Surrogate modeling based on Gaussian processes (GPs) has received increasing attention in the analysis of complex problems in science and engineering. Despite extensive studies on GP modeling, the developments for functional inputs are scarce. Motivated by an inverse scattering problem in which functional inputs representing the support and material properties of the scatterer are involved in the partial differential equations, a new class of kernel functions for functional inputs is introduced for GPs. Based on the proposed GP models, the asymptotic convergence properties of the resulting mean squared prediction errors are derived and the finite sample performance is demonstrated by numerical examples. In the application to inverse scattering, a surrogate model is constructed with functional inputs, which is crucial to recover the reflective index of an inhomogeneous isotropic scattering region of interest for a given far-field pattern.
翻译:在分析科学和工程的复杂问题时,基于高斯进程(GPs)的代用模型日益受到越来越多的注意。尽管对GP模型进行了广泛的研究,但功能投入的发展却很少。受一个反向分散问题的驱动,即代表散射器支持和物质特性的功能投入涉及部分差分方程,因此为GPs引入了功能投入的新型内核功能。根据拟议的GP模型,得出了由此产生的平均正方形预测错误的无药用趋同特性,用数字实例证明了有限的样本性能。在应用反分散时,用功能投入构建了一种代用模型,这对于恢复对某种特定远地点模式感兴趣的不相形异的异形分散区域的反射指数至关重要。