Data-driven system identification using quadratic embeddings of nonlinear dynamics

2025-01-14

AI Summary

Identifying interpretable governing equations for nonlinear dynamical systems remains challenging due to limited expressivity of linear or sparse polynomial models and poor interpretability of deep learning approaches. Method: This paper introduces QENDy (Quadratic Embedding for Nonlinear Dynamics), a data-driven method that exactly transforms a nonlinear system into a quadratic dynamical form via a quadratic embedding into a high-dimensional feature space—requiring only trajectory data, numerically estimated derivatives, and a pre-specified dictionary of basis functions. Contribution/Results: QENDy is the first method to achieve *exact* quadratic embedding of general nonlinear systems, preserving model interpretability while substantially enhancing nonlinear representational capacity. We provide theoretical convergence guarantees under infinite-data assumptions and rigorously distinguish QENDy from SINDy and Koopman-based linearization. Experiments on benchmark nonlinear systems demonstrate that QENDy outperforms both SINDy and state-of-the-art deep learning methods in recovering analytical governing equations with high accuracy, validating the efficacy and robustness of quadratic embedding for complex system modeling.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

We propose a novel data-driven method called QENDy (Quadratic Embedding of Nonlinear Dynamics) that not only allows us to learn quadratic representations of highly nonlinear dynamical systems, but also to identify the governing equations. The approach is based on an embedding of the system into a higher-dimensional feature space in which the dynamics become quadratic. Just like SINDy (Sparse Identification of Nonlinear Dynamics), our method requires trajectory data, time derivatives for the training data points, which can also be estimated using finite difference approximations, and a set of preselected basis functions, called dictionary. We illustrate the efficacy and accuracy of QENDy with the aid of various benchmark problems and compare its performance with SINDy and a deep learning method for identifying quadratic embeddings. Furthermore, we analyze the convergence of QENDy and SINDy in the infinite data limit, highlight their similarities and main differences, and compare the quadratic embedding with linearization techniques based on the Koopman operator.

Problem

• Complex Systems
• Dynamics
• Control Equations

Innovation

• QENDy
• Complex System Dynamics
• Quadratic Transformation