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.