We present a dimension-reduced KRnet map approach (DR-KRnet) for high-dimensional Bayesian inverse problems, which is based on an explicit construction of a map that pushes forward the prior measure to the posterior measure in the latent space. Our approach consists of two main components: data-driven VAE prior and density approximation of the posterior of the latent variable. In reality, it may not be trivial to initialize a prior distribution that is consistent with available prior data; in other words, the complex prior information is often beyond simple hand-crafted priors. We employ variational autoencoder (VAE) to approximate the underlying distribution of the prior dataset, which is achieved through a latent variable and a decoder. Using the decoder provided by the VAE prior, we reformulate the problem in a low-dimensional latent space. In particular, we seek an invertible transport map given by KRnet to approximate the posterior distribution of the latent variable. Moreover, an efficient physics-constrained surrogate model without any labeled data is constructed to reduce the computational cost of solving both forward and adjoint problems involved in likelihood computation. With numerical experiments, we demonstrate the accuracy and efficiency of DR-KRnet for high-dimensional Bayesian inverse problems.

翻译：我们提出了一种降维KRnet映射方法（DR-KRnet）用于求解高维贝叶斯反问题，该方法基于在潜在空间中显式构造从先验测度到后验测度的推动映射。该方法包含两个核心组件：数据驱动的VAE先验以及潜在变量后验的密度近似。实际应用中，先验分布的初始化往往难以与可用先验数据保持一致，即复杂的先验信息通常超出手工构造简单先验的范畴。我们采用变分自编码器（VAE）通过潜在变量与解码器来逼近先验数据集的潜在分布。借助VAE先验提供的解码器，我们将问题重新表述于低维潜在空间中。具体而言，我们通过KRnet构造可逆映射来近似潜在变量的后验分布。此外，为降低似然计算中正问题与伴随问题的求解成本，我们构建了无需标注数据的高效物理解约束代理模型。数值实验验证了DR-KRnet在高维贝叶斯反问题中的精度与效率。