A perspective on regression and Bayesian approaches for system identification of pattern formation dynamics

• Published in: Theoretical and applied mechanics letters 2020-03, Vol.10 (3), p.188-194
• Main Authors: Wang, Zhenlin, Wu, Bowei, Garikipati, Krishna, Huan, Xun
• Format: Article
• Language: English
• Subjects: Bayesian inference; Computational mechanics; Inverse problem; Materials physics; Pattern formation
• ISSN: 2095-0349
• DOI: 10.1016/j.taml.2020.01.028

•Our manuscript presents mathematical frameworks and numerical approaches for identifying the partial differential equations that govern pattern formation in biophysics and materials physics.•In particular, we describe a stepwise regression variational system identification (VSI) method, and a Bayesian inference approach for system identification.•We demonstrate the usage and results of these methods through an easy-to-understand model problem, and discuss the advantages and disadvantages of each.•Regression with VSI is computationally fast and scalable, but has more strict data type requirements.•Bayesian inference provides uncertainty quantification, suitable for noisy and sparse data, flexible for different quantities of interest, but can be very computationally expensive and difficult to scale. We present two approaches to system identification, i.e. the identification of partial differential equations (PDEs) from measurement data. The first is a regression-based variational system identification procedure that is advantageous in not requiring repeated forward model solves and has good scalability to large number of differential operators. However it has strict data type requirements needing the ability to directly represent the operators through the available data. The second is a Bayesian inference framework highly valuable for providing uncertainty quantification, and flexible for accommodating sparse and noisy data that may also be indirect quantities of interest. However, it also requires repeated forward solutions of the PDE models which is expensive and hinders scalability. We provide illustrations of results on a model problem for pattern formation dynamics, and discuss merits of the presented methods.