The development of efficient surrogates for PDEs is a critical step towards scalable modeling of complex, multiscale systems-of-systems. We use transfer learning with multilevel data to train a deep convolutional NN (CNN)-based surrogate model, which significantly reduces the cost of data generation relative to a conventional approach. We show that transfer learning on a mixture of high- and low-fidelity training data—obtained with a two-dimensional PDE and its one-dimensional approximation, respectively—reduces the cost of data generation without reducing performance of the surrogate.