Inpainting Computational Fluid Dynamics with Deep Learning

TL;DR

Complete: The paper tackles the ill-posed problem of reconstructing full fluid-flow fields from partial observations by introducing a two-stage vector-quantized autoencoder (VQ-VAE). The method first learns a discrete latent representation of full 2D turbulent data and then fine-tunes for masked-region completion with a fixed decoder, guided by $\mathcal{L}_{\text{VQ}}$, $\mathcal{L}_{\text{percept}}$, and $\mathcal{L}_{\text{GAN}}$, evaluated on $256\times256$ Kolmogorov flow at $Re=1000$. On mask configurations ranging from a single central patch to many small patches, the approach consistently outperforms Fourier Neural Operator and FactFormer in both point-wise $L^2$ accuracy and preservation of the energy spectrum and vorticity distribution, demonstrating stable and accurate data completion under partial observations. The work provides a practical framework to reduce sensor count and enable coarser CFD meshes, while also highlighting the limits imposed by turbulence data ill-posedness and suggesting avenues like progressive inpainting and latent-space optimization for further improvements.

Abstract

Fluid data completion is a research problem with high potential benefit for both experimental and computational fluid dynamics. An effective fluid data completion method reduces the required number of sensors in a fluid dynamics experiment, and allows a coarser and more adaptive mesh for a Computational Fluid Dynamics (CFD) simulation. However, the ill-posed nature of the fluid data completion problem makes it prohibitively difficult to obtain a theoretical solution and presents high numerical uncertainty and instability for a data-driven approach (e.g., a neural network model). To address these challenges, we leverage recent advancements in computer vision, employing the vector quantization technique to map both complete and incomplete fluid data spaces onto discrete-valued lower-dimensional representations via a two-stage learning procedure. We demonstrated the effectiveness of our approach on Kolmogorov flow data (Reynolds number: 1000) occluded by masks of different size and arrangement. Experimental results show that our proposed model consistently outperforms benchmark models under different occlusion settings in terms of point-wise reconstruction accuracy as well as turbulent energy spectrum and vorticity distribution.

Figures (9)

• Figure 1: An overview of our two-stage method for data completion. Our model (denoted as $f_{\Theta}$) is trained for data construction in the first stage to learn a latent representation of the complete data in a latent space the discretized by vector quantization (as shown in the lower portion of the figure). The model is then fine-tuned for data completion task in the second stage, with a post-quantization convolution module applied to the prediction in the discrete latent space to reduce the artifact caused by quantization. In comparison, a model without a VQ module (denoted as $g_{\Theta}$) directly learns to predict complete data from incomplete input in some continuous latent space (as shown in the upper portion of the figure).
• Figure 2: Model architecture and training procedure of VQ-VAE for data completion. The major components of a VQ-VAE model includes an encoder, a decoder, a vector quantization module and the codebook associated with it, and a discriminator to implement GAN loss. The decoder module is trainable during Stage 1 and frozen during Stage 2 such that the data completion is eventually performed in the VQ space.
• Figure 3: Examples of 2D turbulent flow data considered in our data completion problem, where the first three examples from left to right are incomplete data samples with masked out regions used as model input and the right-most example is a complete data sample used as the ground truth reference.
• Figure 4: Data completion samples from 1-mask experiment by proposed model.
• Figure 5: Data completion samples from 4-mask experiment by proposed model.