Source-linked AI summary
Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels
Han Gao, Luning Sun, Jian-Xun Wang
TL;DR
High-resolution fluid-flow information is difficult to access, while deep-learning super-resolution commonly depends on abundant high-resolution labels. This paper proposes a physics-informed CNN-SR model trained from low-resolution samples, achieving 400× refinement for new inlet boundary conditions in a 20-dimensional parameter space.
Problem
High-resolution fluid-flow information is difficult to access, and existing deep-learning super-resolution models mainly rely on large amounts of high-resolution training data.
Method
The physics-informed CNN-SR model leverages fluid physical laws and boundary conditions to refine low-resolution noisy velocity fields without requiring high-resolution counterparts.
Results
400× refinement was achieved from only 15 low-resolution input samples for flow fields with new inlet boundary conditions sampled in a 20-dimensional parameter space, with higher accuracy and robustness.
Takeaways & Limitations
The approach supports high-resolution flow-field super-resolution using low-resolution samples instead of high-resolution labels.
Takeaways & Limitations
Future work will extend the framework from 2D flow-field spatial super-resolution to spatial and temporal super-resolution of unsteady flows in 3D complex domains.
Abstract
from arXiv · showhide
High-resolution (HR) information of fluid flows, although preferable, is usually less accessible due to limited computational or experimental resources. In many cases, fluid data are generally sparse, incomplete, and possibly noisy. How to enhance spatial resolution and decrease the noise level of flow data is essential and practically useful. Deep learning (DL) techniques have been demonstrated to be effective for super-resolution (SR) tasks, which, however, primarily rely on sufficient HR labels for training. In this work, we present a novel physics-informed DL-based SR solution using convolutional neural networks (CNN), which is able to produce HR flow fields from low-resolution (LR) inputs in high-dimensional parameter space. By leveraging the conservation laws and boundary conditions of fluid flows, the CNN-SR model is trained without any HR labels. Moreover, the proposed CNN-SR solution unifies the forward SR and inverse data assimilation for the scenarios where the physics is partially known, e.g., unknown boundary conditions. Several flow SR problems relevant to cardiovascular applications have been studied to demonstrate the proposed method's effectiveness and merit. Both Gaussian and non-Gaussian MRI noises are investigated to illustrate the denoising capability.
1. Introduction
Fluid-flow data are often costly to obtain at high resolution and may be sparse, incomplete, or noisy, while existing super-resolution methods commonly depend on HR labels or computationally expensive models. The paper proposes physics-informed CNN super-resolution that uses fluid physics and boundary conditions to upscale and denoise LR data without HR training labels.
- Motivation: High-resolution flow information is important for qualitative and quantitative analyses, but measurements and simulations can be spatially limited, noisy, or computationally expensive.Flow MR data have insufficient spatial resolution and signal-to-noise ratio, while fully resolved DNS can require days or weeks and generate very large files.
- Motivation: Parameter-space super-resolution can provide a cost-effective surrogate by learning to upscale efficient LR simulations or experiments to target resolution.This approach is motivated by the difficulty of querying many boundaries, geometries, and operating configurations for uncertainty quantification and optimization.
- Prior approaches: Earlier fluid super-resolution methods use reduced-basis representations, CFD with data assimilation, or data-driven neural networks, but each has important constraints.Reduced-basis methods are limited by linearity assumptions, CFD-based assimilation can be computationally prohibitive, and DL approaches generally require HR labels.
- Research gap: Recent data-driven upsampling can add visual complexity without guaranteeing physical fidelity, motivating the incorporation of prior physics into deep learning.The paper identifies label scarcity and physical-law fidelity as central challenges, especially for irregular-domain and label-free super-resolution.
- Proposed framework: The proposed physics-informed CNN framework super-resolves and denoises LR noisy flow fields without HR labels by using known conservation laws and boundary conditions.It also supports simultaneous inference of unknown conditions, such as boundary conditions, when the governing physics is only partially known.
2. Methodology
The methodology uses a composite CNN to map noisy low-resolution velocity fields to high-resolution, noise-reduced flow solutions. Physics-informed losses, hard boundary enforcement, and sparse-data assimilation enable label-free training and simultaneous super-resolution and inference of unknown physical parameters.
- Overview: The CNN maps possibly noisy LR velocity fields on coarse meshes to HR noise-free flow fields on fine meshes.The HR field represents a higher-dimensional flow solution with more spatial detail.
- Learning architecture: The composite architecture uses separate convolutional sub-networks for individual solution fields, decoupling trainable parameters across state variables.Each decoder uses a three-hidden-layer CNN structure for single-image super-resolution.
- Physics-informed training: Physics-informed training learns the LR-to-HR mapping from governing equations, conservation laws, and boundary conditions instead of HR labels.PDE residuals are minimized, while boundary conditions are imposed hard through padding operations.
- Physics-informed training: The framework is not constrained by HR-data availability, allowing training-space exploration with many inexpensive LR samples.This addresses the high cost and limited accessibility of HR labels in conventional data-driven training.
- Sparse observation assimilation: The LR-data loss should be used only with noise-free LR data because noisy inputs can cause the network to overfit measurement noise.The framework also targets noisy cardiovascular flow measurements, including limited-resolution and low-SNR MRI data.
- Sparse observation assimilation: For partially known physics, sparse observations are assimilated to support forward super-resolution and simultaneous inference of unknown physical parameters.Unknowns such as boundary conditions are parameterized as trainable variables and observations are enforced at sparse locations.
3. Result
The physics-informed CNN-SR model denoises and super-resolves sparse or noisy low-resolution flow data using fluid equations and boundary conditions, including cases with unknown boundary conditions and parameterized inputs. Across cardiovascular flow examples, it matches high-resolution references, supports large refinement factors, and can serve as a surrogate for high-dimensional queries.
- Physics-informed super-resolution: The CNN-SR model uses conservation-law and boundary-condition information to super-resolve low-resolution flow fields without high-resolution labels.It is evaluated on internal flows relevant to cardiovascular applications, including deterministic, partially known, and parametric settings.
- Non-parametric super-resolution: 30× refinement maps a 126-cell vascular-flow mesh to a 3773-cell high-resolution mesh while retaining detailed boundary-layer and velocity-development structure.The deterministic case assumes known governing equations and boundary conditions.
- Known boundary condition: Relative errors are 0.067 for CNN-SR and 0.520 for bicubic-SR under 100% Gaussian noise, with CNN-SR substantially reducing the noise.The CNN-SR field agrees with the high-resolution reference, whereas bicubic interpolation becomes highly unphysical.
- Parametric super-resolution: In the parametric setting, CNN-SR refines inputs by 400×, generalizes to unseen inlet samples, and agrees well with high-resolution references despite noisy inputs.The trained model is described as a cost-effective surrogate for massive queries in high-dimensional input spaces, with potential use in ensemble-based uncertainty quantification.
4. Conclusion
The paper presents a physics-informed CNN-SR framework that learns flow super-resolution from LR data, supports unknown boundary-condition assimilation, and shows strong accuracy, robustness, and speed in 2D flow applications.
- 4. Conclusion: The CNN-SR model uses fluid physical laws and boundary conditions to train from LR samples without HR labels.It produces spatially refined flow fields from noisy LR inputs in parameter space.
- 4. Conclusion: When boundary conditions are unknown, the framework assimilates sparse observations to perform forward super-resolution and infer unknown boundary conditions.This extends the method to partially known physics.
- 4. Conclusion: Compared with bicubic interpolation, CNN-SR achieves significantly higher accuracy and robustness for the studied flow super-resolution tasks.The conclusion also covers cardiovascular applications with Gaussian and non-Gaussian MRI noise.
- 4. Conclusion: A single sample is more than 10^3 times faster than standard FV simulation, indicating potential for many-query applications.The reported speedup applies to the single-sample comparison.
- 4. Conclusion: The current framework is limited to spatial super-resolution of 2D flow fields; future work targets spatial and temporal super-resolution in 3D complex domains.This is the stated scope boundary and extension direction.
Appendix A. Convolution operators for gradient and Laplacian terms
The appendix documents convolution-filter representations used for gradient and Laplacian operators in the physics-informed CNN implementation.
- Appendix A. Convolution operators for gradient and Laplacian terms: Gradient and Laplacian convolution filters are stored as 4D tensors.The appendix displays the tensor entries associated with these operators.
- Appendix A. Convolution operators for gradient and Laplacian terms: The displayed operator tensors include rows containing the values 0 0 16 0 0.These entries are reproduced as shown in the supplied passage.
Appendix B. The KL modes for the spatially-varying inlets
The appendix presents the first 10 KL modes of Gaussian random fields used for spatially varying inlet conditions.
- Appendix B. The KL modes for the spatially-varying inlets: The first 10 KL modes represent Gaussian random fields for the spatially varying inlet construction.The supplied figure passage identifies these modes but does not specify their plotted axes or individual interpretations.
- Appendix B. The KL modes for the spatially-varying inlets: The broader study considers inlet boundary conditions as part of the flow setup.The supplied passage only identifies inlet boundary conditions without further construction details.
Appendix C. Physics-informed training history
The appendix reports training histories for nonparametric and parametric super-resolution cases and gives an iteration-time measurement for the training procedure.
- Appendix C. Physics-informed training history: Training histories are shown for both nonparametric and parametric super-resolution cases.The supplied passage identifies the figure’s scope but does not report a numerical convergence result.
- Appendix C. Physics-informed training history: Each training iteration takes about 2 seconds on an NVIDIA 2080 GPU.This is the stated hardware-specific timing.