APIK：基于偏微分方程的主动物理信息克立格模型
APIK: Active Physics-Informed Kriging Model with Partial Differential Equations
摘要：
Kriging（或高斯过程回归）是一种流行的机器学习方法，其灵活性和封闭形式的预测表达式。然而，在应用克里格工程系统的关键挑战之一是，可用的测量数据是稀缺的，由于测量的限制和高传感成本。另一方面，工程系统的物理知识通常可用偏微分方程（PDE）的形式表示。在这项工作中，我们提出了一个PDE知情克里格模型（PIK），它通过一组PDE点引入PDE信息，并进行类似于标准克里格方法的后验预测。所提出的PIK模型可以结合线性和非线性偏微分方程的物理知识。为了进一步提高学习性能，我们提出了一个主动PIK框架（APIK），设计PDE点，利用PDE信息的PIK模型和测量数据的基础上。所选择的PDE点不仅探索整个输入空间，而且还利用PDE信息在降低预测不确定性方面至关重要的位置。最后，提出了一种期望最大化算法用于参数估计。我们证明了APIK的有效性，在两个合成的例子，冲击波的案例研究，和激光加热的案例研究。
英文摘要：
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.
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