On spike-and-slab priors for Bayesian equation discovery of nonlinear dynamical systems via sparse linear regression

TL;DR

This paper introduces spike-and-slab priors for Bayesian equation discovery in nonlinear dynamical systems, demonstrating improved variable selection and model accuracy over existing methods through MCMC sampling and application to engineering systems.

Contribution

The paper develops and applies three variants of spike-and-slab priors for sparse Bayesian linear regression in discovering nonlinear system equations, with validation on real-world benchmarks.

Findings

SS priors outperform RVM in model selection and accuracy

Effective identification of nonlinearities in engineering systems

Validated methodology on Silverbox benchmark

Abstract

This paper presents the use of spike-and-slab (SS) priors for discovering governing differential equations of motion of nonlinear structural dynamic systems. The problem of discovering governing equations is cast as that of selecting relevant variables from a predetermined dictionary of basis variables and solved via sparse Bayesian linear regression. The SS priors, which belong to a class of discrete-mixture priors and are known for their strong sparsifying (or shrinkage) properties, are employed to induce sparse solutions and select relevant variables. Three different variants of SS priors are explored for performing Bayesian equation discovery. As the posteriors with SS priors are analytically intractable, a Markov chain Monte Carlo (MCMC)-based Gibbs sampler is employed for drawing posterior samples of the model parameters; the posterior samples are used for variable selection and parameter estimation in equation discovery. The proposed algorithm has been applied to four systems of engineering interest, which include a baseline linear system, and systems with cubic stiffness, quadratic viscous damping, and Coulomb damping. The results demonstrate the effectiveness of the SS priors in identifying the presence and type of nonlinearity in the system. Additionally, comparisons with the Relevance Vector Machine (RVM) - that uses a Student's-t prior - indicate that the SS priors can achieve better model selection consistency, reduce false discoveries, and derive models that have superior predictive accuracy. Finally, the Silverbox experimental benchmark is used to validate the proposed methodology.