Advanced representation learning for flow field analysis and reconstruction

In the field of fluid dynamics, obtaining high-fidelity and high-resolution flow field data is crucial for understanding complex fluid behaviors, with applications spanning aerospace, energy, and biomedical engineering. However, experimental and numerical simulations often yield low-resolution data, which limits the ability to capture detailed flow structures, especially in turbulent flows. Traditional computational methods such as Navier-Stokes-based solvers face challenges in computational cost and scalability, especially for high-fidelity simulations [1, 2]. To address this challenge, researchers have explored various techniques to enhance the resolution of flow field data. Traditional methods, such as interpolation, often fail to accurately reconstruct the intricate details of flow fields. Recent advancements in deep learning offer promising solutions, particularly through the use of diffusion models and convolutional neural networks (CNNs) [3, 4] and present transformative opportunities to overcome these limitations [5, 6, 7].

This paper focuses on several interrelated topics based on flow field analysis and reconstruction:

Flow field super-resolution reconstruction. Super-resolution reconstruction involves enhancing the spatial resolution of flow data while preserving physical fidelity. Traditional interpolation methods often fail to capture intricate flow structures. Machine learning (ML) and deep learning (DL) models, including CNNs and generative adversarial networks (GANs) [8], have shown remarkable potential in this area [9].

Flow field inpainting. Flow field inpainting addresses the problem of missing or corrupted data. Techniques such as physics-informed neural networks (PINNs) [10] enable reconstruction while adhering to governing equations like continuity and Navier-Stokes laws.

Fluid-structure interaction (FSI). Coupled FSI systems require simultaneous resolution of solid deformation and fluid motion. DL techniques are being employed to accelerate iterative coupling processes and predict structural responses in real time.

Transient and internal flow analyses. They focus on studying time-dependent dynamic characteristics of fluid flow and internal behaviors. These analyses are widely applied in engineering and natural phenomena, aiming to enhance the understanding and prediction of complex flow dynamics.

Reduced-order modeling (ROM) and fast computing. Reduced-order models are essential for approximating high-dimensional systems in a computationally efficient manner. DL-based ROMs leverage latent space representations to approximate flow dynamics.

By leveraging ML and DL, we aim to enhance both predictive accuracy and computational efficiency in complex flow analyses.

In fluid dynamics, obtaining high-fidelity and high-resolution flow field data is crucial for understanding complex fluid behaviors (see Figure 1 for original flow data) [11, 12, 13]. Traditional methods like interpolation often fail to accurately reconstruct intricate details of flow fields. Deep learning, particularly diffusion models and CNNs, offers promising solutions. Super-resolution reconstruction in flow fields involves enhancing spatial resolution while preserving physical fidelity. DL models like CNNs and GANs have shown potential, but often lack physical constraints, leading to inaccuracies.

Convolutional Neural Networks (CNNs), known for their spatial feature extraction capabilities, are applied for high-resolution flow reconstructions and transient analyses. Diffusion models are generative techniques that progressively add noise to data and then learn to reverse the noise process. These models are particularly effective for stochastic flow field reconstructions. Physics-Informed Neural Networks (PINNs) integrate physical laws into the loss function, ensuring physically consistent predictions, even in sparse data environments.

Researchers use a multi-scale attention mechanism to capture global and local flow features in flow field super-resolution. The model is trained on flow datasets, focusing on maintaining turbulence characteristics. To reconstruct incomplete flow fields in flow field inpainting, one utilizes GANs conditioned on existing data. Physics constraints are incorporated to maintain continuity and energy conservation. Transient flow fields are analyzed using sequence-to-sequence models, such as recurrent neural networks (RNNs) and attention mechanisms, for time-dependent predictions.

Diffusion models, a type of generative model, have been adapted for flow field reconstruction by modeling the process of adding and removing noise in a flow field. Diffusion models gradually corrupt data with noise and then learn to reverse this process. The training involves a forward diffusion process and a reverse denoising process, using a parameterized Markov chain and a neural network to approximate the conditional distribution.

Image restoration, particularly image inpainting, is crucial in various fields [14]. Traditional methods rely on the low-rank property of images but often fail to preserve fine details and edges. This paper introduces the transformed sparsity-boosted low-rank (TSBLR) model [15], incorporating non-convex gamma-norm regularization and non-local prior to enhance image inpainting accuracy and quality (see Figure 6).

Matrix completion techniques are widely used for image restoration. The nuclear norm, as a convex relaxation of the rank function, is popular but limits flexibility. Non-convex rank approximations like the truncated nuclear norm (TNN) offer better control over target rank. The gamma-norm provides a more accurate approximation of the rank function than the nuclear norm. It is defined as a function of the singular values of the matrix, with a parameter gamma controlling the approximation behavior. Replacing the nuclear norm with the gamma-norm in the optimization problem leads to a more precise low-rank approximation.

Transform domain sparse regularization exploits the sparsity of natural images in transform domains like the discrete cosine transform (DCT). By adding a sparsity constraint, the model better reconstructs fine details and textures, resulting in higher quality inpainting. The non-local prior leverages the self-similarity of natural images, searching for similar patches in the neighborhood of the target block to guide the inpainting process. This approach is effective in restoring consistent textures and patterns across the image.

The optimization problem is solved using an alternating direction method of multipliers (ADMM) framework, decomposing the problem into a series of sub-problems. The ADMM algorithm iteratively updates the variables involved, ensuring convergence (see Algorithm 1).

The TSBLR model is evaluated on tasks like pixel inpainting, patch removal and noise removal. It outperforms many low-rank methods in both visual quality and quantitative evaluation. The TSBLR model restores images with sharper edges and more accurate textures, achieving higher PSNR and SSIM values [16]. It effectively removes loss and restores entire data with fewer artifacts (see Figure 7). In the ablation study removing the sparse regularization component leads to a significant decrease in restoration performance, highlighting the importance of the sparse prior in capturing local details and texture information. The model’s performance is sensitive to parameters like truncation, penalty, and regularization. Optimal values are determined through extensive experiments to ensure best performance in different scenarios.

The paper introduces advanced techniques for flow field analysis and reconstruction, with the flow diffusion model achieving superior super-resolution performance, even for complex turbulent flows, by effectively reconstructing detailed features and enabling real-time applications. Additionally, the transformed sparsity-boosted low-rank model significantly enhances flow field accuracy and quality, restoring images with sharper edges and accurate textures, outperforming traditional methods through its non-convex gamma-norm regularization and non-local prior, which effectively capture local details and minimize artifacts.