Antenna synthesis is becoming increasingly challenging given the tight requirements for C-SWAP (cost, size, weight and power) reduction while maintaining stringent electromagnetic performance specifications. While recent reports have investigated developments in AI (primarily deep learning) to address this challenge, such methods have limited capability for multi-objective constrained inverse design and the ability to handle large shape sets. We present a multi-objective antenna synthesis methodology employing deep neural network surrogate-driven evolutionary algorithm. Specifically, we propose a branched-output convolutional neural network architecture for creating the surrogate model and showing its ability to predict multiple electromagnetic parameters over an operating frequency range. The proposed methodology and its synthesis capabilities are showcased by considering the set of polygonal-patch antennas for operation in the 2 to 3 GHz spectral band. Specifically, we report the design, fabrication and experimental characterization of three polygon-shaped patch antennas, each fulfilling different objectives, one being a single band with resonance at 2.45 GHz, the second antenna is dual-band with resonance at 2.2 GHz and 2.9 GHz, and the third one being wide-band centered around 2.7 GHz. The reported methodology enables rapid synthesis (in seconds), produces verifiable sound designs and is promising for furthering data-driven design methodologies for electromagnetic wave device synthesis.