It is computationally intensive and time-consuming to perform a large number of CFD simulations in the process of airfoil optimization. This paper develops an airfoil inverse design method using the proper orthogonal decomposition (POD) and back propagation based neural network (BPNN). The optimization process of this method is as follows: First a sample set of airfoil shapes in the design space are generated through Hicks-Henne parameterization, and the flow fields of the sample airfoils are solved by Xfoil and Fluent. Then two POD models of the airfoil pressure coefficients and the geometric shapes are respectively built, and the corresponding base modal coefficients are obtained. Finally, the BPNN is used to map the base modal coefficients of the pressure coefficients to the base modal coefficients of the geometric shapes, in order to achieve rapid prediction of the specified geometric shape under a given pressure coefficient distribution. The results of the test example at subsonic and transonic shows: a two-layer POD+BPNN model based on 200 samples can realize the prediction of the airfoil with target pressure coefficient distribution, and meet the precision requirement of airfoil inverse design.