Title: Enhanced uncertainty quantification variational autoencoders for the solution of Bayesian inverse problems Show Chinese title

Title: 用于贝叶斯反问题求解的增强不确定性量化变分自编码器

Abstract: Among other uses, neural networks are a powerful tool for solving deterministic and Bayesian inverse problems in real-time, where variational autoencoders, a specialized type of neural network, enable the Bayesian estimation of model parameters and their distribution from observational data allowing real-time inverse uncertainty quantification. In this work, we build upon existing research [Goh, H. et al., Proceedings of Machine Learning Research, 2022] by proposing a novel loss function to train variational autoencoders for Bayesian inverse problems. When the forward map is affine, we provide a theoretical proof of the convergence of the latent states of variational autoencoders to the posterior distribution of the model parameters. We validate this theoretical result through numerical tests and we compare the proposed variational autoencoder with the existing one in the literature both in terms of accuracy and generalization properties. Finally, we test the proposed variational autoencoder on a Laplace equation, with comparison to the original one and Markov Chains Monte Carlo. Show Chinese abstract

Abstract: 在其他用途中，神经网络是解决实时确定性和贝叶斯反问题的强大工具，其中变分自编码器作为一种特殊的神经网络，能够从观测数据中实现模型参数及其分布的贝叶斯估计，从而实现实时反问题的不确定性量化。 在本工作中，我们基于现有的研究[Goh, H. 等，机器学习研究会议论文集，2022]，提出了一种新颖的损失函数，用于训练变分自编码器解决贝叶斯反问题。 当正向映射是仿射的时候，我们提供了变分自编码器潜在状态收敛到模型参数后验分布的理论证明。 我们通过数值测试验证了这一理论结果，并在准确性和泛化性能方面将所提出的变分自编码器与文献中的现有方法进行了比较。 最后，我们将所提出的变分自编码器应用于拉普拉斯方程，并与原始方法和马尔可夫链蒙特卡洛方法进行了比较。