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

In 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.