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

• Published in: Mechanical systems and signal processing 2021-12, Vol.161, p.107986, Article 107986
• Main Authors: Nayek, R., Fuentes, R., Worden, K., Cross, E.J.
• Format: Article
• Language: English
• Subjects: Bayesian variable selection; Equation discovery; Model selection; Nonlinear system identification; Sparse Bayesian learning; Spike-and-slab priors
• ISSN: 0888-3270; 1096-1216
• DOI: 10.1016/j.ymssp.2021.107986

•A novel sparse Bayesian approach to equation discovery for nonlinear system identification using spike-and-slab priors is presented.•Three different variants of spike-and-slab priors are introduced, and a tailored Gibbs sampler is employed for Bayesian inference.•The approach is demonstrated and validated via several simulated case studies of nonlinear dynamic systems, and on the Silverbox experimental benchmark.•The results have been compared with the Sparse Bayesian Learning (SBL) in terms of efficacy and sparsity.•The proposed method demonstrates better model selection consistency, reduced rate of false discoveries and superior predictive accuracy. 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 functions 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 basis function 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 Sparse Bayesian (SBL) – 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.