Let us machine learn!

Machine-learning-based turbulence generator (MLTG) (available on GitHub)

1. • The combination of a convolutional neural network-based bottleneck model and multi-layer perceptron provides an inflow condition of high-fidelity turbulent flow simulations instead of numerical driver or white Gaussian noise-based inflow.

   • x6000 acceleration with TESLA K40 GPU compared to a conventional driver.  

   • Reference: K. Fukami, Y. Nabae, K. Kawai, K. Fukagata, “Synthetic turbulent inflow generator using machine learning,” Physical Review Fluids, 4 (064603), 2019 (preprint: arXiv:1806.08903 [physics.flu-dyn]), [Animation1] [Animation2]

Super-resolution analysis with hybrid downsampled skip-connection/multi-scale model (hDSC/MS model) (available on UCLA Taira Lab)

1. • A customized CNN supported by the multi-size filters and skip connections achieves a reasonable spatial reconstruction of fluid flow data from extremely coarse low-resolution data. 

   • Reference 1: K. Fukami, K. Fukagata, K. Taira, “Super-resolution reconstruction of turbulent flows with machine learning,” Journal of Fluid Mechanics, 870, 106-120, 2019 (preprint: arXiv:1811.11328 [physics.flu-dyn])

   • Reference 2: K. Fukami, K. Fukagata, K. Taira, “Machine-learning-based spatio-temporal super resolution reconstruction of turbulent flows,” Journal of Fluid Mechanics, 909, A9, 2021 (preprint, arXiv:2004.11566 [physics.flu-dyn])

   • Reference 3: K. Fukami, K. Fukagata, K. Taira, “Super-resolution analysis via machine learning: A survey for fluid flows,” Theoretical and Computational Fluid Dynamics, 37, 421--444 (invited), 2023 (preprint, arXiv:2301.10937 [physics.flu-dyn])

Standard convolutional neural network-based autoencoder (CNN-AE) (available on GitHub)

1. • A sample code for nonlinear CNN-AE-based compression of unsteady flows is provided.

   • Example data sets of laminar cylinder wake at Re = 100 are also given.

   • Reference: K. Fukagata, K. Fukami, “Compressing fluid flows with nonlinear machine learning: mode decomposition, latent modeling, and flow control,” in Review, 2025

Mode-decomposing convolutional neural network-based autoencoder (MD-CNN AE) (available on Keio University Fukagata Lab)

1. • A customized CNN-AE can extract and visualize “nonlinear machine learning mode" contained in fluid flow data sets.

   • For a laminar cylinder wake, AE modes contain several linear POD modes thanks to nonlinear superposition. 

   • Reference: T. Murata, K. Fukami, K. Fukagata, “Nonlinear mode decomposition with convolutional neural networks for fluid dynamics,” Journal of Fluid Mechanics, 882, A13, 2020 (preprint: arXiv:1906.04029 [physics.comp-ph])

Convolutional neural network-based autoencoder and long short-term memory based reduced-order modeling (CNNAE-LSTM ROM) (available on GitHub, laminar and turbulence)

1. • High-resolution data are low-dimensionalized into latent space with CNN-AE. The mapped vector is then predicted by using LSTM, which enables us to allow the construction of ROM since the predicted low-dimensionalized vector can be reversed on the original domain with the CNN decoder.

   • We have investigated the applicability to unsteady laminar wakes and wall-bounded turbulence.

   • Reference 1: K. Hasegawa, K. Fukami, T. Murata, K. Fukagata, “Machine-learning-based reduced-order modeling for unsteady fluid flows with various bluff bodies,” Theoretical and Computational Fluid Dynamics, 34 (4), 367--383 (invited),  2020  (preprint, arXiv:2003.07548 [physics.flu-dyn])

   • Reference 2: K. Hasegawa, K. Fukami, T. Murata, K. Fukagata, “CNN-LSTM based reduced order modeling of two-dimensional unsteady flows around a circular cylinder at different Reynolds numbers,” Fluid Dynamics Research, 52, 065501, 2020

   • Reference 3: T. Nakamura, K. Fukami, K. Hasegawa, Y. Nabae, K. Fukagata, “Convolutional neural network and long short-term memory based reduced order surrogate for minimal turbulent channel flow," Physics of Fluids, 33, 025116, 2021 (preprint, arXiv:2010.13351 [physics.flu-dyn])

Hierarchical convolutional neural network-based autoencoder (available on Keio University Fukagata Lab)

1. • A customized CNN-AE based on a transfer learning idea is able to achieve a much more efficient low-dimensionalization compared to linear PCA and conventional AEs for both laminar and turbulent flows. 

   • Reference: K. Fukami, T. Nakamura, K. Fukagata, “Convolutional neural network based hierarchical autoencoder for nonlinear mode decomposition of fluid field data,” Physics of Fluids, 32, 095110, 2020 (preprint, arXiv:2006.06977 [physics.comp-ph])

Interpretable AI with probabilistic neural network (PNN) (available on GitHub)

1. • A fully-connected network trained with maximization of log-likelihood can tell us a confidence interval of its estimation in addition to the ML estimation.

   • Demonstrated with several flow data sets.

   • Reference: R. Maulik, K. Fukami, N. Ramachandra, K. Fukagata, K. Taira, “Probabilistic neural networks for fluid flow surrogate modeling and data recovery,” Physical Review Fluids, 5 (104401), 2020 (preprint, arXiv:2005.04271 [physics.flu-dyn])

Interpretable reduced-order modeling with sparse identification of nonlinear dynamics (SINDy) for low-dimensionalized flow representations (available on GitHub)

1. • Temporal evolution of low-dimensional form of flows (e.g., CNN-AE modes, temporal coefficients of linear models, etc...) is represented as a form of ODE with the support of SINDy.

   • The choice of regression function is also carefully examined.

   • Reference: K. Fukami, T. Murata, K. Zhang, K. Fukagata, “Sparse identification of nonlinear dynamics with low-dimensionalized flow representations," Journal of Fluid Mechanics, 926, A10, 2021 (preprint, arXiv:2010.12177 [physics.flu-dyn])

Grad-CAM for visualization of internal procedure of neural networks (available on GitHub)

1. • Grad-CAM is able to identify the `reason' of machine-learning-based estimation by taking a gradient of outputs at hidden layers.

   • The applicability of Grad-CAM for fluid flow regressions is examined. Both single and multi-dimension output problems are considered.

   • Reference: M. Morimoto, K. Fukami, K. Zhang, K. Fukagata, “Generalization techniques of neural networks for fluid flow estimation," Neural Computing and Applications, 34, 3647-3669, 2022 (preprint, arXiv:2011.11911 [physics.flu-dyn])

Convolutional neural network-based robust flow reconstruction from moving sensors assisted with Voronoi tessellation (available on GitHub)

1. • Sparse sensor measurements are projected onto Voronoi tessellation before being fed into a convolutional neural network.

   • This pre-processing with CNN can handle arbitrary numbers of sensors and moving sensors with a single machine learning model.

   • Reference: K. Fukami, R. Maulik, N. Ramachandra, K. Fukagata, K. Taira, “Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning," Nature Machine Intelligence, 3, 945-951, 2021 (preprint, arXiv:2101.00554 [physics.flu-dyn])

Convolutional neural network-based fluid flow modeling with the assistance of supplemental scalar inputs (available on GitHub)

1. • Focusing on the application of CNN for fluid flow analyses including reduced-order modeling and metamodeling, the influence of supplemental scalars such as Reynolds number is examined.  

   • Sample codes of scalar-input-assisted CNN-AE and CNN-MLP are available.

   • Reference: M. Morimoto, K. Fukami, K. Zhang, A. G. Nair, K. Fukagata, “Convolutional neural networks for fluid flow analysis: toward effective metamodeling and low dimensionalization," Theoretical and Computational Fluid Dynamics, 35 (5), 633-658, 2021 (preprint, arXiv:2101.02535 [physics.flu-dyn])

ML-PIV: machine learning-based experimental velocity estimator from particle image (available on GitHub)

1. • Autoencoder-style convolutional neural network is used to estimate a velocity field from experimental particle images.  

   • Codes for the automatic generation of artificial particle image from DNS data are available.

   • Reference: M. Morimoto, K. Fukami, K. Fukagata, “Experimental velocity data estimation for imperfect particle images using machine learning,” Physics of Fluids, 33, 087121, 2021 

2D-3D CNN: volumetric data reconstruction from sectional fluid flow data (available on GitHub)

1. • Coupling of two- and three-dimensional convolutional neural networks is performed to reconstruct a three-dimensional fluid flow realization from two-dimensional sections. 

   • Towards extra data saving, the input two-dimensional sections can also be replaced with low-resolution images by combining with super-resolution analysis.

   • A code for generating “adaptive-sampled" low-resolution fluid flow snapshots is also available. 

   • Reference: M. Matsuo, K. Fukami, T. Nakamura, M. Morimoto, K. Fukagata, “Reconstructing three-dimensional bluff body wake from sectional flow fields with convolutional neural networks," SN Computer Science, 5, 306, 2024 (preprint, arXiv:2103.09020 [physics.flu-dyn])

Observable-augmented autoencoder for physically-interpretable manifold identification (available on GitHub)

1. • While nonlinear autoencoder-based fluid flow compression is powerful, the data distribution in a low-dimensional latent space has not often been cared.

   • Connecting the latent vectors and physical observables promotes physically-coherent manifold identification, demonstrated with both numerical (flows around an airfoil with extremely strong gust) and experimental (flows around a flat plate with transient gust encounter) aerodynamic data sets. 

   • Reference 1: K. Fukami, K. Taira, “Grasping extreme aerodynamics on a low-dimensional manifold," Nature Communications, 14, 6480, 2023 (preprint, arXiv:2305.08024 [physics.flu-dyn])

   • Reference 2: K. Fukami, H. Nakao, K. Taira, “Data-driven transient lift attenuation for extreme vortex gust-airfoil interactions," Journal of Fluid Mechanics, 992, A17, 2024 (preprint, arXiv:2403.00263 [physics.flu-dyn])

Phase autoencoder for nonlinear dynamics (available on GitHub)

• A range of nonlinear dynamics can often be expressed with a single-phase variable, although it is challenging to identify them, especially when the equations of motion are not available. 

• Focusing on limit-cycle oscillators, this customized autoencoder can encode the phase variable from a given physical variable.

• Reference: K. Yawata, K. Fukami, K. Taira, H. Nakao, “Phase autoencoder for limit-cycle oscillators," Chaos, 34, 063111, 2024 (preprint, arXiv:2403.06992 [nlin.AO]) (Selected as an Editor's Pick.)