Towards extraction of orthogonal and parsimonious non-linear modes from turbulent flows

1. Introduction

• Modal decomposition provides low-dimensional coordinates for dominant flow features, supporting reduced-order modeling, flow analysis, and flow control.Source: §1 · p. 2
• AE-based modes are neither orthogonal nor ranked, which may limit their interpretability and robustness despite their reconstruction advantages over POD.Hierarchical CNN-AEs rank modes by their contribution to reconstruction, but interpretability and non-uniqueness issues remain.Source: §1 · p. 3
• The proposed CNN-βVAE penalizes latent-variable correlation and latent-vector size to learn a minimal set of near-orthogonal nonlinear modes from turbulent flows.The method is applied to high-fidelity flow data from well-resolved LES of a simplified urban environment and is compared with CNN-AE, hierarchical CNN-AE, and POD.Source: §1 · p. 3
• Urban-flow prediction and physical understanding matter for urban planning, air quality management, and pollutant dispersion.Source: §1 · p. 4

2. Modal decomposition

• Modal decomposition extracts energetically and dynamically important flow features as modes, providing low-dimensional coordinates for analysis, reduced-order modeling, and flow control.Source: §2 · p. 4

2.1. Training data

• The training data are 1,200 interpolated instantaneous streamwise-velocity fluctuation fields from a well-resolved LES of flow around two wall-mounted obstacles.The input fields are x–z cross-sections at y = 0.5h over the specified region around the obstacles, interpolated to a 96 × 192 grid.Source: §2.1 · p. 5

2.2. Proper-orthogonal decomposition (POD)

• POD is optimal among linear decompositions for a given mode count and yields uncorrelated coefficients, but its optimality applies only relative to other linear representations.Source: §2.2 · p. 6
• 247 POD modes are required to capture 99% of the energy in the urban-flow data.The figure plots eigenvalues and cumulative eigenvalue spectrum; its red line marks the mode count required for 99% energy.Source: §2.2 · p. 7, Figure 1 · p. 7

3.1. CNN-based autoencoders

• The CNN-AE maps input data to a low-dimensional latent vector and decodes it back, learning features needed to minimize reconstruction loss.The input here is the fluctuating streamwise velocity component; nonlinear activation functions allow nonlinear mappings.Source: §3 · p. 7

3. CNN-based autoencoders for modal decomposition

• CNN layers process multiscale coherent features; kernels reduce learned parameters, pooling compresses data, and upsampling expands it.Training uses mean-squared reconstruction error, Adam, early stopping, and a validation set comprising 20% of the data.Source: §3 · pp. 7–8

3.1. CNN-based autoencoders

• The CNN-AE uses five convolution-and-pooling steps before flattening features into a size-d latent vector, then reconstructs through upsampling and convolution.The input/output may be streamwise velocity alone or all three velocity components.Source: §3.1 · pp. 8–9, Figure 2 · p. 9

3.2. CNN-based hierarchical autoencoders

• The CNN-HAE trains subnetworks successively, each adding a latent variable to the previously learned variables for reconstruction.The first subnetwork uses d = 1; subsequent subnetworks reconstruct from [r1 r2], then [r1 r2 r3], and so on.Source: §3.2 · p. 9, Figure 3 · p. 10

3.3. CNN-based β-variational autoencoders

• VAEs map data to a latent distribution from which new meaningful samples can be generated.The section presents a modified VAE for turbulent-flow modal decomposition.Source: §3.3 · p. 9

3.3.1. Marginal likelihood

• The marginal likelihood represents the data distribution approximation parameterized by θ, but its integral is typically computationally infeasible.The described maximum-likelihood objective uses a latent prior distribution.Source: §3.3.1 · p. 10

3.3.2. Evidence lower bound (ELBO)

• The ELBO is a tractable lower bound on marginal likelihood and is used as the neural-network training cost.Its reconstruction log-likelihood term encourages accurate reconstruction.Source: §3.3.2 · p. 11

3.3.4. Disentangled representation

• The CNN-βVAE aims to minimize latent-variable correlation and latent-vector size, motivating near-orthogonal modes in a compact representation.Its decoder is the same as the CNN-AE decoder.Source: §3.3.4 · p. 12

4. Results and discussion

• The KL regularizer and β constrain latent information capacity, motivating independent variables and near-orthogonal, parsimonious modes; the evaluation tracks reconstruction Ek and independence detR.detR is 100 for completely uncorrelated variables and zero for completely correlated variables.Source: §4 · pp. 12–13, §4.1 · p. 13

4.1. Learning orthogonal and parsimonious modes

• With β = 2.5 × 10^-3, only 3 latents are active (detR = 99.84; Ek = 82.17%); reducing β increases reconstruction accuracy while reducing independence.At β = 10^-4, all 10 latents are significant and detR = 94.55, while Ek = 92.78%.Source: §4.1 · p. 14, Figure 5 · p. 14
• At β = 10^-3, increasing the specified latent-vector size from 5 to 10 or 20 still yields only 5 significant modes and does not improve performance.For d = 5, 10, and 20, respectively, detR is 99.20, 98.38, and 99.22, while Ek is 87.36%, 86.93%, and 87.15%.Source: §4.1 · pp. 14–15

4.2. Ranking the CNN-βVAE modes

• The proposed ranking selects modes sequentially by their maximum reconstruction Ek, adding each chosen mode to those already selected.The procedure encodes one latent variable at a time, decodes it, and evaluates reconstruction energy before selecting the next mode.Source: §4.2 · pp. 15–16
• The first two ranked CNN-βVAE modes resemble POD modes and represent large-scale vortex shedding from the obstacles into the wake.The comparison shows ranked modes for CNN-βVAE, POD, CNN-AE, and CNN-HAE with d = 5.Source: Figure 6 · p. 15, §4.2 · p. 16

4.3. Comparison of CNN-AE-based models

• CNN-AE achieves the best reconstruction (Ek = 94.22%), while 5 POD modes capture 32.41%; all three CNN-AE models reach about 90% Ek using 5 modes.CNN-HAE and CNN-βVAE obtain Ek of 91.84% and 87.36%, respectively.Source: Figure 7 · p. 16, §4.3 · p. 17
• In the zoomed region between obstacles, all three CNN-AE models preserve dominant turbulent structures despite losing some small-scale features; 5-mode POD does not reconstruct the flow properly.Figure 8 shows the reconstruction comparison in the area marked between the obstacles.Source: Figure 8 · p. 17, §4.3 · p. 18
• CNN-βVAE latent variables are less correlated than CNN-AE variables (detR 99.20 versus 87.59), while CNN-HAE has the lowest detR among the compared methods.The correlation matrices compare CNN-HAE, CNN-AE, CNN-βVAE, and 5 POD modes.Source: §4.3 · p. 18, Figure 9 · p. 18

• The β-VAE/CNN architecture minimizes latent correlations and penalizes latent-vector size, while a separate algorithm ranks modes by reconstruction contribution.Source: §5 · pp. 18–19
• With d=5 and β=10^-3, CNN-βVAE achieves Ek=87.36% versus POD's 32.41% and detR=99.20 on turbulent urban flow.Source: §5 · p. 19
• CNN-βVAE's Ek=87.36% trails CNN-AE (93.93%) and CNN-HAE (91.84%), but it has the highest detR among AE-based models.Source: §5 · p. 19
• POD requires 247 modes to obtain 99% of reconstruction energy, indicating the difficulty of representing turbulent flow with a few linear modes.Source: §5 · p. 19
• Increasing latent size does not change the penalization-driven minimal mode set; lighter capacity constraints slightly improve reconstruction while permitting more latent correlations.Source: §5 · p. 19
• The first CNN-βVAE mode represents large-scale vortex shedding from around obstacles into the wake.Source: §5 · p. 19