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Convolutional neural networks for fluid flow analysis: toward effective metamodeling and low-dimensionalization
Masaki Morimoto, Kai Fukami, Kai Zhang, Aditya G. Nair, Koji Fukagata
TL;DR
Fluid-flow CNN design involves many parameter choices and still lacks clear guidelines. This paper tests metamodels and autoencoders while varying CNN design choices; its findings show that effects depend on the flow and operation, with scalar-input placement and dimensional extension especially consequential.
Problem
Fluid-flow CNN modeling requires parameter tuning and prior knowledge, yet users rely on trial and error because design guidelines are lacking.
Source: §1 · p. 2Method
The paper studies CNN-MLP metamodeling for unsteady laminar wakes and CNN autoencoders for turbulence, examining scalar-input placement and CNN operations.
Source: §1 · p. 3, §4 · pp. 17–19Results
Earlier scalar-input placement generally improves estimation; CNNs are less sensitive to dimensional reduction than extension, and zero padding gives reasonable accuracy while periodic padding performs best.
Source: §4 · pp. 18–19Takeaways & Limitations
CNN design choices, including scalar-input placement, should be selected with the target flow and problem setting in mind.
Source: §4 · pp. 18–19The conventional CNN studied cannot be applied directly to fluid-flow data on unstructured meshes.
Source: §4 · p. 19
Abstract
from arXiv · showhide
We focus on a convolutional neural network (CNN), which has recently been utilized for fluid flow analyses, from the perspective on the influence of various operations inside it by considering some canonical regression problems with fluid flow data. We consider two types of CNN-based fluid flow analyses; 1. CNN metamodeling and 2. CNN autoencoder. For the first type of CNN with additional scalar inputs, which is one of the common forms of CNN for fluid flow analysis, we investigate the influence of input placements in the CNN training pipeline. As an example, estimation of force coefficients of an inclined flat plate and two side-by-side cylinders in laminar flows is considered. We find that care should be taken for the placement of additional scalar inputs depending on the problem setting and the complexity of flows that users handle. We then discuss the influence of various parameters and operations on the CNN performance, with the utilization of autoencoder (AE). A two-dimensional turbulence is considered for the demonstration of AE. The results of AE highly rely on the decaying nature. Investigation on the influence of padding operation at a convolutional layer is also performed. The zero padding shows reasonable ability compared to other methods which account for the boundary conditions assumed in the numerical data. Moreover, the effect of the dimensional reduction/extension methods inside CNN is also examined. The CNN model is robust against the difference in dimension reduction operations, while it is sensitive to the dimensional extension methods. The findings of this paper will help us better design a CNN architecture for practical fluid flow analysis
1 Introduction
The introduction frames fluid-flow CNN design as a trial-and-error problem without clear guidelines, then presents a study of CNN metamodeling and autoencoders to examine design choices.
1 Introduction
- Fluid-flow CNN design still relies on trial and error because tuning many parameters and incorporating prior knowledge lack clear guidelines.The paper frames its question around how to choose strategies for better CNN performance in fluid-flow analysis.Source: §1 · pp. 2–3
- Prior fluid-flow CNN work spans turbulence modeling [2] [6], LES and RANS applications [7], and closure modeling.The introduction also describes CNN use in fluid-data reconstruction [12] and autoencoder-based low-dimensionalization [20].Source: §1 · pp. 2–3
- The design question includes how to select CNN strategies such as multi-size filters and additional scalar inputs tied to flow phenomena.The introduction cites multi-size filters for representing a wide range of turbulence scales and scalar inputs such as angle of attack and Reynolds number.Source: §1 · p. 3
- The study examines CNN-MLP metamodels for unsteady laminar wakes and CNN autoencoders for turbulence to investigate CNN design choices.Both are framed as regression tasks, y = F(x, φ), with additional scalar inputs φ.Source: §1 · p. 3
2 Methods
The study examines CNN-MLP metamodeling and CNN-AE for fluid flows, focusing on CNN operations and scalar-input placement. Its cases use flat-plate and side-by-side-cylinder wakes and decaying two-dimensional turbulence.
2.1.1 Basic principle of CNN
- CNN operations include convolution for feature extraction, pooling for downsampling, and upsampling or transposed convolution for dimensional expansion.The study focuses on how downsampling choices and dimensional-expansion techniques influence CNN ability.Source: §2.1.1 · pp. 3–5
2.1.2 Covered CNN models
- CNN-MLP maps flow fields to scalar coefficients and tests four scalar-input placements; CNN-AE reconstructs turbulence, with scalar inputs placed at four network depths.All networks use three-fold cross-validation; CNN-MLP tests use configurations excluded from training, while CNN-AE tests come from the training range.Source: §2.1.2 · pp. 5–6
2.2.1 Flat plate wake
- Flat-plate wake cases use DNS at two Reynolds numbers and two angles of attack; greater angle and Reynolds number accompany more complex vortices, while drag rises and shedding frequency falls with angle.The dataset contains 1,200 snapshots for each flow case.Source: §2.2.1 · p. 7, Figure 5 · p. 7
2.2.2 Wake of two side-by-side cylinders
- Side-by-side-cylinder wakes vary size ratio and gap ratio; low gaps yield generally chaotic wakes, while larger gaps restore order with synchronization or quasi-periodicity depending on cylinder size ratio.For r = 1.15, the streets synchronize at g = 1.5, whereas g = 2.0 and 2.5 are quasi-periodic.Source: §2.2.2 · pp. 7–9, Figure 6 · p. 8
2.2.3 Decaying homogeneous isotropic turbulence
- The turbulence AE uses vorticity fields from decaying isotropic homogeneous turbulence at three initial Reynolds numbers, with the instantaneous Taylor Reynolds number as an additional scalar input.The vorticity data are interpolated to a common 256² grid for AE input, and 1,000 snapshots per initial Reynolds number are used.Source: Figure 4 · p. 6, §2.2.3 · p. 9, Figure 7 · p. 9
3 Results and Discussion
The results show that scalar-input placement matters differently across flow problems, while CNN-AE performance depends on the decay characteristics of the turbulence data and on architectural choices. Zero padding performs comparably to periodic padding in the tested setting, and dimensional expansion is more sensitive than reduction methods.
3.1 Example 1: scalar input-aided convolutional neural networks
- Upstream scalar inputs generally improve flat-plate coefficient estimates, while placement has less influence for the two-cylinder flows, making the preferred placement problem-dependent.For the flat plate, higher Reynolds number and angle of attack are less accurate; for the cylinders, errors are higher at g < 1.0, corresponding to chaotic flows.Source: Figure 8 · p. 10, §3.1 · p. 11, Figure 10 · p. 12
- With noisy cylinder-flow inputs, first-layer scalar placement gives the lowest error, although placements show no clear difference on clean inputs.The authors recommend considering both clean-data accuracy and noise robustness when choosing placement.Source: §3.1 · pp. 11–12, Figure 11 · p. 12
3.2.1 Size of the filter for the convolutional operation
- Larger filters lower reconstruction error, while errors track snapshot similarity: distinct early decay fields reconstruct less well than later fields with common vortex structures.The study uses two-dimensional decaying homogeneous isotropic turbulence with biperiodic boundaries; sampling more snapshots near the beginning and end may reduce errors for less-similar fields.Source: §3.2.1 · pp. 13–14, Figure 12 · p. 13
3.2.2 Compression rate of CNN-AE
- Below 10^5 weights, reconstruction error decreases as weight count increases; above 2 × 10^5 weights, the trend varies, consistent with possible optimization instability.The study adjusts weight counts by changing filter size and recommends considering weight count alongside latent-mode count.Source: §3.2.2 · p. 14, Figure 13 · p. 14
3.2.3 Padding operation for the convolutional operation
- Zero padding matches periodic padding in overall error, whereas replication padding becomes unstable at H = 11 and has the largest errors near boundaries.Zero-padding error rises at the boundary but approaches the periodic-padding level for db > 0.05; the best padding choice may depend on flow boundary conditions.Source: §3.2.3 · pp. 15–16, Figure 14 · p. 15
3.2.4 Input placements of supplemental scalar value
- For CNN-AE, adding supplemental scalars improves reconstruction; first-layer placement performs best and latent-space placement worst among the tested placements.The authors suggest that sufficient weights are needed before merging scalar inputs with the CNN-AE.Source: §3.2.4 · p. 16, Figure 15 · p. 16
3.2.5 Dimensional reducing/expanding methods
- Max and average pooling give similar errors, but transposed convolution becomes unstable for H ≥ 9; up-sampling error decreases as filter size grows.Large-filter transposed-convolution models show checkerboard artifacts, while transposed convolution generally performs better with fewer weights at H ≤ 7.Source: §3.2.5 · pp. 16–17, Figure 16 · p. 17
4 Conclusions
The conclusions assess CNN metamodeling with scalar inputs and autoencoder-based low-dimensionalization, identifying architecture choices that affect performance. The findings are bounded by the studied settings, and the authors identify unstructured meshes and practical interpretability as remaining issues.
4 Conclusions
- The conclusions draw on force-coefficient metamodeling for an inclined flat plate and two side-by-side cylinders, and on CNN-AE low-dimensionalization of fluid flows.The metamodel estimates drag and lift coefficients from vorticity fields; the AE investigation considers two-dimensional decaying turbulence.Source: §4 · p. 17
- Earlier-layer scalar inputs generally improve force-coefficient estimates and robustness, especially with less training data or noisy inputs.The authors suggest that merging supplemental information with biases and nonlinear activations may explain the robustness trend.Source: §4 · p. 18
- For CNN-AE, larger models can estimate better, but too many weights can destabilize training; periodic padding performs best for the studied bi-periodic turbulence, while zero padding remains reasonable.The authors advise choosing padding according to the flow-data boundary conditions.Source: §4 · p. 18
- CNN-AE also benefits more from scalar inputs at earlier layers, is less sensitive to dimensional-reduction methods, and is sensitive to dimensional-extension choices.The authors note that transposed convolution uses more weights than up-sampling and may encounter the curse of dimensionality.Source: §4 · p. 19
- The conventional CNN used in the study cannot be applied directly to unstructured-mesh data, which the authors identify as a remaining issue for practical fluid-flow analysis.They also point to interpretability and generalization to test situations unlike the training regime as continuing research directions.Source: §4 · p. 19
Appendix A: Details of neural networks
The appendix describes the network structures used for force-coefficient estimation and two-dimensional decaying-turbulence CNN-AE. Scalar-input branches expand supplemental values to match the target layer before concatenation.
Appendix A: Details of neural networks
- Tables 1 and 2 describe the CNN-MLP model structure for force-coefficient estimation, with the main stream on the left and an MLP-CNN branch on the right that expands supplemental scalars into two-dimensional data for concatenation.Source: Appendix A · p. 19
- In the CNN-MLP models, an MLP-CNN branch expands supplemental scalar values into two-dimensional data sized for concatenation with the mainstream.The branch’s last layer is concatenated with the main stream network.Source: Appendix A · p. 19
- Table 3 gives the CNN-AE structure for two-dimensional decaying turbulence, where supplemental Reynolds numbers are expanded and concatenated at four placements.The Reynolds-number branch is the MLP-CNN part shown in the table’s right half.Source: Appendix A · p. 19
Appendix B: Detailed observation for the robustness of the CNN-MLP model against noisy input
The appendix reports a robustness analysis of CNN-MLP predictions under noisy inputs, including an aggregate error view and individual flow-configuration results. It also supplies network-structure tables for the model configurations used in the study.
Appendix B: Detailed observation for the robustness of the CNN-MLP model against noisy input
- Tables 1 and 2 describe the CNN-MLP structures for flat-plate and side-by-side-cylinder force-coefficient estimation, while Table 3 describes the turbulence CNN-AE.The table captions identify each model’s application and distinguish the mainstream from the scalar-expansion branch.Source: Table 1 · p. 20, Table 2 · p. 20, Table 3 · p. 21
- Earlier-layer scalar insertion generally gives a better solution in the noisy-input robustness results, including the individual configurations and coefficients.Figure 11 summarizes ensemble L2 error norms averaged over both cylinders’ coefficients across distance factors; Figures 18–20 show the configurations and coefficients separately.Source: Appendix B · p. 21
- The aggregate robustness view averages the L2 error norm over both cylinders’ coefficients across the considered distance factors.The appendix also presents results individually by flow configuration and coefficient.Source: Appendix B · p. 21