  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.
