Phased array weather radars, particularly with high temporal resolution, essentially need a robust and fast beamformer to accurately estimate precipitation profiles such as reflectivity and Doppler velocity. In this paper, we introduce a neural-network-based beamformer to address this problem. In particular, the optimum weight vector is computed by modeling the problem as a three-layer radial basis function neural network (RBFNN), which is trained with I/O pairs obtained from the optimum Wiener solution. The RBFNN was chosen because of its characteristic of accurate approximation and good generalization, and its robustness against interference and noise. The proposed RBFNN beamforming method is compared with traditional beamforming methods, namely, Fourier beamforming (FR), Capon beamforming, and the flower pollination algorithm (FPA), which is a recently proposed nature-inspired optimization algorithm. It is shown that the RBFNN approach has nearly optimal performance in various precipitation radar signal simulations relative to the rival methods. The validity of the RBFNN beamformer is demonstrated by using real weather data collected by the phased array radar (PAR) at Osaka University, and compared with, in addition to the FR and FPA methods, the minimum mean square error beamforming method. It is shown that the RBFNN method estimates the reflectivity of the PAR at Osaka University with less clutter level than those of the other three methods.