As a general uncertainty   quantification framework of inverse problems of partial differential   equations (IPofPDEs), the Bayesian inverse method has attracted many   researchers' attention. One of the critical obstacles to applying the   Bayesian inverse approach is how to efficiently compute the statistical   quantities (e.g., posterior mean and variances). Similar difficulties also   meet in the investigations of uncertainty quantification of machine learning   models. In the machine learning community, the researchers proposed the   variational inference (VI) approach, which balances accuracy and efficiency.   However, due to the infinite-dimensional formulation of the IPofPDEs, there   are few investigations on VI approaches to IPofPDEs. In this talk, we briefly   introduce the main ideas of VI methods. Then we construct the   infinite-dimensional mean-field based VI approach for general linear problems   and the infinite-dimensional transformation based VI approach for nonliear   inverse problems. Both of the classical and neural network related VI methods   are discussed under the circumstances of solving IPofPDEs.