Quantifying Model Form Uncertainty in Spring-Mass-Damper Systems

Models built from coupled ordinary differential equations are common in mechanics, chemical kinetics, electrodynamics, and many other fields. A canonical example, in both theory and experiments, is a system of linked spring-mass-dampers. Modeling all interactions between these objects often becomes intractable either due to computational expense or incomplete knowledge of the system. Common reduced models may involve only interactions between a small subset of the spring-mass-dampers. But these simplifications can lead to high model error, rendering the model useless for prediction. In this work, we explore decreasing model error through interpretable model correction: an inadequacy operator augments the reduced model to form an enriched model. We calibrate the enriched model with hierarchical Bayesian inference and validate it with posterior predictive assessments. Physical theory informs the inadequacy operator, which contains terms to capture the effect of the omitted objects on the reduced model. Several analytical and numerical examples are given. Results show that most of the model error can be recovered with a simple time-varying inadequacy operator.