APIK: Active Physics-Informed Kriging Model with Partial Differential Equations

Kriging (or Gaussian process regression) becomes 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 are scarce due to the measurement limitations or 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 paper a PDE-informed Kriging model (PIK) that 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.