Gaussian process (GP) is a very popular machine learning method for online surrogate model-assisted antenna design optimization. Despite many successes, two improvements are important for GP-based antenna global optimization methods, including (1) the convergence speed (i.e., the number of necessary electromagnetic simulations to obtain a high-performance design), and (2) the GP model training cost when there are several tens of design variables and/or specifications. In both aspects, state-of-the-art GP-based methods show practical but not desirable performance. Therefore, a new method, called self-adaptive Bayesian neural network surrogate model-assisted differential evolution for antenna optimization (SB-SADEA), is presented in this paper. The key innovations include: (1) The introduction of the Bayesian neural network (BNN)-based antenna surrogate modeling method into this research area, replacing GP modeling, and (2) a bespoke self-adaptive lower confidence bound method for antenna design landscape making use of the BNN-based antenna surrogate model. The performance of SB-SADEA is demonstrated by two challenging design cases, showing considerable improvement in terms of both convergence speed and machine learning cost compared to state-of-the-art GP-based antenna global optimization methods.