SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics (2004.02322v2)

Abstract: Accurately modeling the nonlinear dynamics of a system from measurement data is a challenging yet vital topic. The sparse identification of nonlinear dynamics (SINDy) algorithm is one approach to discover dynamical systems models from data. Although extensions have been developed to identify implicit dynamics, or dynamics described by rational functions, these extensions are extremely sensitive to noise. In this work, we develop SINDy-PI (parallel, implicit), a robust variant of the SINDy algorithm to identify implicit dynamics and rational nonlinearities. The SINDy-PI framework includes multiple optimization algorithms and a principled approach to model selection. We demonstrate the ability of this algorithm to learn implicit ordinary and partial differential equations and conservation laws from limited and noisy data. In particular, we show that the proposed approach is several orders of magnitude more noise robust than previous approaches, and may be used to identify a class of complex ODE and PDE dynamics that were previously unattainable with SINDy, including for the double pendulum dynamics and the Belousov Zhabotinsky (BZ) reaction.

• The paper introduces a convex optimization approach that overcomes null space issues, improving noise robustness by up to three orders of magnitude compared to previous methods.
• It demonstrates the identification of both ODEs and PDEs with rational nonlinearities, effectively modeling complex systems like the double pendulum and the BZ reaction.
• The study provides robust model selection guidelines and extends its applicability to systems with control inputs, paving the way for real-world dynamic system analysis.

Review of "SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics"

The paper introduces SINDy-PI, a significant evolution of the Sparse Identification of Nonlinear Dynamics (SINDy) framework aimed at improving the identification of implicit dynamical systems, particularly those characterized by rational nonlinearities. This framework addresses the sensitivity of existing methodologies, such as implicit-SINDy, to noise, establishing itself as a robust method capable of handling noisy datasets while identifying complex dynamical systems.

Summary and Contributions

SINDy-PI builds on the foundational SINDy algorithm, which seeks parsimonious dynamical models from data by optimizing sparsity within a predefined function library. While SINDy is adept at identifying systems expressible in an explicit form, its extension, implicit-SINDy, struggles with noise when representing implicit functions or rational nonlinearities due to the ill-conditioned null space problem. SINDy-PI overcomes this limitation by utilizing a convex optimization approach that allows it to be much more noise-tolerant while maintaining the capability to identify implicit dynamics.

Key Contributions:

1. Parallel and Implicit Sparse Identification: The paper recasts the identification problem to bypass reliance on null space calculations, significantly enhancing noise robustness. It employs a convex formulation, parallelizing the task of testing candidate terms in the dynamics, which makes the algorithm compatible with high-performance computing resources.
2. Application Scope: SINDy-PI effectively identifies ordinary differential equations (ODEs) and partial differential equations (PDEs) with rational nonlinearities. Examples include complex systems such as the double pendulum and chemical kinetics represented by the Belousov–Zhabotinsky (BZ) reaction.
3. Model Selection Framework: The authors introduce robust guidelines for model selection and validation, deploying advanced criteria such as test error minimization to ensure that models are both concise and accurate.
4. Inclusion of Control Inputs: SINDy-PI is extended to systems with external forces, broadening its application to real-world dynamic systems, including robotics and physical systems influenced by active control measures.

Key Results

The framework demonstrates superior performance across several metrics, including but not limited to enhanced noise robustness—up to three orders of magnitude better than implicit-SINDy. SINDy-PI effectively identifies models using significantly less data compared to its predecessors. For example, in the chaotic dynamics of the double pendulum, SINDy-PI successfully captured the system's inherent rational nonlinearities under realistic noise levels.

Implications and Future Directions

The advancements presented in SINDy-PI have profound implications for the field of system identification and model inference, particularly in domains where data noise and model complexity impede the application of traditional identification methods. By addressing noise sensitivity, SINDy-PI opens up opportunities for the discovery and control of complex systems in fields ranging from fluid dynamics to chemical engineering and beyond.

Future Research Directions:

• Library Optimization: The authors highlight the exponential growth of library size with polynomial terms' order as a challenge for robustness. Future work could explore more systematic, potentially data-driven approaches to library construction.
• Integration of Learning Techniques: There is potential in combining SINDy-PI with machine learning methods to automate the identification of library functions and parameters further.
• Enhanced Model Validation: Advanced validation techniques could be integrated to better assess model generalizability across disparate datasets.

SINDy-PI marks a significant step forward in the identification of nonlinear dynamical systems, offering a robust toolset to researchers looking to uncover governing equations from observational data in a noise-tolerant manner. As these techniques are refined and expanded, their potential to impact scientific and engineering practices continues to grow, aligning with the increasing emphasis on data-driven discovery in complex system dynamics.