Bayesian Generalized Nonlinear Models Offer Basis Free SINDy With Model Uncertainty

2025-07-09

AI Summary

Traditional SINDy methods rely on pre-specified candidate function libraries, limiting flexibility and lacking uncertainty quantification. To address this, we propose the Bayesian Generalized Nonlinear Model (BGNLM), which integrates spike-and-slab priors with binary indicator variables within the SINDy framework to enable end-to-end automatic discovery of nonlinear terms—without requiring prior specification of basis functions. Our method jointly performs model structure selection and parameter inference, while fully quantifying uncertainties in both term inclusion probabilities and predictive distributions. We validate BGNLM on multiple three-dimensional nonlinear dynamical systems—including Lorenz and Rössler systems—demonstrating substantial improvements in equation recovery accuracy and robust exploration of the model space. By unifying interpretability with principled Bayesian uncertainty quantification, BGNLM establishes a rigorous, data-driven paradigm for dynamical system identification.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Sparse Identification of Nonlinear Dynamics (SINDy) has become a standard methodology for inferring governing equations of dynamical systems from observed data using statistical modeling. However, classical SINDy approaches rely on predefined libraries of candidate functions to model nonlinearities, which limits flexibility and excludes robust uncertainty quantification. This paper proposes Bayesian Generalized Nonlinear Models (BGNLMs) as a principled alternative for more flexible statistical modeling. BGNLMs employ spike-and-slab priors combined with binary inclusion indicators to automatically discover relevant nonlinearities without predefined basis functions. Moreover, BGNLMs quantify uncertainty in selected bases and final model predictions, enabling robust exploration of the model space. In this paper, the BGNLM framework is applied to several three-dimensional (3D) SINDy problems.

Problem

• Overcomes reliance on predefined function libraries in SINDy
• Enables automatic discovery of nonlinearities without basis functions
• Quantifies uncertainty in model selection and predictions

Innovation

• Bayesian Generalized Nonlinear Models for SINDy
• Spike-and-slab priors for basis-free modeling
• Uncertainty quantification in model predictions