Modern radio observatories often take the form of a large-N-small-D interferometer, where a large number of modest size reflector antennas are used to provide both high resolution (through the large spacing between them) as well as high sensitivity (due to the large collecting area). This topology only really became viable recently due to the enormous amount of computing power needed to correlate the signals between all the baselines becoming realistic with the growth in computing power of modern-day electronics. Since a substantial part of the cost of such a telescope is the individual reflector antenna systems, every effort should be made to maximize the sensitivity of each (nominally identical) antenna. Each percent of improvement in the sensitivity is roughly equal to having one more reflector available in a system of 100 antennas (when the experiment is sensitivity limited). To achieve the maximum possible sensitivity out of a reflector antenna design, numerical optimization of the reflector and feed geometry is required. The optimization goal should be the sensitivity of the system, and not some intermediate proxy, to guarantee the final design is near optimal. This requires fast evaluation of the simulated sensitivity of the antenna for each geometric perturbation requested by the optimization routine. The only way to achieve a tractable simulation time is by approximation of the sensitivity – which in this context must be done to an extremely high degree of accuracy to ensure that the last few percent of performance can be extracted from a nearly flat fitness function space around the optimum. This talk will discuss some of the surrogate-based approaches we used to design the reflector antennas of two of the largest radio telescope observatories ever developed; the Square Kilometre Array (SKA) in South Africa and the Next Generation Very Large Array (ngVLA) in the USA. We focus on two main breakthroughs: fast and accurate approximations of antenna temperature [1], as well as prediction of the presence of trapped modes in the feed antenna [2]. These serve to illustrate how there is still no substitute for good physical insight into the problem to achieve tractable high-quality designs of complex engineering devices.