Designing composite materials according to the need of applications is fundamentally a challenging and time-consuming task. A deep neural network-based computational framework is developed in this work to solve the forward (predictive) and the inverse (generative) composite design problem. The predictor model is based on the popular convolution neural network architecture and trained with the help of finite element simulations. Conventionally, a large amount of training data is required for accurate prediction from neural network models. A data augmentation strategy is proposed in this study which significantly saves computational resources in the training phase. It shown that the data augmentation approach is general and can be used in any setting involving periodic microstructures. We next use, the property predictor model as a feedback mechanism in the neural network-based generator model. The proposed predictive-generative model is used to obtain the composite microstructure for various requirements such as maximization of elastic properties, specified elastic constants, etc. The efficacy of the proposed predictive-generative model is demonstrated by solving certain class of problems. It is envisaged that the developed model coupled with data augmentation strategy will significantly reduce the cost and time associated with the composite material designing process for varying application requirements.