PySINDy: A comprehensive Python package for robust sparse system identification (2111.08481v2)

Abstract: Automated data-driven modeling, the process of directly discovering the governing equations of a system from data, is increasingly being used across the scientific community. PySINDy is a Python package that provides tools for applying the sparse identification of nonlinear dynamics (SINDy) approach to data-driven model discovery. In this major update to PySINDy, we implement several advanced features that enable the discovery of more general differential equations from noisy and limited data. The library of candidate terms is extended for the identification of actuated systems, partial differential equations (PDEs), and implicit differential equations. Robust formulations, including the integral form of SINDy and ensembling techniques, are also implemented to improve performance for real-world data. Finally, we provide a range of new optimization algorithms, including several sparse regression techniques and algorithms to enforce and promote inequality constraints and stability. Together, these updates enable entirely new SINDy model discovery capabilities that have not been reported in the literature, such as constrained PDE identification and ensembling with different sparse regression optimizers.

• The paper presents robust enhancements to PySINDy that enable the discovery of nonlinear dynamics from noisy, limited data.
• It expands the candidate term library to incorporate actuated systems, PDEs, and implicit differential equations for complex system identification.
• Advanced optimization techniques, including sparse regression and inequality constraints, significantly improve model fidelity and stability.

PySINDy: A Comprehensive Python Package for Sparse System Identification

The paper presents a significant update to PySINDy, a Python package designed for sparse identification of nonlinear dynamics (SINDy). This enhancement extends PySINDy's capabilities to handle more complex differential equations and offers advanced features for robust system identification. The improvements allow for the discovery of general differential equations from noisy and limited data, a critical challenge in data-driven modeling.

Key Contributions

The paper highlights several enhancements to the original PySINDy package. These include:

1. Extended Library of Candidate Terms: The authors expand the library to incorporate actuated systems, Partial Differential Equations (PDEs), and implicit differential equations, enabling the identification of more complex dynamical systems.
2. Robust Formulations: The inclusion of integral SINDy formulations and ensembling techniques improves the reliability of model discovery in real-world scenarios. This is particularly beneficial when working with noisy datasets.
3. Advanced Optimization Algorithms: The update introduces new sparse regression techniques and methods for enforcing inequality constraints and promoting stability, enhancing the fidelity of identified models.
4. Broad Applicability: PySINDy now supports various dynamical systems, including control systems and PDEs, by expanding the functionality of the candidate library. This enhancement enables the package to handle a wide range of spatiotemporal data scenarios.

Implications and Applications

The improvements in PySINDy have substantial practical and theoretical implications. Practically, the package offers a robust tool for scientists and engineers to perform automated model discovery from complex datasets. The new features support a broader class of systems, such as those with multiple control inputs and PDE-based models, making the tool more versatile and applicable across diverse scientific domains.

Theoretically, the advancements facilitate the exploration of new theories and models in system dynamics, allowing researchers to validate or refute existing hypotheses using extensive data. The capability to enforce constraints and stability within the optimization process represents a step forward in aligning data-driven models with physical laws.

Numerical Results

Numerical demonstrations, such as the accurate modeling of the Kuramoto-Sivashinsky PDE, underscore the package's enhanced capabilities. The identified model closely matches known dynamics, indicating strong performance in capturing complex system behaviors.

Future Developments

The paper suggests several directions for future research and development:

• Integration with Machine Learning: Combining PySINDy with machine learning frameworks could further enhance its capabilities, especially in scenarios with high-dimensionality and nonlinearity.
• User Community and Contribution: As an open-source project, PySINDy invites contributions from researchers, which could lead to continuous improvement and innovation in the field.
• Simplification for Non-Experts: Simplifying the tool for non-experts while maintaining its robustness could broaden its accessibility and application in various fields of research.

Conclusion

PySINDy represents a significant advancement in the field of data-driven model discovery, offering a comprehensive toolset for sparse system identification. The paper's contributions make it a valuable resource for experienced researchers seeking to explore complex dynamical systems through data. The ongoing development and community involvement promise to keep PySINDy at the forefront of system identification research.