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：高维贝叶斯反问题的高效求解在变分推断框架下具有潜力但仍面临挑战。主要困难体现在两个方面：首先，变分推断方法采用简单解析分布近似后验分布，这使得在实际应用中难以估计复杂的空间变参数；其次，变分推断通常依赖基于梯度的优化，当应用于含偏微分方程的反问题时，计算开销可能过高甚至难以实现。针对这些挑战，我们提出一种新型近似方法用于估计高维后验分布。该方法利用深度生成模型学习可生成空间变参数的先验模型，从而将后验近似从复杂参数域转移到隐变量空间，显著提升估计精度。同时，为加速梯度计算，我们采用可微物理约束代理模型替代伴随方法。所提方法可完全通过自动微分方式实现。数值算例展示了非均质介质中两类对数渗透率估计问题，结果表明该方法具有有效性、准确性和高效性。