Bayesian Generalized Nonlinear Models Offer Basis Free SINDy With Model Uncertainty

TL;DR

This paper introduces Bayesian Generalized Nonlinear Models (BGNLMs) as a flexible, basis-free alternative to classical SINDy, enabling automatic discovery of nonlinearities and quantification of model uncertainty in dynamical systems.

Contribution

The paper presents BGNLMs that automatically identify relevant nonlinear functions and quantify uncertainty, overcoming limitations of predefined libraries in traditional SINDy methods.

Findings

Successfully applied to 3D SINDy problems

Automatically discovers relevant nonlinearities

Provides uncertainty quantification in model predictions

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.