# PiRD: Physics-informed Residual Diffusion for Flow Field Reconstruction

arXiv: https://arxiv.org/abs/2404.08412 · HTML: https://mlsanity.com/paper/2404.08412/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 19 of its 43 references that are in the map.
- First posted on arXiv 12 Apr 2024.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

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

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Image Super-Resolution Using Deep Convolutional Networks](https://mlsanity.com/paper/1501.00092/) (2014)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Activating More Pixels in Image Super-Resolution Transformer](https://mlsanity.com/paper/2205.04437/) (2022)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [ResShift: Efficient Diffusion Model for Image Super-resolution by Residual Shifting](https://mlsanity.com/paper/2307.12348/) (2023)
- [Learning Texture Transformer Network for Image Super-Resolution](https://mlsanity.com/paper/2006.04139/) (2020)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Blind Super-Resolution Kernel Estimation using an Internal-GAN](https://mlsanity.com/paper/1909.06581/) (2019)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [CT Super-resolution GAN Constrained by the Identical, Residual, and Cycle Learning Ensemble(GAN-CIRCLE)](https://mlsanity.com/paper/1808.04256/) (2018)
- [Encoding physics to learn reaction-diffusion processes](https://mlsanity.com/paper/2106.04781/) (2021)
- [Image Reconstruction: From Sparsity to Data-adaptive Methods and Machine Learning](https://mlsanity.com/paper/1904.02816/) (2019)
- [Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning](https://mlsanity.com/paper/2101.00554/) (2021)
- [BSRT: Improving Burst Super-Resolution with Swin Transformer and Flow-Guided Deformable Alignment](https://mlsanity.com/paper/2204.08332/) (2022)
- [Synthetic Lagrangian Turbulence by Generative Diffusion Models](https://mlsanity.com/paper/2307.08529/) (2023)
- [Deep Learning for Efficient Reconstruction of High-Resolution Turbulent DNS Data](https://mlsanity.com/paper/2010.11348/) (2020)

## Built on by (most cited first, 10 of 10)

- [Mesh-based Super-Resolution of Fluid Flows with Multiscale Graph Neural Networks](https://mlsanity.com/paper/2409.07769/) (2024)
- [Guiding diffusion models to reconstruct flow fields from sparse data](https://mlsanity.com/paper/2510.19971/) (2025)
- [RED-DiffEq: Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion](https://mlsanity.com/paper/2509.21659/) (2025)
- [Contrastive Diffusion Alignment: Learning Structured Latents for Controllable Generation](https://mlsanity.com/paper/2510.14190/) (2025)
- [Divergence-Free Diffusion Models for Incompressible Fluid Flows](https://mlsanity.com/paper/2601.19368/) (2026)
- [Decoupled Latent Optimization of Diffusion Models for Full Waveform Inversion](https://mlsanity.com/paper/2606.14139/) (2026)
- [Patch-Based 3D Variational Autoencoder for Super-Resolution of Turbulent Channel Flow](https://mlsanity.com/paper/2507.22082/) (2025)
- [Distributed physics-informed neural networks via domain decomposition for fast flow reconstruction](https://mlsanity.com/paper/2602.15883/) (2026)
- [Reconstructing Multi-Scale Physical Fields from Extremely Sparse Measurements with an Autoencoder-Diffusion Cascade](https://mlsanity.com/paper/2512.01572/) (2025)
- [Coupled Diffusion-Encoder Models for Reconstruction of Flow Fields](https://mlsanity.com/paper/2601.07946/) (2026)

## Abstract (arXiv)

The use of machine learning in fluid dynamics is becoming more common to expedite the computation when solving forward and inverse problems of partial differential equations. Yet, a notable challenge with existing convolutional neural network (CNN)-based methods for data fidelity enhancement is their reliance on specific low-fidelity data patterns and distributions during the training phase. In addition, the CNN-based method essentially treats the flow reconstruction task as a computer vision task that prioritizes the element-wise precision which lacks a physical and mathematical explanation. This dependence can dramatically affect the models' effectiveness in real-world scenarios, especially when the low-fidelity input deviates from the training data or contains noise not accounted for during training. The introduction of diffusion models in this context shows promise for improving performance and generalizability. Unlike direct mapping from a specific low-fidelity to a high-fidelity distribution, diffusion models learn to transition from any low-fidelity distribution towards a high-fidelity one. Our proposed model - Physics-informed Residual Diffusion, demonstrates the capability to elevate the quality of data from both standard low-fidelity inputs, to low-fidelity inputs with injected Gaussian noise, and randomly collected samples. By integrating physics-based insights into the objective function, it further refines the accuracy and the fidelity of the inferred high-quality data. Experimental results have shown that our approach can effectively reconstruct high-quality outcomes for two-dimensional turbulent flows from a range of low-fidelity input conditions without requiring retraining.