We introduce a conditional pseudo-reversible normalizing flow for constructing surrogate models of a physical model polluted by additive noise to efficiently quantify forward and inverse uncertainty propagation. Existing surrogate modeling approaches usually focus on approximating the deterministic component of physical model. However, this strategy necessitates knowledge of noise and resorts to auxiliary sampling methods for quantifying inverse uncertainty propagation. In this work, we develop the conditional pseudo-reversible normalizing flow model to directly learn and efficiently generate samples from the conditional probability density functions. The training process utilizes dataset consisting of input-output pairs without requiring prior knowledge about the noise and the function. Our model, once trained, can generate samples from any conditional probability density functions whose high probability regions are covered by the training set. Moreover, the pseudo-reversibility feature allows for the use of fully-connected neural network architectures, which simplifies the implementation and enables theoretical analysis. We provide a rigorous convergence analysis of the conditional pseudo-reversible normalizing flow model, showing its ability to converge to the target conditional probability density function using the Kullback-Leibler divergence. To demonstrate the effectiveness of our method, we apply it to several benchmark tests and a real-world geologic carbon storage problem.

翻译：本文提出一种条件伪可逆归一化流，用于构建受加性噪声污染的物理模型的代理模型，以高效量化正向和逆向不确定性传播。现有代理建模方法通常侧重于逼近物理模型的确定性分量，但这种策略需要预先了解噪声特性，并借助辅助采样方法进行逆向不确定性传播的量化。本研究开发的条件伪可逆归一化流模型可直接学习条件概率密度函数并高效生成样本。训练过程利用由输入-输出对组成的数据集，无需预先了解噪声及函数的相关信息。训练后的模型能够从任何高概率区域被训练集覆盖的条件概率密度函数中生成样本。此外，伪可逆特性允许使用全连接神经网络架构，从而简化实现过程并支持理论分析。我们对该模型进行了严格的收敛性分析，证明其能够通过库尔贝克-莱布勒散度收敛至目标条件概率密度函数。为验证方法的有效性，我们将其应用于多个基准测试问题及一个真实的地质碳封存问题。