MODULO: a python toolbox for data-driven modal decomposition

MODULO is a modal decomposition package developed at the von Karman Institute for Fluid Dynamics (VKI). It offers a wide range of decomposition techniques, allowing users to choose the most appropriate method for their specific problem. MODULO can efficiently handle large datasets natively, thanks to a memory-saving feature that partitions the data and processes the decomposition in chunks (ninni2020modulo). Moreover, it supports non-uniform meshes through a weighted inner product formulation. Currently, MODULO heavily relies on NumPy routines and does not offer additional parallel computing capabilities beyond those naturally provided by NumPy.

While the discontinued MATLAB version of MODULO (ninni2020modulo) is accessible in the “Old_Matlab_Implementation” branch, it is no longer maintained. The latest decomposition techniques are exclusively available in the current Python version.

As a part of the MODULO project, we provide a series of lectures on data-driven modal decomposition, and its applications. These are available at the MODULO YouTube channel.

Table of contents

• MODULO: a python toolbox for data-driven modal decomposition
• Modal decompositions
• Release Notes
• Installation
  • Installation via pip
  • Installation from source
• Documentation
• Example
  • Example 1: POD decomposition
  • Example 2: Memory Saving option
  • Example 3: non-uniform grid
• Computational Cost Estimates
• Community guidelines
  • Contributing to MODULO
  • Report bugs
  • Ask for help
• Citation
• References
• Related projects

Modal decompositions

Modal decompositions aim to describe the data as a linear combination of modes, obtained by projecting the data onto a suitable set of basis. For instance, consider a matrix $D(x, t)$, where $x$ and $t$ are the spatial and temporal coordinates, respectively, the modal decomposition can be written as:

$D(x_i, t_k) = \phi(x_i) \Sigma \psi(t_k)^T$

where $\phi(x_i)$ is the spatial basis, $\psi(t_k)$ is the temporal basis, and $\Sigma$ is the modal coefficients. Different decompositions employ different bases, such as prescribed Fourier basis ($\psi_\mathcal{F}$) for the Discrete Fourier Transform (DFT), or data-driven basis, i.e. tailored on the dataset at hand, for the Proper Orthogonal Decomposition (POD).

We refer to (mendez2022statistical, mendez2022generalizedmultiscalemodalanalysis, Mendez_2023) for an introduction to the topic.

MODULO currently features the following decompositions:

• Discrete Fourier Transform (DFT) (briggs1995dft)
• Proper Orthogonal Decomposition (POD) (sirovich1987turbulence, berkooz1993proper)
• Multi-Scale Proper Orthogonal Decomposition (mPOD) (mendez2019multi)
• Dynamic Mode Decomposition (DMD) (schmid2010dynamic)
• Spectral Proper Orthogonal Decomposition (SPOD) (csieber2016spectral, towne2018spectral), note that the two are different formulations, and both are available in MODULO.
• Kernel Proper Orthogonal Decomposition (KPOD) (mika1998kernel)

We remind the curious reader to the respective references for a detailed description of each decomposition, and to the documentation for a practical guide on how to use them in MODULO.

Release Notes

The latest version of MODULO (v2.0) includes the following updates:

1. Faster EIG/SVD algorithms, using powerful randomized svd solvers from scikit_learn

   (see here and here.) It is now possible to select various options as "eig_solver" and "svd_solver", offering different trade-offs in terms of accuracy vs computational time.

2. Computation the POD directly via SVD, using any of the four "svd_solver" options. This is generally faster but requires more memory.

3. Faster subscale estimators for the mPOD: the previous version used the rank of the correlation matrix in each scale to define the number of modes to be computed in each portion of the splitting vector before assembling the full basis. This is computationally very demanding. This estimation has been replaced by a frequency-based threshold (i.e. based on the frequency bins within each portion) since one can show that the frequency-based estimator is always more "conservative" than the rank-based estimator.

4. Major improvement on the memory saving option : the previous version of modulo always required in input the matrix D. Then, if the memory saving option was active, the matrix was partitioned and stored locally to free the RAM before computing the correlation matrix (see this tutorial by D. Ninni). In the new version, it is possible to initialize a modulo object without the matrix D (see exercise 5 in the examples). Instead, one can create the partitions without loading the matrix D.

5. Implementation of Dynamic Mode Decomposition (DMD) from (Schmid, P.J 2010)
6. Implementation of the two Spectral POD formulations, namely the one from (Sieber et al 2016), and the one from (Towne et al 2018).
7. Implementation of a kernel version of the POD, in which the correlation matrix is replaced by a kernel matrix. This is described in Lecture 15 of the course Hands on Machine Learning for Fluid dynamics 2023. We refer also to: Mendez, 2022.

8. Implementation of a formulation for non-uniform meshes, using a weighted matrix for all the relevant inner products. This is currently available only for POD and mPOD but allows for handling data produced from CFD simulation without resampling on a uniform grid (see exercise 4). It can be used both with and without the memory-saving option.

Installation

Installation via pip

You can access the latest update of the modulo python package on PyPI using the command line:

$ pip install modulo_vki

Installation from source

Alternatively, you can clone the repository and install the package locally:

$ git clone https://github.com/mendezVKI/MODULO.git

$ cd MODULO

$ python setup.py install

or, if you have pip installed in your environment,

$ pip install .

Documentation

The documentation of MODULO is available here. It contains a comprehensive guide on how to install and use the package, as well as a detailed description of the decompositions required inputs and outputs. A list of YouTube videos is also available to guide the introduce the user to modal decomposition and MODULO.

Example

Example 1: POD decomposition

The following example illustrates how to decompose a data set (D) using the POD decomposition.

from modulo_vki import ModuloVKI
import numpy as np

# Create a random dataset
D = np.random.rand(100, 1000)

# Initialize the ModuloVKI object
m = ModuloVKI(D)

# Compute the POD decomposition
phi_POD, Sigma_POD, psi_POD = m.Compute_POD_K()

which returns the spatial basis ($phi$), the temporal basis ($psi$), and the modal amplitudes ($Sigma$) of the POD decomposition.

Example 2: Memory Saving option

For the Memory Saving option, MODULO decomposes $D$ in N_partitions, defined by the user (refer to examples/ex_04_Memory_Saving.py).

from modulo_vki import ModuloVKI
import numpy as np

# Create a random dataset
D = np.random.rand(100, 1000)

# Initialize the ModuloVKI object
m = ModuloVKI(D, N_PARTITIONS=10)

# Compute the POD decomposition
phi_POD, Sigma_POD, psi_POD = m.Compute_POD_K()

Example 3: non-uniform grid

If you are dealing with non-uniform grid (e.g. output of a Computational Fluid Dynamic (CFD) simulation), you can use the weighted inner product formulation (refer to examples/ex_05_nonUniform_POD.py).

from modulo_vki import ModuloVKI
import numpy as np

# Create a random dataset
D = np.random.rand(100, 1000)

# Get the area of the grid
a_dataSet = gridData.compute_cell_sizes()
area = a_dataSet['Area']

# Compute weights
areaTot = np.sum(area)
weights = area/areaTot # sum should be equal to 1

# Initialize the ModuloVKI object
m = ModuloVKI(D, weights=weights)

# Compute the POD decomposition
phi_POD, Sigma_POD, psi_POD = m.Compute_POD_K()

Computational Cost Estimates

We here provide a rough estimate of the amoung of RAM required to decompose a test case with and without the memory saving option. This option reduces the memory usage at the cost of increasing the computational time (see https://www.youtube.com/watch?v=LclxO1WTuao)

Given a dataset $D \in \mathbb{R}^{n_s \times n_t}$, we consider the computation of $n_r$ modes. When using the memory saving option, we refer to $n_t' = n_t / n_p$ as the number of time steps in each partition, and to $n_s' = n_s / n_p$ as the number of spatial points in each partition.

	Phase 1: $D$	Phase 2: $K$	Phase 3: $\Psi$	Phase 4: $\Phi$
No Memory Saving	$n_s \times n_t$	$n_t^2$	$n_t^2 + n_t \times n_r$	$n_s \times n_t + n_t \times n_r + n_s \times n_r$
Memory Saving	/	$n_s \times n_t' + n_t' \times n_t'$	$n_t^2 + n_t \times n_r$	$n_s \times n_t' + n_s' \times n_t + n_s \times n_r$

If the memory saving option is active, the memory requirements are mostly linked to the storage of the correlation matrix $K$ in Phase 2. This table can be used to estimate if a dataset is too large for the available RAM, recalling that data in single precision requires 4 bytes (or 32 bits).

For example, for a dataset with n_s=1 000 000 and n_t = 5000 the following table estimates the RAM required in the two cases, considering n_b=10 partitions in the case of memory saving:

	Phase 1: $D$	Phase 2: $K$	Phase 3: $\Psi$	Phase 4: $\Phi$
No Memory Saving	18.6 GB	0.093 GB	≈0.112 GB	≈22.39 GB
Memory Saving	/	≈1.86 GB	≈ 0.0026 GB	≈ 4.20 GB

Community guidelines

Contributing to MODULO

We welcome contributions to MODULO.

It is recommended to perform a shallow clone of the repository to avoid downloading the entire history of the project:

$ git clone --depth 1 https://github.com/mendezVKI/MODULO.git

This will download only the latest version of the repository, which is sufficient for contributing to the project, and will save you time and disk space.

To create a new feature, please submit a pull request, specifying the proposed changes and providing an example of how to use the new feature (that will be included in the examples/ folder).

The pull request will be reviewed by the MODULO team before being merged into the main branch, and your contribution duly acknowledged.

Report bugs

If you find a bug, or you encounter unexpected behaviour, please open an issue on the MODULO GitHub repository.

Ask for help

If you have troubles using MODULO, or you need help with a specific decomposition, please open an issue on the MODULO GitHub repository.

Citation

If you use MODULO in your research, please cite it as follows:

``Poletti, R., Schena, L., Ninni, D. Mendez, M. A. (2024). MODULO: A Python toolbox for data-driven modal decomposition. Journal of Open Source Software, 9(102), 6753, https://doi.org/10.21105/joss.06753

@article{Poletti2024,
  doi = {10.21105/joss.06753},
  url = {https://doi.org/10.21105/joss.06753},
  year = {2024},
  publisher = {The Open Journal},
  volume = {9},
  number = {102},
  pages = {6753},
  author = {R. Poletti and L. Schena and D. Ninni and M. A. Mendez},
  title = {MODULO: A Python toolbox for data-driven modal decomposition},
  journal = {Journal of Open Source Software} }

and

Ninni, D., & Mendez, M. A. (2020). MODULO: A software for Multiscale Proper Orthogonal Decomposition of data. SoftwareX, 12, 100622.

@article{ninni2020modulo,
    title={MODULO: A software for Multiscale Proper Orthogonal Decomposition of data},
    author={Ninni, Davide and Mendez, Miguel A},
    journal={SoftwareX},
    volume={12},
    pages={100622},
    year={2020},
    publisher={Elsevier}
}

We are currently working on a Journal of Open Source article that will be available soon.

References

• Mendez, Miguel Alfonso. "Statistical Treatment, Fourier and Modal Decomposition." arXiv preprint arXiv:2201.03847 (2022).
• Mendez, M. A. (2023) "Generalized and Multiscale Modal Analysis". In : Mendez M.A., Ianiro, A., Noack, B.R., Brunton, S. L. (Eds), "Data-Driven Fluid Mechanics: Combining First Principles and Machine Learning". Cambridge University Press, 2023:153-181. https://doi.org/10.1017/9781108896214.013. The pre-print is available at https://arxiv.org/abs/2208.12630.
• Ninni, Davide, and Miguel A. Mendez. "MODULO: A software for Multiscale Proper Orthogonal Decomposition of data." SoftwareX 12 (2020): 100622.
• Mendez, Miguel A. "Linear and nonlinear dimensionality reduction from fluid mechanics to machine learning." Measurement Science and Technology 34.4 (2023): 042001.
• Briggs, William L., and Van Emden Henson. The DFT: an owner's manual for the discrete Fourier transform. Society for Industrial and Applied Mathematics, 1995.
• Berkooz, Gal, Philip Holmes, and John L. Lumley. "The proper orthogonal decomposition in the analysis of turbulent flows." Annual review of fluid mechanics 25.1 (1993): 539-575.
• Sirovich, Lawrence. "Turbulence and the dynamics of coherent structures. III. Dynamics and scaling." Quarterly of Applied mathematics 45.3 (1987): 583-590.
• Mendez, M. A., M. Balabane, and J-M. Buchlin. "Multi-scale proper orthogonal decomposition of complex fluid flows." Journal of Fluid Mechanics 870 (2019): 988-1036.
• Schmid, Peter J. "Dynamic mode decomposition of numerical and experimental data." Journal of fluid mechanics 656 (2010): 5-28.
• Sieber, Moritz, C. Oliver Paschereit, and Kilian Oberleithner. "Spectral proper orthogonal decomposition." Journal of Fluid Mechanics 792 (2016): 798-828.
• Towne, Aaron, Oliver T. Schmidt, and Tim Colonius. "Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis." Journal of Fluid Mechanics 847 (2018): 821-867.
• Mika, Sebastian, et al. "Kernel PCA and de-noising in feature spaces." Advances in neural information processing systems 11 (1998).

Related projects

MODULO encapsulates a wide range of decomposition techniques, but not all of them. We refer to the project below for an additional set of decomposition techniques:

• ModRed, https://github.com/belson17/modred

There are also decomposition-specific projects, some of which are listed below:

• Rogowski, Marcin, Brandon CY Yeung, Oliver T. Schmidt, Romit Maulik, Lisandro Dalcin, Matteo Parsani, and Gianmarco Mengaldo. "Unlocking massively parallel spectral proper orthogonal decompositions in the PySPOD package." Computer Physics Communications 302 (2024): 109246.
• Lario, A., Maulik, R., Schmidt, O.T., Rozza, G. and Mengaldo, G., 2022. Neural-network learning of SPOD latent dynamics. Journal of Computational Physics, 468, p.111475.
• Ichinaga, Andreuzzi, Demo, Tezzele, Lapo, Rozza, Brunton, Kutz. PyDMD: A Python package for robust dynamic mode decomposition. arXiv preprint, 2024.
• Rogowski, Marcin, et al. "Unlocking massively parallel spectral proper orthogonal decompositions in the PySPOD package." Computer Physics Communications 302 (2024): 109246.