On the Integration of Physics-Based Machine Learning with Hierarchical Bayesian Modeling Techniques

TL;DR

This paper introduces a physics-informed Gaussian Process model that integrates mechanics-based models with hierarchical Bayesian techniques to improve prediction accuracy and scalability in physical systems modeling.

Contribution

It proposes embedding physics-based models into Gaussian Processes and employs hierarchical Bayesian methods to handle non-stationarity and improve scalability.

Findings

Enhanced modeling of physical systems with integrated physics and ML.

Improved scalability and computational efficiency for sequential data.

Demonstrated effectiveness in structural dynamics inverse problems.

Abstract

Machine Learning (ML) has widely been used for modeling and predicting physical systems. These techniques offer high expressive power and good generalizability for interpolation within observed data sets. However, the disadvantage of black-box models is that they underperform under blind conditions since no physical knowledge is incorporated. Physics-based ML aims to address this problem by retaining the mathematical flexibility of ML techniques while incorporating physics. In accord, this paper proposes to embed mechanics-based models into the mean function of a Gaussian Process (GP) model and characterize potential discrepancies through kernel machines. A specific class of kernel function is promoted, which has a connection with the gradient of the physics-based model with respect to the input and parameters and shares similarity with the exact Autocovariance function of linear dynamical systems. The spectral properties of the kernel function enable considering dominant periodic processes originating from physics misspecification. Nevertheless, the stationarity of the kernel function is a difficult hurdle in the sequential processing of long data sets, resolved through hierarchical Bayesian techniques. This implementation is also advantageous to mitigate computational costs, alleviating the scalability of GPs when dealing with sequential data. Using numerical and experimental examples, potential applications of the proposed method to structural dynamics inverse problems are demonstrated.