PyDMD: A Python package for robust dynamic mode decomposition (2402.07463v1)

Abstract: The dynamic mode decomposition (DMD) is a simple and powerful data-driven modeling technique that is capable of revealing coherent spatiotemporal patterns from data. The method's linear algebra-based formulation additionally allows for a variety of optimizations and extensions that make the algorithm practical and viable for real-world data analysis. As a result, DMD has grown to become a leading method for dynamical system analysis across multiple scientific disciplines. PyDMD is a Python package that implements DMD and several of its major variants. In this work, we expand the PyDMD package to include a number of cutting-edge DMD methods and tools specifically designed to handle dynamics that are noisy, multiscale, parameterized, prohibitively high-dimensional, or even strongly nonlinear. We provide a complete overview of the features available in PyDMD as of version 1.0, along with a brief overview of the theory behind the DMD algorithm, information for developers, tips regarding practical DMD usage, and introductory coding examples. All code is available at https://github.com/PyDMD/PyDMD .

• The paper introduces significant enhancements to PyDMD by integrating advanced DMD variants such as OptDMD and BOP-DMD, which improve noise suppression and data robustness.
• It presents a modular, user-friendly architecture that simplifies data pre-processing, model fitting, and visualization for effective spatiotemporal analysis.
• Experimental results on synthetic data validate the package's ability to accurately capture dynamic patterns, paving the way for broad applications in scientific research and engineering.

PyDMD: A Python Package for Robust Dynamic Mode Decomposition

The paper "PyDMD: A Python package for robust dynamic mode decomposition" presents a comprehensive update to the PyDMD software, offering an array of enhancements to the existing dynamic mode decomposition (DMD) framework. DMD is a well-established technique that extracts coherent spatiotemporal structures from data. Its linear algebra foundation facilitates various optimizations, making it a valuable tool across scientific disciplines. PyDMD, a Python implementation, has been extended to include advanced DMD methods tailored for complex data scenarios such as noise, nonlinearity, and multiscale features.

Overview of PyDMD Enhancements

The paper details the expanded functionality of PyDMD up to version 1.0, including support for several sophisticated DMD variants:

• Optimized DMD (OptDMD): This variant improves noise suppression by formulating the DMD process as a nonlinear optimization problem using variable projection. OptDMD can handle unevenly sampled data and integrate constraints to enhance robustness.
• Bagging Optimized DMD (BOP-DMD): Built upon OptDMD, BOP-DMD employs statistical techniques to further stabilize results, making it a robust choice for noisy datasets.
• Coherent Spatiotemporal Scale Separation (CoSTS): Extends DMD for systems with multiscale dynamics, enabling effective separation of different spatial and temporal scales.
• Parametric and Randomized DMD: These variants are tailored for parameterized systems and data compression, respectively, expanding DMD application to larger and more complex datasets.
• Physics-Informed DMD: This extension integrates physical constraints into the DMD framework, ensuring that the resulting models adhere to underlying physical laws.
• Hankel and Higher-Order DMD: These approaches use time-delayed data embedding, essential for systems where data lacks full observability.

PyDMD Structure and Usage

PyDMD's modular architecture allows easy integration of various DMD methods. The package's core, the DMDBase class, encapsulates fundamental functionalities, while specific DMD variants inherit and extend these base capabilities. This design supports seamless user interaction through a simple yet powerful interface.

Practically, PyDMD enables users to harness the strengths of DMD through an intuitive workflow that involves initializing the appropriate DMD variant, pre-processing data if necessary (e.g., using time-delays), fitting the model, and finally employing built-in visualization tools for analysis. This workflow facilitates accurate extraction of spatiotemporal modes, crucial for tasks like future-state prediction and system control.

Experimental Validation

The paper illustrates PyDMD's capability through an example involving synthetic data comprising known spatiotemporal patterns. The package effectively identifies these patterns, demonstrating its utility in revealing underlying dynamics even in complex systems. Users are advised to leverage expert knowledge and carefully select DMD variants based on data characteristics to ensure optimal results.

Implications and Future Directions

PyDMD's updates significantly broaden its applicability, enabling robust application of DMD in real-world scenarios across various scientific and engineering domains. The inclusion of advanced methods like BOP-DMD enhances noise resilience, which is critical for many industrial applications.

As a central repository of DMD algorithms, PyDMD aims to foster continuous development and integration of novel DMD techniques. Future research may further enhance PyDMD by incorporating additional robust and adaptive DMD methodologies, potentially exploring tensor decompositions and integration with machine learning paradigms for enriched data analysis capabilities.

In conclusion, this paper demonstrates PyDMD's robust and versatile structure, making it a crucial tool for researchers and practitioners seeking to analyze dynamical systems efficiently and effectively.