PySINDy: A Python package for the Sparse Identification of Nonlinear Dynamics from Data (2004.08424v1)

Abstract: PySINDy is a Python package for the discovery of governing dynamical systems models from data. In particular, PySINDy provides tools for applying the sparse identification of nonlinear dynamics (SINDy) (Brunton et al. 2016) approach to model discovery. In this work we provide a brief description of the mathematical underpinnings of SINDy, an overview and demonstration of the features implemented in PySINDy (with code examples), practical advice for users, and a list of potential extensions to PySINDy. Software is available at https://github.com/dynamicslab/pysindy.

• The paper presents PySINDy as a novel, accessible Python package implementing the SINDy methodology for sparse identification of nonlinear dynamics from data.
• It details a three-step process—numerical differentiation, library construction, and sparse regression—to extract interpretable governing equations.
• The package mitigates challenges like noise sensitivity with customizable differentiation schemes and flexible candidate function libraries, enhancing practical model discovery.

Overview of PySINDy: A Python Package for Sparse Identification of Nonlinear Dynamics

The academic paper introduces PySINDy, a Python package designed to discover governing dynamical systems models from empirical data using the Sparse Identification of Nonlinear Dynamics (SINDy) methodology. This package represents an advancement in the practical application of the SINDy framework by providing accessible tools to implement sparse regression techniques for model discovery, particularly when a system’s underlying dynamics are described by nonlinear differential equations.

Core Features and Motivation

PySINDy provides tools for applying the SINDy methodology, which frames the model discovery task as a sparse regression problem. It involves identifying significant terms in the system dynamics from a broader library of candidate functions. The main advantage of this approach lies in the interpretability of the resulting models and its versatility across different domains ranging from fluid dynamics to plasma physics. By focusing on sparsity, SINDy leverages the prior knowledge encapsulated in the candidate functions, many of which are historically grounded in the physics of the systems being modeled.

The package is designed for broad accessibility, adhering to well-known Python standards such as scikit-learn, enabling both novice users and seasoned researchers to apply SINDy effectively. This makes PySINDy an ideal tool for scientific model discovery, particularly in disciplines where measurement data is abundant, and there is a need for automated methods to discern underlying dynamic models.

Technical Methodology

The package executes the SINDy methodology through three primary stages:

1. Numerical Differentiation: This involves the calculation of derivatives from empirical data, a critical step since these derivatives serve as the regression targets. PySINDy includes various differentiation schemes, such as finite difference methods, which are robust but sensitive to noise, prompting the inclusion of smoothed finite difference options.
2. Library Construction: The selection of candidate functions, forming the basis of potential model structures. PySINDy includes libraries for polynomial terms, trigonometric functions, and allows for custom functions.
3. Sparse Regression: Solving the regression problem to identify which terms in the library are active in describing the system. Multiple optimization techniques are available, such as sequentially thresholded least squares (STLSQ) and sparse relaxed regression (SR3).

Practical Implications and Future Directions

The paper outlines potential pitfalls in practical applications, such as handling noisy data and selecting appropriate basis functions. These are mitigated by strategic choices in numerical differentiation and library construction. Moreover, PySINDy incorporates modular components, allowing custom techniques to be integrated as required by specific use cases.

The implications of PySINDy are substantial in both research and practical applications. It simplifies the process of inferring dynamical systems directly from data, making advanced model discovery accessible to experimentalists in fields like biology, chemistry, and engineering without extensive computational expertise.

Future developments could focus on extending PySINDy for partial differential equations, improving noise robustness via alternative formulations like integral methods, and refining the choice of library functions. The package could also be expanded to include automatic identification of coordinate systems or the integration of physical constraints directly within the learning process.

Conclusion

PySINDy marks a valuable contribution to the field of data-driven model discovery. By combining mathematical rigor with computational accessibility, it facilitates the integration of sparse regression techniques into the broader workflow of scientific inquiry. As the toolkit evolves, it is likely to foster significant advancements in how complex systems are modeled and understood from empirical data, enabling vast opportunities for exploration across varied scientific domains.