A Survey of Bayesian Calibration and Physics-informed Neural Networks in Scientific Modeling

Abstract

Computer simulations are used to model of complex physical systems. Often, these models represent the solutions (or at least approximations) to partial differential equations that are obtained through costly numerical integration. This paper presents a survey of two important statistical/machine learning approaches that have shaped the field of scientific modeling. Firstly we survey the developments on Bayesian calibration of computer models since the seminal work by Kennedy and O’Hagan. In their paper, the authors proposed an elegant way to use the Gaussian processes to extend calibration beyond parameter and observation uncertainty and include model-form and data size uncertainty. Secondly, we also survey physics-informed neural networks, a topic that has been receiving growing attention due to the potential reduction in computational cost and modeling flexibility. In addition, in order to help the interested reader to familiarize with these topics and venture into custom implementations, we present a summary of applications and software tools. Finally, we close the paper with suggestion for future research directions and a thought provoking call for action.