In this paper, we develop a surrogate modelling approach for capturing the output field (e.g. the pressure head) from groundwater flow models involving a stochastic input field (e.g. the hydraulic conductivity). We use a Karhunen–Loève expansion for a log-normally distributed input field and apply manifold learning (local tangent space alignment) to perform Gaussian process Bayesian inference using Hamiltonian Monte Carlo in an abstract feature space, yielding outputs for arbitrary unseen inputs. We also develop a framework for forward uncertainty quantification in such problems, including analytical approximations of the mean of the marginalized distribution (with respect to the inputs). To sample from the distribution, we present Monte Carlo approach. Two examples are presented to demonstrate the accuracy of our approach: a Darcy flow model with contaminant transport in 2-d and a Richards equation model in 3-d.

中文翻译：

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一种基于非线性降维的替代建模方法，用于地下水流模型中的不确定性量化

在本文中，我们开发了一种替代建模方法，用于从涉及随机输入场（例如水力传导率）的地下水流模型中捕获输出场（例如压头）。我们对对数正态分布的输入字段使用 Karhunen-Loève 扩展，并应用流形学习（局部切线空间对齐）在抽象特征空间中使用哈密顿蒙特卡罗来执行高斯过程贝叶斯推理，产生任意看不见的输入的输出。我们还为此类问题中的前向不确定性量化开发了一个框架，包括边缘化分布（相对于输入）均值的分析近似值。为了从分布中采样，我们提出了 Monte Carlo 方法。提供了两个示例来证明我们方法的准确性：