

Closest ancestors · 10

Same direction, newest first

  1. 2024FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI7 Nov
  2. 2024How Numerical Precision Affects Arithmetical Reasoning Capabilities of LLMs17 Oct
  3. 2024MathCoder2: Better Math Reasoning from Continued Pretraining on Model-translated Mathematical Code10 Oct
  4. 2024GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models7 Oct
  5. 2024Iterative Reasoning Preference Optimization30 Apr
  6. 2024Don't Trust: Verify -- Grounding LLM Quantitative Reasoning with Autoformalization26 Mar
  7. 2024Large Language Models for Mathematical Reasoning: Progresses and Challenges31 Jan
  8. 2023TRIGO: Benchmarking Formal Mathematical Proof Reduction for Generative Language Models16 Oct
  9. 2022A Survey of Deep Learning for Mathematical Reasoning20 Dec
  10. 2022Lila: A Unified Benchmark for Mathematical Reasoning31 Oct

Other references · 27

  1. 2023GPT-4 Technical Reportcited by 18,396
  2. 2021LoRA: Low-Rank Adaptation of Large Language Modelscited by 14,380
  3. 2021Training Verifiers to Solve Math Word Problemscited by 10,109
  4. 2021Evaluating Large Language Models Trained on Codecited by 8,320
  5. 2021Measuring Mathematical Problem Solving With the MATH Datasetcited by 5,371
  6. 2020The Pile: An 800GB Dataset of Diverse Text for Language Modelingcited by 2,411
  7. 2023Robots That Ask For Help: Uncertainty Alignment for Large Language Model Plannerscited by 308
  8. 2024DeepSeek-Prover: Advancing Theorem Proving in LLMs through Large-Scale Synthetic Datacited by 234
  9. 2024RedPajama: an Open Dataset for Training Large Language Modelscited by 182
  10. 2020Learning to Control PDEs with Differentiable Physicscited by 132
  11. 2024Large Language Models for Robotics: Opportunities, Challenges, and Perspectivescited by 89
  12. 2024A Survey on Deep Learning for Theorem Provingcited by 57
15 more
  1. 2021Optimal control of PDEs using physics-informed neural networkscited by 52
  2. 2024Decision-Making Behavior Evaluation Framework for LLMs under Uncertain Contextcited by 41
  3. 2023Multilingual Mathematical Autoformalizationcited by 39
  4. 2023Data-Efficient Learning of Natural Language to Linear Temporal Logic Translators for Robot Task Specificationcited by 38
  5. 2023Decomposing the Enigma: Subgoal-based Demonstration Learning for Formal Theorem Provingcited by 37
  6. 2024Easy Problems That LLMs Get Wrongcited by 30
  7. 2024Autoformalizing Euclidean Geometrycited by 27
  8. 2022Physics-informed neural networks for PDE-constrained optimization and controlcited by 19
  9. 2024DiffPhyCon: A Generative Approach to Control Complex Physical Systemscited by 17
  10. 2022Fast and Precise: Adjusting Planning Horizon with Adaptive Subgoal Searchcited by 12
  11. 2024Explain Like I'm Five: Using LLMs to Improve PDE Surrogate Models with Textcited by 7
  12. 2024VernaCopter: Disambiguated Natural-Language-Driven Robot via Formal Specificationscited by 4
  13. 2024CLMASP: Coupling Large Language Models with Answer Set Programming for Robotic Task Planningcited by 3
  14. 2023Actor-Critic Methods using Physics-Informed Neural Networks: Control of a 1D PDE Model for Fluid-Cooled Battery Packscited by 2
  15. 2024Closed-form Solutions: A New Perspective on Solving Differential Equationscited by 2

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 11; 2026: 12.

23 papers.

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  1. 2025CodePDE: An Inference Framework for LLM-driven PDE Solver Generationcited by 39
  2. 2025Autoformalization in the Era of Large Language Models: A Surveycited by 21
  3. 2025Enhancing Physics-Informed Neural Networks Through Feature Engineeringcited by 8
  4. 2025Lean Finder: Semantic Search for Mathlib That Understands User Intentscited by 8
  5. 2026AutoNumerics: An Autonomous, PDE-Agnostic Multi-Agent Pipeline for Scientific Computingcited by 8
  6. 2026PDEAgent-Bench: A Multi-Metric, Multi-Library Benchmark for PDE Solver Generationcited by 8
  7. 2025Text-Trained LLMs Can Zero-Shot Extrapolate PDE Dynamics, Revealing a Three-Stage In-Context Learning Mechanismcited by 7
  8. 2025Towards a Common Framework for Autoformalizationcited by 7
  9. 2025SciML Agents: Write the Solver, Not the Solutioncited by 6
  10. 2026OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inferencecited by 5
  11. 2026Self-Evolving Scientific Agent Designs Physically Reasoned White-Box Fluid Controlcited by 4
  12. 2025Integrating Expert Knowledge into Logical Programs via LLMscited by 3
11 more
  1. 2025PDE-SHARP: PDE Solver Hybrids through Analysis and Refinement Passescited by 3
  2. 2026AutoPDE: Reliable Agentic PDE Solving via Explicitly Represented Solver Strategiescited by 3
  3. 2026Reinforcement Learning with Verifiable Physics: Post-training LLMs with Continuous Rewardscited by 2
  4. 2025TopoAlign: A Framework for Aligning Code to Math via Topological Decompositioncited by 0
  5. 2025Towards Autoformalization of LLM-generated Outputs for Requirement Verificationcited by 0
  6. 2026Evaluating the Formal Reasoning Capabilities of Large Language Models through Chomsky Hierarchycited by 0
  7. 2026Discovering Ordinary Differential Equations with LLM-Based Qualitative and Quantitative Evaluationcited by 0
  8. 2026MioFFAn: an Annotation Software for Formula Formalization with LLM Automation Capabilitiescited by 0
  9. 2026Large language models for partial differential equation workflowscited by 0
  10. 2026DiffPDE: Masked Diffusion Language Models as PDE Solvercited by 0
  11. 2026GPUPhysBench: Benchmarking Coding Agents for Correct and Efficient GPU Physics Simulationcited by 0

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  1. 2025Automated Code Development for PDE Solvers Using Large Language Models
  2. 2025PDE-Agent: A toolchain-augmented multi-agent framework for PDE solving
  3. 2026AutoPDEBench: Benchmarking LLM Auto-Research for Neural PDE Solver Design
  4. 2024LLMs for Mathematical Modeling: Towards Bridging the Gap between Natural and Mathematical Languages
  5. 2025Towards Reasoning for PDE Foundation Models: A Reward-Model-Driven Inference-Time-Scaling Algorithm
  6. 2024Capabilities of Large Language Models in Control Engineering: A Benchmark Study on GPT-4, Claude 3 Opus, and Gemini 1.0 Ultra
  7. 2025A Survey on Mathematical Reasoning and Optimization with Large Language Models
  8. 2026Mathematical Reasoning in Large Language Models: Benchmarks, Architectures, Evaluation, and Open Challenges