Solving high-dimensional Bayesian inverse problems (BIPs) with the variational inference (VI) method is promising but still challenging. The main difficulties arise from two aspects. First, VI methods approximate the posterior distribution using a simple and analytic variational distribution, which makes it difficult to estimate complex spatially-varying parameters in practice. Second, VI methods typically rely on gradient-based optimization, which can be computationally expensive or intractable when applied to BIPs involving partial differential equations (PDEs). To address these challenges, we propose a novel approximation method for estimating the high-dimensional posterior distribution. This approach leverages a deep generative model to learn a prior model capable of generating spatially-varying parameters. This enables posterior approximation over the latent variable instead of the complex parameters, thus improving estimation accuracy. Moreover, to accelerate gradient computation, we employ a differentiable physics-constrained surrogate model to replace the adjoint method. The proposed method can be fully implemented in an automatic differentiation manner. Numerical examples demonstrate two types of log-permeability estimation for flow in heterogeneous media. The results show the validity, accuracy, and high efficiency of the proposed method.

翻译：以变式推断法解决高维贝雅反向问题(BIPs)很有希望,但仍然具有挑战性。主要困难来自两个方面。首先,六种方法使用简单和分析性的变异分布法来近似后部分布,因此难以在实际中估计复杂的空间变化参数。第二,六种方法通常依赖基于梯度的优化,在应用涉及部分差异方程的BIP(PDEs)时,这种优化在计算上可能是昂贵的或棘手的。为了应对这些挑战,我们提出了一个新的近似法来估计高维的后部分布。这一方法利用了一种深层基因化模型来学习能够生成空间变化参数的先前模型。这样,使后部近似于潜在变量而不是复杂的参数,从而提高了估计的准确性。此外,为了加速梯度计算,我们采用了一种不同的物理学约束的代谢模型来取代部分差异式方法。拟议的方法可以以自动区分的方式完全实施。数字示例显示了两种类型的对不同媒体流动的逻辑-渗透性估计方法。