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 available measurement data are scarce due limitations or high sensing costs. On other hand, physical knowledge system often represented form partial differential equations (PDEs). We present this paper PDE-informed model (PIK) introduces PDE information via set points conducts posterior similar standard method. The proposed PIK can incorporate from both linear nonlinear PDEs. To further improve performance, we propose an active framework (APIK) designs leverage based on data. selected not only explore whole input space but also exploit locations where critical reducing predictive uncertainty. Finally, expectation-maximization algorithm developed parameter estimation. demonstrate effectiveness APIK two synthetic examples: shock wave case study laser heating study.