Coded-illumination can enable quantitative phase microscopy of transparent samples with minimal hardware requirements. Intensity images are captured with different source patterns and a non-linear phase retrieval optimization reconstructs the image. The non-linear nature of the processing makes optimizing the illumination pattern designs complicated. Traditional techniques for experimental design (e.g. condition number optimization, spectral analysis) consider only linear measurement formation models and linear reconstructions. Deep neural networks (DNNs) can efficiently represent the non-linear process and can be optimized over via training in an end-to-end framework. However, DNNs typically require a large amount of training examples and parameters to properly learn the phase retrieval process, without making use of the known physical models. Here, we aim to use both our knowledge of the physics and the power of machine learning together. We develop a new data-driven approach to optimizing coded-illumination patterns for a LED array microscope for a given phase reconstruction algorithm. Our method incorporates both the physics of the measurement scheme and the non-linearity of the reconstruction algorithm into the design problem. This enables efficient parameterization, which allows us to use only a small number of training examples to learn designs that generalize well in the experimental setting without retraining. We show experimental results for both a well-characterized phase target and mouse fibroblast cells using coded-illumination patterns optimized for a sparsity-based phase reconstruction algorithm. Our learned design results using 2 measurements demonstrate similar accuracy to Fourier Ptychography with 69 measurements.