Manufacturing of microstructures using a microfluidic device is a largely empirical effort due to the multi-physical nature of the fabrication process. As such, in moving toward autonomous manufacturing, models are desired that will predict microstructure attributes (e.g., size, porosity, and stiffness) based on known inputs, such as sheath and core fluid flow rates. Potentially more useful is the prospect of inputting desired microfiber features into a design model to extract appropriate manufacturing parameters. In this study, we demonstrate that deep neural networks (DNNs) trained with sparse datasets augmented by synthetic data can produce accurate predictive and design models to accelerate materials development. For our predictive model with known sheath and core flow rates and bath solution percentage, calculated solid microfiber dimensions are shown to be greater than 95% accurate, with porosity and Young's modulus exhibiting greater than 90% accuracy for a majority of conditions. Likewise, the design model is able to recover sheath and core flow rates with 95% accuracy when provided values for microfiber dimensions, porosity, and Young's modulus. As a result, DNN-based modeling of the microfiber fabrication process demonstrates high potential for reducing time to manufacture of microstructures with desired characteristics.