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Convolutional neural network based hierarchical autoencoder for nonlinear mode decomposition of fluid field data
Kai Fukami, Taichi Nakamura, Koji Fukagata
TL;DR
The paper tackles the lack of contribution ordering and interpretability in nonlinear autoencoder modes for fluid-field reduction. It develops a CNN-based hierarchical autoencoder that sequentially adds latent modes, finding ordered representations for laminar wakes and efficient turbulent-flow mapping with mode families. The authors suggest applications in reduced-order modeling and combinations with linear mode-decomposition methods.
Problem
Conventional autoencoder modes are nonlinear and generally non-orthogonal, so their individual energy contributions cannot be arranged like POD modes.
Method
The paper extends hierarchical autoencoders to CNNs, training sequential subnetworks that stack previously learned latent modes with new modes before reconstruction.
Results
The method extracts contribution-ordered modes for laminar cylinder wakes and efficiently maps turbulent channel-flow fields using ordered autoencoder mode families.
Takeaways & Limitations
The ordered nonlinear modes support reduced-order modeling and may be combined with linear theory-based methods that use mode ordering.
Takeaways & Limitations
The hierarchical decomposition is affected by lower-order subnetworks, and neural-network training can produce non-unique results.
Abstract
from arXiv · showhide
We propose a customized convolutional neural network based autoencoder called a hierarchical autoencoder, which allows us to extract nonlinear autoencoder modes of flow fields while preserving the contribution order of the latent vectors. As preliminary tests, the proposed method is first applied to a cylinder wake at $Re_D$ = 100 and its transient process. It is found that the proposed method can extract the features of these laminar flow fields as the latent vectors while keeping the order of their energy content. The present hierarchical autoencoder is further assessed with a two-dimensional $y-z$ cross-sectional velocity field of turbulent channel flow at $Re_τ$ = 180 in order to examine its applicability to turbulent flows. It is demonstrated that the turbulent flow field can be efficiently mapped into the latent space by utilizing the hierarchical model with a concept of ordered autoencoder mode family. The present results suggest that the proposed concept can be extended to meet various demands in fluid dynamics including reduced order modeling and its combination with linear theory-based methods by using its ability to arrange the order of the extracted nonlinear modes.
I. INTRODUCTION
The paper addresses limitations of linear and conventional autoencoder-based flow-field reduction by extending hierarchical autoencoders to CNNs. It evaluates the approach on laminar cylinder wakes and turbulent channel-flow velocity fields.
- I. INTRODUCTION: Linear methods such as POD and DMD extract dominant coherent structures, while autoencoders provide nonlinear low-dimensional mappings for fluid flows.Autoencoder activation functions can account for nonlinearities in reduced-order representations.
- I. INTRODUCTION: The proposed CNN-based hierarchical autoencoder extracts modes in contribution order while supporting more efficient compression than conventional autoencoders.The method extends an existing hierarchical autoencoder concept to convolutional neural networks.
- I. INTRODUCTION: The method is tested on a two-dimensional cylinder wake at Re_D = 100, its transient process, and y-z cross-sectional turbulent channel flow at Re_τ = 180.The turbulent-flow assessment introduces an ordered autoencoder mode family for higher-dimensional problems.
II. CONVOLUTIONAL NEURAL NETWORK BASED HIERARCHICAL AUTOENCODER
The hierarchical CNN autoencoder maps high-dimensional flow fields into a low-dimensional latent space through sequential subnetworks. Each new latent mode is added to previously learned modes, producing an ordered reconstruction hierarchy and mode families for turbulent flows.
- II. CONVOLUTIONAL NEURAL NETWORK BASED HIERARCHICAL AUTOENCODER: A CNN autoencoder is trained to reproduce its input while mapping high-dimensional flow data into a lower-dimensional latent vector.The weights minimize a prescribed reconstruction error, with encoder and decoder functions forming the latent representation.
- II. CONVOLUTIONAL NEURAL NETWORK BASED HIERARCHICAL AUTOENCODER: CNN layers use shared filters, nonlinear activation functions, max pooling in the encoder, and upsampling in the decoder.The paper uses the hyperbolic tangent activation function and notes similar performance with ReLU or sigmoid functions.
- Motivation: The hierarchical design targets the interpretability problem caused by non-orthogonal autoencoder modes, which lack eigenvalue- or singular-value-like contribution indicators.This ordering is intended to make nonlinear latent modes more comparable in contribution structure to linear decompositions.
- Hierarchical mode construction: The first subnetwork learns γ1, whose decoder reconstructs the field using only the first latent mode.This establishes the lowest-order mode in the hierarchical sequence.
- Hierarchical mode construction: Each subsequent subnetwork stacks previously obtained modes with a new latent vector before decoding, thereby learning remaining features.For example, the third subnetwork decodes {γ1, γ2, γ3} after using {γ1, γ2} as hierarchical inputs.
- Hierarchical mode construction: Repeating the procedure yields autoencoder modes ordered by their contributions to reconstructed fields.The M-mode construction uses M subnetworks, with each higher-order subnetwork depending on lower-order latent vectors while training independently.
- Ordered autoencoder mode family: An ordered autoencoder mode family groups multiple modes within each subnetwork to reduce the number of subnetworks needed for complex flows such as turbulence.The original one-mode-per-subnetwork design is used for laminar examples, while mode families are used for turbulent channel flow.
A. Example 1: Periodic wake behind a two-dimensional cylinder
The hierarchical autoencoder is evaluated on a periodic cylinder wake at Re_D = 100, where it extracts nonlinear modes in contribution order while reconstructing the flow from latent variables.
- Setup: The periodic wake is generated from two-dimensional DNS at Re_D = 100 and represented using the vorticity field as autoencoder input and output.The periodic training data cover a domain of 8.2 ≤ x ≤ 17.8 and −2.4 ≤ y ≤ 2.4, sampled at approximately 23 snapshots per period.
- Mode decomposition: Two POD modes cannot sufficiently reconstruct the periodic wake, whereas the hierarchical autoencoder improves reconstruction through nonlinear activation.The comparison includes POD, MD-CNN-AE, and the present hierarchical AE, with two modes used for each method.
- Mode decomposition: The hierarchical autoencoder places the wake’s dominant POD-like structures in AE mode 1, while AE mode 2 contains no structural POD modes.AE mode 1 contains eight POD mode-like structures, whereas the two MD-CNN-AE modes together contain six.
- Interpretation: The results suggest that hierarchical training extracts features according to their reconstruction contribution, consistent with the periodic wake being representable by a scalar phase variable.The paper states that one variable is sufficient to represent this time-periodic example.
B. Example 2: Transient wake behind a two-dimensional cylinder
The hierarchical autoencoder is tested on a non-periodic transient cylinder wake and compared with POD, MD-CNN-AE, and a linear hierarchical autoencoder. It provides accurate nonlinear reconstruction while preserving contribution ordering across modes.
- Transient wake: The transient cylinder wake provides a non-periodic test case for assessing the hierarchical autoencoder beyond periodic flow.The training data are prepared from DNS using an extended domain compared with the periodic example.
- Reconstruction: Two POD modes are insufficient for transient-wake reconstruction, while autoencoder methods achieve lower L2 error through nonlinear activation.The transient field requires more modes than the periodic example, and hierarchical-AE reconstructions closely agree with the DNS reference.
- Mode ordering: In the transient case, hierarchical AE mode 2 contains additional structures that the first subnetwork cannot extract, unlike the periodic-wake mode 2.Nonlinear activation prevents a clean POD-like interpretation of mode 2, but the hierarchical concept still orders the AE modes.
- Mode ordering: The hierarchical autoencoder assigns dominant POD modes clearly to AE mode 1, whereas MD-CNN-AE modes blend higher-order structures without proper ordering.The comparison identifies blended structures such as POD mode 3 within MD-CNN-AE mode 1.
- Comparison: The linear hierarchical autoencoder extracts fields analogous to POD modes 1 and 2 and preserves their energy-contribution order, highlighting the role of nonlinear activation.The nonlinear model extracts richer blended structures, while the linear model remains limited by its analogy to POD.
C. Example 3: y −z cross-section of turbulent channel flow
The hierarchical autoencoder is evaluated on turbulent channel-flow velocity fields using ordered mode families, yielding efficient compression and lower reconstruction error than POD and conventional autoencoders across most latent-space sizes.
- Flow configuration: The study uses y-z cross-sectional turbulent channel-flow velocity fields at Re_τ = 180, with training data obtained from validated direct numerical simulation.The computational domain is (4πδ, 2δ, 2πδ) with (256, 96, 256) grid points.
- Ordered mode families: Four ordered autoencoder mode families divide M total modes into M/4 dimensions per subnetwork for turbulent-field reconstruction.The number of families can affect compression because higher-level decomposition depends on lower-order subnetworks.
- Reconstruction performance: At nγ = 288, hierarchical AE reconstruction has lower L2 error and finer-scale recovery than POD and conventional AE.The comparison uses streamwise-velocity contours and ensemble L2 error norms of the three velocity components.
- Statistical assessment: The hierarchical AE provides better RMS velocity-fluctuation distributions than POD and conventional AE at nγ = 288, while conventional AE distributions are non-smooth for especially v and w.The ordered-mode concept is associated with smoother distributions for the POD and hierarchical models.
- Limitation: Machine-learning reconstructions can show a slight trend change near y+ ≈25 because they do not necessarily satisfy continuity.The paper suggests incorporating physics into the loss function as a possible mitigation.
- Mode-family contributions: Hierarchical AE mode families separate large-scale structures into lower families and finer structures across near-wall and channel-center regions into higher families.By contrast, high-order POD modes represent fine structure mainly near the wall.
- Computational cost: Training the conventional and hierarchical AE models takes approximately 10 h and 7 h, respectively, using 10000 snapshots.After training, the AE models provide better compression with faster reconstruction than POD.
- Latent-space dependence: Hierarchical AE outperforms POD and conventional AE in L2 error across the tested latent-space range except nγ = 8, where all methods reconstruct poorly.At nγ = 72, hierarchical AE captures finer scales despite the other methods' low fidelity; differences diminish at nγ = 3072.
IV. CONCLUSIONS
The study introduces a CNN-based hierarchical autoencoder that orders nonlinear modes by their contribution and extends this framework with mode families for more complex flow fields. Applications to laminar and turbulent flows indicate strong potential for nonlinear low-dimensional mapping, while non-uniqueness and hierarchical dependence remain unresolved issues.
- The CNN-based hierarchical autoencoder orders nonlinear modes according to their contributions to reconstructed flow fields while achieving efficient order reduction.
- For the cylinder wake, the first AE mode contained dominant structures appearing in high-order POD modes when two latent vectors were used.
- The mode-family concept groups AE modes by subnetwork, enabling reconstruction of more complex flows requiring high-order spatial modes.
- The hierarchical AE with mode families showed strong potential as a nonlinear low-dimensional mapping function for turbulent flow fields.
- Neural-network training produces non-unique decompositions, and higher-level hierarchical results depend on subnetworks responsible for lower-order modes.