Toward the end-to-end optimization of the SWGO array layout

n this document we consider the problem of finding the optimal layout for the array of water Cherenkov detectors proposed by the SWGO collaboration to study very-high-energy gamma rays in the southern hemisphere. We develop a continuous model of the secondary particles produced by atmospheric showers initiated by high-energy gamma rays and protons, and build an optimization pipeline capable of identifying the most promising configuration of the detector elements. The pipeline employs stochastic gradient descent to maximize a utility function aligned with the scientific goals of the experiment. We demonstrate how the software is capable of finding the global maximum in the high-dimensional parameter space, and discuss its performance and limitations.