Source-linked AI summary
Machine-learning-based reduced order modeling for unsteady flows around bluff bodies of various shapes
Kazuto Hasegawa, Kai Fukami, Takaaki Murata, Koji Fukagata
TL;DR
The paper addresses reduced order modeling for complex, high-dimensional unsteady flow data, where linear methods may require many modes. It sequentially combines a CNN-AE for spatial reduction with an LSTM for latent-space time prediction, testing flows around unseen bluff-body shapes. The ML-ROM reproduces flow statistics well for unseen shapes, with performance depending on latent-vector and LSTM time-step choices.
Problem
Linear reduced order models can require too many modes to represent nonlinear flow phenomena, motivating nonlinear dimension reduction for complex unsteady flows.
Method
A CNN-AE maps DNS flow fields into a low-dimensional latent space, while an LSTM recursively predicts the latent vectors' temporal evolution before CNN decoding.
Results
The ML-ROM shows statistically good agreement with DNS for unseen bluff-body shapes, including mean centerline velocity, force coefficients, and Strouhal number.
Takeaways & Limitations
The ML-ROM can reproduce unsteady wakes around unseen bluff-body shapes when its latent-vector size and LSTM time step are appropriately selected.
Takeaways & Limitations
The proof of concept uses a limited range of shapes and laminar periodic flows, so greater shape variability is required for practice.
Abstract
from arXiv · showhide
We propose a method to construct a reduced order model with machine learning for unsteady flows. The present machine-learned reduced order model (ML-ROM) is constructed by combining a convolutional neural network autoencoder (CNN-AE) and a long short-term memory (LSTM), which are trained in a sequential manner. First, the CNN-AE is trained using direct numerical simulation (DNS) data so as to map the high-dimensional flow data into low-dimensional latent space. Then, the LSTM is utilized to establish a temporal prediction system for the low-dimensionalized vectors obtained by CNN-AE. As a test case, we consider flows around a bluff body whose shape is defined using a combination of trigonometric functions with random amplitudes. The present ML-ROMs are trained on a set of 80 bluff body shapes and tested on a different set of 20 bluff body shapes not used for training, with both training and test shapes chosen from the same random distribution. The flow fields are confirmed to be well reproduced by the present ML-ROM in terms of various statistics. We also focus on the influence of two main parameters: (1) the latent vector size in the CNN-AE, and (2) the time step size between the mapped vectors used for the LSTM. The present results show that the ML-ROM works well even for unseen shapes of bluff bodies when these parameters are properly chosen, which implies great potential for the present type of ML-ROM to be applied to more complex flows
1 Introduction
Reduced order models address the difficulty of handling complex, high-dimensional flow data by mapping flow fields into lower-dimensional spaces. This study proposes a sequential CNN-AE–LSTM ML-ROM and evaluates it on unseen bluff-body shapes while examining key model parameters.
- High-resolution simulations and image-based measurements produce complex fluid big data that is difficult to understand and control directly.
- 7260 POD modes are required to reconstruct 95% of the total energy in a low-Reynolds-number turbulent channel flow, illustrating the burden of linear reduction.
- Machine learning is considered a nonlinear dimension-reduction candidate for reducing the number of modes needed to represent flow phenomena.
- Prior work applied machine learning to nonlinear reduced order modeling, including extreme learning machines, multilayer perceptrons with LSTM, and CNN autoencoders.
- The proposed ML-ROM sequentially combines a CNN-AE for spatial compression with an LSTM for temporal prediction, and tests generalization across randomly defined bluff-body shapes.
2 Methods
The method builds an ML-ROM for unsteady bluff-body flows by combining DNS-generated training data, a multi-scale CNN autoencoder, and an LSTM for latent-space temporal prediction.
- 2.1 Training data: DNS generates two-dimensional flow fields around randomly shaped bluff bodies for ML-ROM training, validation, and assessment.The bodies are rescaled to unit frontal length and simulated at Re_D = 100; rotated fields expand the dataset to 100 flow types.
- 2.1 Training data: The study tests whether one ML-ROM can approximate the time-discretized Navier–Stokes evolution for unseen bluff-body shapes.The target mapping is q^(n+1)∆t = F(q^n∆t), with q = [u, v, p]^T.
- 2.2.1 Convolutional neural network autoencoder (CNN-AE): The CNN-AE encoder maps high-dimensional flow data into a latent vector, while its decoder reconstructs the physical-space field.The extracted learning data contain velocity and pressure in a subdomain around the body, with dimensions (384, 192, 3).
- 2.2.1 Convolutional neural network autoencoder (CNN-AE): The proposed MS-CNN-AE uses three filter scales—3 × 3, 5 × 5, and 9 × 9—to capture multi-scale spatial information.Its encoder uses convolution, batch normalization, ReLU, and max pooling; the decoder uses convolution, batch normalization, ReLU, and upsampling.
- 2.2.1 Convolutional neural network autoencoder (CNN-AE): The CNN-AE is trained by minimizing a loss combining mean squared error and gradient difference loss.Adam optimization, four-fold cross-validation, and learning curves showing convergence without observed overfitting are used during training.
- 2.2.2 Long Short-Term Memory (LSTM): The trained LSTM recursively predicts future latent vectors from initial CNN-encoded fields, while retaining temporal information in its cell state.The combined ML-ROM encodes DNS fields, predicts the next latent vector, and reconstructs the predicted field through the CNN decoder.
3 Results and Discussion
The ML-ROM combines a sequentially trained MS-CNN-AE and LSTM to reconstruct and predict unsteady wakes around unseen random bluff-body shapes. Its fidelity depends on selecting suitable latent-vector and LSTM time-step sizes.
- ML-ROM construction: The MS-CNN-AE maps high-dimensional DNS flow fields into a latent space, after which the LSTM predicts temporal evolution for the ML-ROM.The baseline latent space is R∈6×3×4, with latent-vector size nz = 72.
- Assessment on unseen shapes: The MS-CNN-AE was trained on 80 bluff-body shapes and evaluated on 20 different shapes excluded from training.The test shapes are shown as a separate evaluation set.
- Assessment on unseen shapes: Reconstructed instantaneous fields agree well with DNS, with time-averaged local squared error concentrated near the bluff body and sufficiently small in the wake region.These comparisons were reported for multiple bluff-body shapes.
- Assessment on unseen shapes: Mean centerline velocities, mean squared errors, and time-averaged drag and lift coefficients indicate close agreement between reconstructed fields and DNS.The assessments cover shapes 1, 3, 5, 7, 9, 11, 13, and 15 at Re_D = 100.
- Assessment on unseen shapes: The ML-ROM reproduces mean centerline velocity, force coefficients, and Strouhal number for unseen shapes, indicating that its temporal wake structure is well predicted.The error remains periodic but does not grow over 1000 recursive inputs.
- Parameter dependence: A latent-vector size of nz = 72 is the minimum tested size with appropriate fidelity and can outperform nz = 4608, while smaller sizes underpredict some velocities and force coefficients.Pooling operations are identified as a likely source of greater robustness for unknown wakes at nz = 72.
- Parameter dependence: The LSTM input-history length also affects ML-ROM results, with more than five time steps showing no significant dependence in a preliminary test for this periodic flow.The authors state that the appropriate number should depend on user requirements.
4 Conclusions
The ML-ROM reproduced unsteady flows around unseen bluff-body shapes with statistically good agreement, while parameter studies identified conditions affecting reconstruction and temporal accuracy. The study remains a proof of concept limited to a narrow range of laminar periodic shapes.
- The ML-ROM showed statistically good agreement with DNS data for bluff-body shapes excluded from training.The result suggests robustness beyond the specific flow fields used during training under different geometrical configurations.
- Larger time steps between LSTM input vectors increased prediction error.A time step of ∆t = 0.25 can be recommended for accurately reproducing the Strouhal number.
- The CNN-AE latent-vector size had relatively small influence on reconstruction ability for this temporally periodic problem.The authors caution that this weak dependence might be specific to the present setting.
- The study was a proof of concept using a limited range of bluff-body shapes and laminar periodic flows.More variability is required in practice, and extensions to three-dimensional high-Reynolds-number flows are proposed.
- Applying this ML-ROM approach to more complex flows requires a more efficient, preferably interpretable, low-dimensionalization method.The authors identify low-dimensionalization as the key issue for broader applications.