Among the problems that prevent free-space optical communication systems from becoming a truly mainstream technology is beam wander, which is especially important for structured light beams since beam misalignment introduces additional crosstalk at the receiver. The paper suggests a recurrent neural network-based (RNN) solution to predict beam wander in free space optics (FSO). The approach uses past beam center of mass positions to predict future movement, significantly outperforming various prediction types. The proposed approach is demonstrated using under-sampled experimental data over a 260 m link as a worst-case and over-sampled simulated data as a best-case scenario. In addition to conventional Gaussian beams, Hermite- and Laguerre-Gaussian beam wander is also investigated. With a 20 to 40% improvement in error over naive and linear predictions, while predicting multiple samples ahead in typical situations and overall matching or outperforming considered predictions across all studied scenarios, this method could help mitigate turbulence-induced fading and has potential applications in intelligent re-transmits, quality of service, optimized error correction, maximum likelihood-type algorithms, and predictive adaptive optics.