The problem of beam alignment and tracking in high mobility scenarios such as high-speed railway (HSR) becomes extremely challenging, since large overhead cost and significant time delay are introduced for fast time-varying channel estimation. To tackle this challenge, we propose a learning-aided beam prediction scheme for HSR networks, which predicts the beam directions and the channel amplitudes within a period of future time with fine time granularity, using a group of observations. Concretely, we transform the problem of high-dimensional beam prediction into a two-stage task, i.e., a low-dimensional parameter estimation and a cascaded hybrid beamforming operation. In the first stage, the location and speed of a certain terminal are estimated by maximum likelihood criterion, and a data-driven data fusion module is designed to improve the final estimation accuracy and robustness. Then, the probable future beam directions and channel amplitudes are predicted, based on the HSR scenario priors including deterministic trajectory, motion model, and channel model. Furthermore, we incorporate a learnable non-linear mapping module into the overall beam prediction to allow non-linear tracks. Both of the proposed learnable modules are model-based and have a good interpretability. Compared to the existing beam management scheme, the proposed beam prediction has (near) zero overhead cost and time delay. Simulation results verify the effectiveness of the proposed scheme.