In order to solve the expensive multi-objective optimization problems, whose accurate mathematical models are difficult to obtain and which have high experimental cost in evaluation experiments, an expensive multi-objective evolutionary algorithm based on the $R2$ indicator ($R2$-EMO) is proposed, which uses the Gaussian process to replace the real model to evaluate the performance of the individual by calculating the $R2$ indicator. In selecting the evaluation points, the $R2$ indicator considers the convergence and the diversity of population. And it takes the expectation and mean square error into account, which strengthens the ability of exploration. Meanwhile, a double-archive management strategy is carried out and updated in each iteration. One is used to store the non-dominated individuals and the other is used to build the surrogate process. Compared with the ParEGO, KRVEA, MOEAD and NSGAIII, $R2$-EMO algorithm has achieved a better performance and can converge to the Pareto front rapidly in dealing with expensive multi-objective optimization problems.