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Phase-field modeling is an effective but computationally expensive method for capturing the mesoscale morphological and microstructure evolution in materials. Hence, fast and generalizable surrogate models are needed to alleviate the cost of computationally taxing processes such as in optimization and design of materials. The intrinsic discontinuous nature of the physical phenomena incurred by the presence of sharp phase boundaries makes the training of the surrogate model cumbersome. We develop a framework that integrates a convolutional autoencoder architecture with a deep neural operator (DeepONet) to learn the dynamic evolution of a two-phase mixture and accelerate time-to-solution in predicting the microstructure evolution. We utilize the convolutional autoencoder to provide a compact representation of the microstructure data in a low-dimensional latent space. DeepONet, which consists of two sub-networks, one for encoding the input function at a fixed number of sensors locations (branch net) and another for encoding the locations for the output functions (trunk net), learns the mesoscale dynamics of the microstructure evolution from the autoencoder latent space. The decoder part of the convolutional autoencoder then reconstructs the time-evolved microstructure from the DeepONet predictions. The trained DeepONet architecture can then be used to replace the high-fidelity phase-field numerical solver in interpolation tasks or to accelerate the numerical solver in extrapolation tasks.