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 the GP-based antenna global optimization methods, including: 1) the convergence speed (i.e., the number of necessary electromagnetic (EM) 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, the state-of-the-art GP-based methods show practical but not desirable performance. Therefore, a new method, called the self-adaptive Bayesian neural network surrogate-model-assisted differential evolution (DE) for antenna optimization (SB-SADEA), is presented in this article. 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 (LCB) 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 with the state-of-the-art GP-based antenna global optimization methods.