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Nonlinear mode decomposition with convolutional neural networks for fluid dynamics
Takaaki Murata, Kai Fukami, Koji Fukagata
TL;DR
The paper addresses the difficulty of representing highly nonlinear flow fields with conventional linear mode-decomposition methods. It introduces the MD-CNN-AE, which maps flow data into two latent values and reconstructs interpretable decomposed fields. Nonlinear activation lowers reconstruction error relative to POD, while tanh modes contain multiple conventional POD bases, although the method has so far been tested mainly on large-scale structures.
Problem
Conventional reduced-order models have limited applicability to highly nonlinear flows because their linear representations may require many modes.
Method
The MD-CNN-AE uses a CNN autoencoder with one encoder and two decoders to map flow fields into two latent values and visualize two decomposed fields.
Results
Nonlinear MD-CNN-AEs achieve lower reconstruction errors than POD, and tanh decomposed fields contain multiple orthogonal bases corresponding to POD modes 1, 3, 5 and 2, 4, 6.
Takeaways & Limitations
The nonlinear MD-CNN-AE can extract flow features in fewer dimensions than POD while retaining interpretable relationships with conventional POD modes.
Takeaways & Limitations
The method has so far been examined mainly for flows with large-scale spatial structures, and additional network-design improvements may be required for turbulence.
Abstract
from arXiv · showhide
We present a new nonlinear mode decomposition method to visualize the decomposed flow fields, named the mode decomposing convolutional neural network autoencoder (MD-CNN-AE). The proposed method is applied to a flow around a circular cylinder at $Re_D=100$ as a test case. The flow attributes are mapped into two modes in the latent space and then these two modes are visualized in the physical space. Because the MD-CNN-AEs with nonlinear activation functions show lower reconstruction errors than the proper orthogonal decomposition (POD), the nonlinearity contained in the activation function is considered the key to improve the capability of the model. It is found by applying POD to each field decomposed using the MD-CNN-AE with hyperbolic tangent activation that a single nonlinear MD-CNN-AE mode contains multiple orthogonal bases, in contrast to the linear methods, i.e., POD and the MD-CNN-AE with linear activation. We further assess the proposed MD-CNN-AE by applying it to a transient process of a circular cylinder wake in order to examine its capability for flows containing high-order spatial modes. The present results suggest a great potential for the nonlinear MD-CNN-AE to be used for feature extraction of flow fields in lower dimension than POD, while retaining interpretable relationships with the conventional POD modes.
1. Introduction
Conventional mode decomposition methods support reduced-order modeling but struggle with highly nonlinear flows, motivating a nonlinear CNN autoencoder that can decompose and visualize low-dimensional flow modes.
- POD and DMD extract low-dimensional flow structures for visualization, comparison, and reduced-order control.
- 7260 POD modes are required to reproduce 95% of the energy in a turbulent channel flow at Reτ = 180, compared with two modes for 99% in a cylinder wake at ReD = 100.
- CNNs use weight sharing to process large flow-field datasets at reasonable computational costs.
- Nonlinear autoencoders improve dimension-reduction capability over linear perceptrons, which behave similarly to POD.
- The proposed MD-CNN-AE decomposes flow fields into nonlinear low-dimensional modes and visualizes each mode using a circular-cylinder flow at ReD = 100.
2. Methods
The method trains convolutional autoencoders to compress cylinder-flow fields into a two-value latent representation and reconstruct them, while MD-CNN-AE uses separate decoders to produce interpretable modes. Training uses DNS data, convolutional/pooling/upsampling layers, and L2 reconstruction-error minimization.
- 2.1. Training data: Training data come from a two-dimensional DNS of incompressible cylinder flow at Re_D = 100.The computational domain uses dimensionless streamwise and transverse extents, a Cartesian grid, and no-slip cylinder boundary conditions.
- 2.1. Training data: The machine-learning inputs are streamwise and transverse velocity fluctuations from periodic-shedding and transient-wake cases.For periodic shedding, the extracted region contains 384 × 192 grid points, with snapshots spaced by 0.25.
- 2.2. Machine learning model: CNN layers reduce flow-field dimensions through convolution and max pooling, then restore them using upsampling.Max pooling compresses each region by (1/P)^2 while retaining local features.
- 2.2. Machine learning model: The conventional CNN autoencoder maps the flow field to a two-value latent vector and decodes it back to the original dimensions.Its encoder and decoder are represented by r = Fenc(q) and q̃ = Fdec(r), and training minimizes the L2 error between input and output.
- 2.2. Machine learning model: MD-CNN-AE divides the latent vector into two variables and uses two decoders to generate separate decomposed fields whose sum reconstructs the output.This design permits each mode to be visualized, unlike the conventional CNN autoencoder.
- 2.2. Machine learning model: The proposed network uses H = 3, K = 2, pooling ratio P = 2, 28 layers, and 9646 trainable parameters.Training uses early stopping and Adam, with randomly selected snapshots for training.
3. Results and discussion
MD-CNN-tanh reconstructs periodic and transient cylinder wakes more accurately than POD while decomposing nonlinear modes into multiple POD-like bases. Its encoded variables remain interpretable, but the decomposition is not unique and does not match the dimensional efficiency of LLE.
- Activation functions: Nonlinear MD-CNN activations produce lower L2 norm errors than linear activation and POD, with tanh and Softsign performing best.Sigmoid trains poorly because of vanishing gradients; linear activation has the same error level as POD.
- Periodic vortex shedding case: MD-CNN-tanh gives the closest reconstruction to reference flow fields and the lowest local time-averaged L2 error except just downstream of the cylinder.POD and MD-CNN-Linear have similar reconstructed fields and error distributions.
- Periodic vortex shedding case: MD-CNN-tanh decomposes the flow into periodic encoded variables whose phases shift relative to POD, while the decomposition trajectory is not unique across validation folds.The non-uniqueness is attributed to the nonlinearity of the activation function.
- Nonlinear mode interpretation: Each nonlinear decomposed field contains multiple POD bases: field 1 resembles modes 1, 3, and 5, while field 2 resembles modes 2, 4, and 6.The decomposed fields also contain nearly equal amounts of kinetic energy, and nonlinear averaging can introduce structures that cancel when fields are recombined.
- Nonlinear mode interpretation: Unlike two-mode POD and MD-CNN-Linear, MD-CNN-tanh retains higher POD modes across its two decomposed fields, lowering reconstruction error while distributing bases similarly to full POD.The result is consistent with analytical nonlinear relations between cylinder-wake POD modes 1–2 and modes 3–6.
- Transient wake case: For the transient wake, MD-CNN-tanh again has lower L2 error than two-mode POD, with one nonlinear field containing modes 1, 3, and 5 and the other containing modes 2, 4, and 7.The method therefore retains features of unsteady flow in lower dimension than POD, although it is less dimensionally efficient than LLE.
4. Conclusions
The MD-CNN-AE decomposes flow fields into two nonlinear, visualizable modes and can reconstruct the cylinder wake more accurately than two-mode POD and linear MD-CNN-AE. Its nonlinear modes combine multiple POD bases, while tests on a transient wake produced broadly similar results; broader turbulent-flow applicability remains unresolved.
- The MD-CNN-AE uses one encoder and two decoders to map a cylinder flow into two latent values and reconstruct it from two decomposed fields.
- Nonlinear activation reduces reconstruction L2 norm error compared with two-mode POD and MD-CNN-Linear, whereas the linear model resembles POD.
- MD-CNN-tanh decomposes the flow into fields whose POD bases correspond to modes 1, 3, 5 and modes 2, 4, 6, respectively.
- A transient cylinder-wake process with higher-order spatial modes produced broadly similar results, supporting the method’s performance beyond the periodic case.
- The method has so far been tested only on flows with large-scale spatial structures, and additional network improvements are required for more complex flows such as turbulence.