  Kriging (or Gaussian process regression) is a popular machine learning method
for its flexibility and closed-form prediction expressions. However, one of the
key challenges in applying kriging to engineering systems is that the available
measurement data is scarce due to the measurement limitations and high sensing
costs. On the other hand, physical knowledge of the engineering system is often
available and represented in the form of partial differential equations (PDEs).
We present in this work a PDE Informed Kriging model (PIK), which introduces
PDE information via a set of PDE points and conducts posterior prediction
similar to the standard kriging method. The proposed PIK model can incorporate
physical knowledge from both linear and nonlinear PDEs. To further improve
learning performance, we propose an Active PIK framework (APIK) that designs
PDE points to leverage the PDE information based on the PIK model and
measurement data. The selected PDE points not only explore the whole input
space but also exploit the locations where the PDE information is critical in
reducing predictive uncertainty. Finally, an expectation-maximization algorithm
is developed for parameter estimation. We demonstrate the effectiveness of APIK
in two synthetic examples, a shock wave case study, and a laser heating case
study.
