IEEEThe awareness and practical benefits of behavioral modeling methods have been steadily growing in the antenna engineering community over the last decade or so. Undoubtedly, the most important advantage thereof is a possibility of a dramatic reduction of computational expenses associated with computer-aided design procedures, especially those relying on full-wave electromagnetic (EM) simulations. In particular, the employment of fast replacement models (surrogates) allows for repetitive evaluations of the antenna structure at negligible cost, thereby accelerating processes such as parametric optimization, multi-criterial design, or uncertainty quantification. Notwithstanding, a construction of reliable data-driven surrogates is seriously hindered by the curse of dimensionality and the need for covering broad ranges of geometry/material parameters, which is imperative from the perspective of design utility. A recently proposed constrained modeling approach with knowledge-based stochastic determination of the model domain addresses this issue to a large extent and has been demonstrated to enable quasi-global modeling capability while maintaining a low setup cost. This work introduces a novel technique that capitalizes on the domain confinement paradigm and incorporates deep-learning-based regression modeling to facilitate handling of highly-nonlinear antenna characteristics. The presented framework is demonstrated using three microstrip antennas and favorably compared to several state-of-the-art techniques. The predictive power of our models reaches remarkable two percent of a relative RMS error (averaged over the considered antenna structures), which is a significant improvement over all benchmark methods.