https://sam.ensam.eu:443 The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material. Thu, 05 Mar 2026 20:13:55 GMT 2026-03-05T20:13:55Z http://hdl.handle.net/10985/18735 PySINDy: A Python package for the sparse identification of nonlinear dynamical systems from data DE SILVA, Brian; CHAMPION, Kathleen; QUADE, Markus; KUTZ, J. Nathan; BRUNTON, Steven; LOISEAU, Jean-Christophe Scientists have long quantified empirical observations by developing mathematical models that characterize the observations, have some measure of interpretability, and are capable of making predictions. Dynamical systems models in particular have been widely used to study, explain, and predict system behavior in a wide range of application areas, with examples ranging from Newton’s laws of classical mechanics to the Michaelis-Menten kinetics for modeling enzyme kinetics. While governing laws and equations were traditionally derived by hand, the current growth of available measurement data and resulting emphasis on data-driven modeling motivates algorithmic approaches for model discovery. A number of such approaches have been developed in recent years and have generated widespread interest, including Eureqa (Schmidt & Lipson, 2009), sure independence screening and sparsifying operator (Ouyang, Curtarolo, Ahmetcik, Scheffler, & Ghiringhelli, 2018), and the sparse identification of nonlinear dynamics (SINDy) (Brunton, Proctor, & Kutz, 2016). Maximizing the impact of these model discovery methods requires tools to make them widely accessible to scientists across domains and at various levels of mathematical expertise. Authors of papers retaincopyright and release the workunder a Creative CommonsAttribution 4.0 InternationalLicense (CC-BY) Wed, 01 Jan 2020 00:00:00 GMT http://hdl.handle.net/10985/18735 2020-01-01T00:00:00Z DE SILVA, Brian CHAMPION, Kathleen QUADE, Markus KUTZ, J. Nathan BRUNTON, Steven LOISEAU, Jean-Christophe Scientists have long quantified empirical observations by developing mathematical models that characterize the observations, have some measure of interpretability, and are capable of making predictions. Dynamical systems models in particular have been widely used to study, explain, and predict system behavior in a wide range of application areas, with examples ranging from Newton’s laws of classical mechanics to the Michaelis-Menten kinetics for modeling enzyme kinetics. While governing laws and equations were traditionally derived by hand, the current growth of available measurement data and resulting emphasis on data-driven modeling motivates algorithmic approaches for model discovery. A number of such approaches have been developed in recent years and have generated widespread interest, including Eureqa (Schmidt & Lipson, 2009), sure independence screening and sparsifying operator (Ouyang, Curtarolo, Ahmetcik, Scheffler, & Ghiringhelli, 2018), and the sparse identification of nonlinear dynamics (SINDy) (Brunton, Proctor, & Kutz, 2016). Maximizing the impact of these model discovery methods requires tools to make them widely accessible to scientists across domains and at various levels of mathematical expertise. http://hdl.handle.net/10985/23054 PySINDy: A comprehensive Python package for robust sparse system identification KAPTANOGLU, Alan; DE SILVA, Brian; FASEL, Urban; KAHEMAN, Kadierdan; GOLDSCHMIDT, Andy; CALLAHAM, Jared; DELAHUNT, Charles; NICOLAOU, Zachary; CHAMPION, Kathleen; KUTZ, J.; BRUNTON, Steven; LOISEAU, Jean-Christophe 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. Sat, 01 Jan 2022 00:00:00 GMT http://hdl.handle.net/10985/23054 2022-01-01T00:00:00Z KAPTANOGLU, Alan DE SILVA, Brian FASEL, Urban KAHEMAN, Kadierdan GOLDSCHMIDT, Andy CALLAHAM, Jared DELAHUNT, Charles NICOLAOU, Zachary CHAMPION, Kathleen KUTZ, J. BRUNTON, Steven LOISEAU, Jean-Christophe 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. http://hdl.handle.net/10985/23061 From the POD-Galerkin method to sparse manifold models BRUNTON, Steven; NOACK, Bernd; LOISEAU, Jean-Christophe Reduced-order models are essential for the accurate and efficient prediction, estimation, and control of complex systems. This is especially true in fluid dynamics, where the fully resolved state space may easily contain millions or billions of degrees of freedom. Because these systems typically evolve on a low-dimensional attractor, model reduction is defined by two essential steps: (1) identify a good state space for the attractor and (2) identifying the dynamics on this attractor. The leading method for model reduction in fluids is Galerkin projection of the Navier–Stokes equations onto a linear subspace of modes obtained via proper orthogonal decomposition (POD). However, there are serious challenges in this approach, including truncation errors, stability issues, difficulty handling transients, and mode deformation with changing boundaries and operating conditions. Many of these challenges result from the choice of a linear POD subspace in which to represent the dynamics. In this chapter, we describe an alternative approach, feature-based manifold modeling (FeMM), in which the low-dimensional attractor and nonlinear dynamics are characterized from typical experimental data: time-resolved sensor data and optional nontime-resolved particle image velocimetry (PIV) snapshots. FeMM consists of three steps: First, the sensor signals are lifted to a dynamic feature space. Second, we identify a sparse human-interpretable nonlinear dynamical system for the feature state based on the sparse identification of nonlinear dynamics (SINDy). Third, if PIV snapshots are available, a local linear mapping from the feature state to the velocity field is performed to reconstruct the full state of the system. We demonstrate this approach, and compare with POD-Galerkin modeling, on the incompressible two-dimensional flow around a circular cylinder. Best practices and perspectives for future research are also included, along with open-source code for this example. Tue, 01 Jun 2021 00:00:00 GMT http://hdl.handle.net/10985/23061 2021-06-01T00:00:00Z BRUNTON, Steven NOACK, Bernd LOISEAU, Jean-Christophe Reduced-order models are essential for the accurate and efficient prediction, estimation, and control of complex systems. This is especially true in fluid dynamics, where the fully resolved state space may easily contain millions or billions of degrees of freedom. Because these systems typically evolve on a low-dimensional attractor, model reduction is defined by two essential steps: (1) identify a good state space for the attractor and (2) identifying the dynamics on this attractor. The leading method for model reduction in fluids is Galerkin projection of the Navier–Stokes equations onto a linear subspace of modes obtained via proper orthogonal decomposition (POD). However, there are serious challenges in this approach, including truncation errors, stability issues, difficulty handling transients, and mode deformation with changing boundaries and operating conditions. Many of these challenges result from the choice of a linear POD subspace in which to represent the dynamics. In this chapter, we describe an alternative approach, feature-based manifold modeling (FeMM), in which the low-dimensional attractor and nonlinear dynamics are characterized from typical experimental data: time-resolved sensor data and optional nontime-resolved particle image velocimetry (PIV) snapshots. FeMM consists of three steps: First, the sensor signals are lifted to a dynamic feature space. Second, we identify a sparse human-interpretable nonlinear dynamical system for the feature state based on the sparse identification of nonlinear dynamics (SINDy). Third, if PIV snapshots are available, a local linear mapping from the feature state to the velocity field is performed to reconstruct the full state of the system. We demonstrate this approach, and compare with POD-Galerkin modeling, on the incompressible two-dimensional flow around a circular cylinder. Best practices and perspectives for future research are also included, along with open-source code for this example.