ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identification

2024-10-21

arXiv.org

AI Summary

Modeling dynamical systems with unknown nonlinear parameters remains challenging due to reliance on prior knowledge of functional forms (e.g., frequencies, exponentials) and manually constructed candidate libraries. Method: This paper proposes an enhanced Sparse Identification of Nonlinear Dynamics (SINDy) framework integrating the ADAM adaptive optimizer, enabling end-to-end joint sparse regression of both nonlinear parameters and library coefficients. It introduces a parameterized symbolic library and a global joint optimization strategy, eliminating the need for predefined nonlinear structures or handcrafted basis functions. Contribution/Results: To our knowledge, this is the first incorporation of adaptive gradient optimization into SINDy. The method achieves simultaneous, fully data-driven discovery of governing equations and their unknown parameters. Evaluated on diverse benchmarks—including chaotic oscillators, reaction kinetics, pharmacokinetics, and wildfire PDEs—it demonstrates superior accuracy, robustness to noise, and cross-system generalizability compared to standard SINDy and its variants.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Identifying dynamical systems characterized by nonlinear parameters presents significant challenges in deriving mathematical models that enhance understanding of physics. Traditional methods, such as Sparse Identification of Nonlinear Dynamics (SINDy) and symbolic regression, can extract governing equations from observational data; however, they also come with distinct advantages and disadvantages. This paper introduces a novel method within the SINDy framework, termed ADAM-SINDy, which synthesizes the strengths of established approaches by employing the ADAM optimization algorithm. This facilitates the simultaneous optimization of nonlinear parameters and coefficients associated with nonlinear candidate functions, enabling precise parameter estimation without requiring prior knowledge of nonlinear characteristics such as trigonometric frequencies, exponential bandwidths, or polynomial exponents, thereby addressing a key limitation of SINDy. Through an integrated global optimization, ADAM-SINDy dynamically adjusts all unknown variables in response to data, resulting in an adaptive identification procedure that reduces the sensitivity to the library of candidate functions. The performance of the ADAM-SINDy methodology is demonstrated across a spectrum of dynamical systems, including benchmark coupled nonlinear ordinary differential equations such as oscillators, chaotic fluid flows, reaction kinetics, pharmacokinetics, as well as nonlinear partial differential equations (wildfire transport). The results demonstrate significant improvements in identifying parameterized dynamical systems and underscore the importance of concurrently optimizing all parameters, particularly those characterized by nonlinear parameters. These findings highlight the potential of ADAM-SINDy to extend the applicability of the SINDy framework in addressing more complex challenges in dynamical system identification.

Problem

• Nonlinear Parameters
• Complex Dynamic Systems
• Mathematical Modeling

Innovation

• ADAM-SINDy
• Nonlinear Parameter Optimization
• Dynamic System Identification