Inverse modeling for computing a high-dimensional spatially-varying property field from indirect sparse and noisy observations is a challenging problem. This is due to the complex physical system of interest often expressed in the form of multiscale PDEs, the high-dimensionality of the spatial property of interest, and the incomplete and noisy nature of observations. To address these challenges, we develop a model that maps the given observations to the unknown input field in the form of a surrogate model. This inverse surrogate model will then allow us to estimate the unknown input field for any given sparse and noisy output observations. Here, the inverse mapping is limited to a broad prior distribution of the input field with which the surrogate model is trained. In this work, we construct a two- and three-dimensional inverse surrogate models consisting of an invertible and a conditional neural network trained in an end-to-end fashion with limited training data. The invertible network is developed using a flow-based generative model. The developed inverse surrogate model is then applied for an inversion task of a multiphase flow problem where given the pressure and saturation observations the aim is to recover a high-dimensional non-Gaussian log-permeability field where the two facies consist of heterogeneous log-permeability and varying length-scales. For both the two- and three-dimensional surrogate models, the predicted sample realizations of the non-Gaussian log-permeability field are diverse with the predictive mean being close to the ground truth even when the model is trained with limited data.

从间接的稀疏和嘈杂的观测值计算高维空间变化属性场的逆建模是一个具有挑战性的问题。这是由于感兴趣的复杂物理系统通常以多尺度PDE的形式表示，感兴趣的空间特性的高维性以及观测的不完整和嘈杂的性质。为了解决这些挑战，我们开发了一个模型，该模型以代理模型的形式将给定的观测值映射到未知的输入字段。然后，该逆代理模型将使我们能够为任何给定的稀疏和嘈杂的输出观测值估计未知的输入字段。在此，逆映射被限制为训练替代模型的输入字段的广泛先验分布。在这项工作中，我们构建了一个二维和三维逆代理模型，该模型由一个可逆的和有条件的神经网络组成，该网络以有限的训练数据进行端到端的训练。可逆网络是使用基于流的生成模型开发的。然后将开发的逆代理模型应用于多相流问题的反演任务，在给定压力和饱和度观测值的情况下，目标是恢复一个高维非高斯对数渗透率场，其中两个相由非均质对数渗透率组成和不同的长度尺度。对于二维和三维替代模型，即使使用有限的数据训练模型，非高斯对数渗透率字段的预测样本实现也各不相同，其预测均值接近于地面真实情况。