标题： 边界Matérn GP：在不规则域上可靠地结合边界信息的高斯过程建模 显示英文标题

标题： The BdryMatérn GP: Reliable incorporation of boundary information on irregular domains for Gaussian process modeling

摘要： 高斯过程（GPs）被广泛用作复杂现象的昂贵计算机模拟器的代理模型。 然而，一个关键瓶颈是其训练数据是从这个昂贵的模拟器生成的，因此可能非常有限。 一种有前途的解决方案是利用科学知识中的边界信息来补充学习模型。 然而，尽管最近有关于集成边界GPs的工作，但这些模型大多无法适应不规则（即非超立方体）域上的边界信息，并且不提供样本路径平滑度控制或近似误差分析，而这两者对于可靠的代理建模非常重要。 因此，我们提出了一种新的BdryMatérn GP建模框架，该框架可以可靠地在具有几乎处处二次可微边界的不规则连通域上整合狄利克雷、诺伊曼和罗宾边界。 我们的模型利用了一种通过随机偏微分方程公式推导出的新BdryMatérn协方差核。 类似于具有Matérn核的GP，我们证明了BdryMatérn GP的样本路径满足所需的边界条件，并对其导数具有平滑度控制。 我们进一步提出了一种使用有限元建模的BdryMatérn核高效近似方法，并进行了严格的误差分析。 最后，我们在一系列数值实验中展示了BdryMatérn GP在将广泛边界纳入不规则域中的有效性。 显示英文摘要

摘要： Gaussian processes (GPs) are broadly used as surrogate models for expensive computer simulators of complex phenomena. However, a key bottleneck is that its training data are generated from this expensive simulator and thus can be highly limited. A promising solution is to supplement the learning model with boundary information from scientific knowledge. However, despite recent work on boundary-integrated GPs, such models largely cannot accommodate boundary information on irregular (i.e., non-hypercube) domains, and do not provide sample path smoothness control or approximation error analysis, both of which are important for reliable surrogate modeling. We thus propose a novel BdryMat\'ern GP modeling framework, which can reliably integrate Dirichlet, Neumann and Robin boundaries on an irregular connected domain with a boundary set that is twice-differentiable almost everywhere. Our model leverages a new BdryMat\'ern covariance kernel derived in path integral form via a stochastic partial differential equation formulation. Similar to the GP with Mat\'ern kernel, we prove that sample paths from the BdryMat\'ern GP satisfy the desired boundaries with smoothness control on its derivatives. We further present an efficient approximation procedure for the BdryMat\'ern kernel using finite element modeling with rigorous error analysis. Finally, we demonstrate the effectiveness of the BdryMat\'ern GP in a suite of numerical experiments on incorporating broad boundaries on irregular domains.