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Abstract:
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Kriging has become a popular machine learning method for its flexibility and closed-form expressions. One key challenge in applying kriging is that the available measurement data can be scarce due to measurement limitations or high costs. On the other hand, physical knowledge of the system is often available and represented in the form of partial differential equations (PDEs). We present 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 the PDEs. To further improve its performance, we propose an Active PIK framework (APIK) that designs PDE points to leverage the PDE information based on the PIK model and 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 examples on shock wave and laser heating. (Joint work with Jialei Chen and Zhehui Chen.)
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