Bayesian inference for high-dimensional inverse problems is challenged by the computational costs of the forward operator and the selection of an appropriate prior distribution. Amortized variational inference addresses these challenges where a neural network is trained to approximate the posterior distribution over existing pairs of model and data. When fed previously unseen data and normally distributed latent samples as input, the pretrained deep neural network -- in our case a conditional normalizing flow -- provides posterior samples with virtually no cost. However, the accuracy of this approach relies on the availability of high-fidelity training data, which seldom exists in geophysical inverse problems due to the heterogeneous structure of the Earth. In addition, accurate amortized variational inference requires the observed data to be drawn from the training data distribution. As such, we propose to increase the resilience of amortized variational inference when faced with data distribution shift via a physics-based correction to the conditional normalizing flow latent distribution. To accomplish this, instead of a standard Gaussian latent distribution, we parameterize the latent distribution by a Gaussian distribution with an unknown mean and diagonal covariance. These unknown quantities are then estimated by minimizing the Kullback-Leibler divergence between the corrected and true posterior distributions. While generic and applicable to other inverse problems, by means of a seismic imaging example, we show that our correction step improves the robustness of amortized variational inference with respect to changes in number of source experiments, noise variance, and shifts in the prior distribution. This approach provides a seismic image with limited artifacts and an assessment of its uncertainty with approximately the same cost as five reverse-time migrations.
翻译:远方操作员的计算成本和选择适当的先前分布方法,对高维反向问题的贝氏推论提出了挑战。当神经网络受过训练,可以对现有模型和数据组合的后方分布进行近似分布,从而应对挑战时,在输入先前不为人知的数据和通常分配的潜伏样本时,预先训练的深神经网络 -- -- 在我们的情况中,一个有条件的正常流流 -- -- 提供了后端样本,而几乎没有任何成本。然而,这一方法的准确性取决于能否获得高易变性培训数据,这些数据很少存在于地球物理的反常问题中。此外,准确的摊销变异性变异性推论要求从培训数据分布中提取观察到的数据。因此,我们提议,在数据分配变化变异性变异性变异性变异时,通过基于物理的校正调整到有条件的顺差分布,而不是用标准的源的潜值分布,我们用高度方法对地球物理变异性分布进行对比,然后用可测量的变异性分布比值进行对比,而以未知的平面变异性变异性变异性变异性变异性变异性分布则在前一级和变异性变异性变异性变异性变异性排序中,以未知性变异性变异性变法的原始变现的原始变法的变现的变法在原始变式变法的变法前变法的变的变变的变的变法中,在变的变的变法中,以未知性变的变式变法中,在变式变式变的变式变式变现的变式变式变的变式变式变式变式变式变的变后变的变的变的变的变的变的变式变式变式变式变式变式变式的变后变的变式变的变后变式变法中,在前变式变式变式变式变式形式和变式的变式的变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变式变