On the Integration of Physics-Based Machine Learning with Hierarchical Bayesian Modeling Techniques
This work addresses the challenge of incorporating physical knowledge into machine learning for improved performance in physical system modeling, particularly in structural dynamics, but it is incremental as it builds on existing physics-based ML and hierarchical Bayesian methods.
The paper tackles the problem of black-box machine learning models underperforming in blind conditions by integrating physics-based models into Gaussian Processes with a specific kernel, and demonstrates its application to structural dynamics inverse problems through numerical and experimental examples.
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.