# DiffPDE: Masked Diffusion Language Models as PDE Solver

arXiv: https://arxiv.org/abs/2608.30532 · HTML: https://mlsanity.com/paper/2608.30532/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Discrete and text diffusion, one of 14 directions in Diffusion models.
- Builds on 19 of its 34 references that are in the map.
- First posted on arXiv 31 Aug 2026.

## Placed in

- Diffusion models / [Discrete and text diffusion](https://mlsanity.com/diffusion/discrete-and-text-diffusion/)

## Builds on (most cited first, 19 of 19)

- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Qwen2.5-Coder Technical Report](https://mlsanity.com/paper/2409.12186/) (2024)
- [Teaching Large Language Models to Self-Debug](https://mlsanity.com/paper/2304.05128/) (2023)
- [Fast-dLLM: Training-free Acceleration of Diffusion LLM by Enabling KV Cache and Parallel Decoding](https://mlsanity.com/paper/2505.22618/) (2025)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [AIDE: AI-Driven Exploration in the Space of Code](https://mlsanity.com/paper/2502.13138/) (2025)
- [LLM and Simulation as Bilevel Optimizers: A New Paradigm to Advance Physical Scientific Discovery](https://mlsanity.com/paper/2405.09783/) (2024)
- [Dream-Coder 7B: An Open Diffusion Language Model for Code](https://mlsanity.com/paper/2509.01142/) (2025)
- [CodePDE: An Inference Framework for LLM-driven PDE Solver Generation](https://mlsanity.com/paper/2505.08783/) (2025)
- [Cross-Modal Fine-Tuning: Align then Refine](https://mlsanity.com/paper/2302.05738/) (2023)
- [On the Benefits of Memory for Modeling Time-Dependent PDEs](https://mlsanity.com/paper/2409.02313/) (2024)
- [UPS: Efficiently Building Foundation Models for PDE Solving via Cross-Modal Adaptation](https://mlsanity.com/paper/2403.07187/) (2024)
- [PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs](https://mlsanity.com/paper/2502.00963/) (2025)
- [Evaluating LLMs' Mathematical and Coding Competency through Ontology-guided Interventions](https://mlsanity.com/paper/2401.09395/) (2024)
- [Exploring the Potential of Diffusion Large Language Models in Code Generation](https://mlsanity.com/paper/2509.11252/) (2025)
- [FrontierCO: Real-World and Large-Scale Evaluation of Machine Learning Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2505.16952/) (2025)
- [Component Fourier Neural Operator for Singularly Perturbed Differential Equations](https://mlsanity.com/paper/2409.04779/) (2024)

## Abstract (arXiv)

Existing approaches for synthesizing Partial Differential Equation (PDE) solvers predominantly rely on autoregressive models, yet their global left-to-right decoding incurs substantial redundancy when addressing inherently localized bugs. In this work, we challenge this inefficient paradigm and propose DiffPDE, a framework leveraging discrete diffusion language models for targeted code repair. By introducing a localized re-masking and infilling strategy, DiffPDE regenerates only erroneous regions while preserving correct context, naturally aligning generation with the sparse nature of PDE errors. Furthermore, to handle coupled bugs requiring sequential interventions, we present Iterative Debugging GRPO (ID-GRPO), a reinforcement learning scheme that enables multi-round debugging within single trajectories via intermediate rewards. Experiments on PDEBench show that DiffPDE achieves competitive accuracy, outperforms same-scale AR models, and significantly accelerates repair.