PiRD: Physics-informed Residual Diffusion for Flow Field Reconstruction

TL;DR

PiRD addresses reconstructing high-fidelity flow fields $oldsymbol{\omega}$ from sparse, noisy low-fidelity data $\tilde{\boldsymbol{\omega}}$ by integrating a physics-informed Residual Shifting Diffusion Model with physics-informed neural networks to enforce the vorticity transport PDE. It learns a Markov chain between LF and HF distributions while simultaneously enforcing the governing equation via a PDE loss, using a tri-channel input to estimate time dynamics and a diffusion-based sampling strategy. On 2D Kolmogorov flows, PiRD achieves lower Mean Relative Error and PDE residual than CNN-based methods and diffusion baselines, while preserving kinetic-energy spectra and vorticity distributions, even under noise or unseenLF patterns. With ~20 sampling steps, PiRD offers improved efficiency and robustness, suggesting potential for real-time flow-field reconstruction.

Abstract

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.

Figures (11)

• Figure 1: The architecture of PiRD. During the forward process, a time-step $t$ is randomly drawn from $(0,T]$, then a UNet is used for predicting $\omega$ from $\omega_{t}$ with the objective function that both penalize the element-wise difference but also the physics loss. During the inference process, the LF field $\Tilde{\omega}$ is first mapped to $\omega_{T}$ then followed by the reverse Markov chain and finally predicted HF field $\omega$.
• Figure 2: Downsampling methods for the experiments. (a) illustrates an evenly down-sampling operation. The original data is initially downsampled evenly, followed by nearest interpolation to upsample it to match the desired input dimension of the model. (b) depicts a random selection down-sampling operation. A certain percentage of data points are randomly selected from the original dataset, after which the tesselation method is employed on the sparse dataset to achieve the desired dimension. Gaussian noise is introduced to the downsampled dataset during the noisy flow field reconstruction experiments.
• Figure 3: Flow field reconstruction from four different low-fidelity inputs (i.e., $4\times$ downsampling, $8\times$ downsampling, $5\%$ random selection, and $1.5625\%$ random selection). A zoomed-in comparison among PiRD, DDPM-based, and UNet methods for three testing cases.
• Figure 4: Comparison of PiRD's performance on (a) kinetic energy spectrum and (b) vorticity distribution.
• Figure 5: PiRD and UNet's performances on MRE and PDE under a progressive density of injected Gaussian noise. (a and b) illustrate the comparisons of MRE and PDE loss for PiRD and UNet on the noisy 4x reconstruction task. (c and d) illustrate the comparisons of MRE and PDE loss for PiRD and UNet on the noisy 5% reconstruction task.