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Coded illumination can enable quantitative phase microscopy of transparent samples with minimal hardware requirements. Intensity images are captured with different source patterns, then a nonlinear phase retrieval optimization reconstructs the image. The nonlinear nature of the processing makes optimizing the illumination pattern designs complicated. The traditional techniques for the experimental design (e.g., condition number optimization, and spectral analysis) consider only linear measurement formation models and linear reconstructions. Deep neural networks (DNNs) can efficiently represent the nonlinear 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. In this paper, we aim to use both our knowledge of the physics and the power of machine learning together. We propose a new data-driven approach for optimizing coded-illumination patterns for an LED array microscope for a given phase reconstruction algorithm. Our method incorporates both the physics of the measurement scheme and the nonlinearity 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 two measurements demonstrate similar accuracy to Fourier ptychography with 69 measurements.