There is increasing interest in learning how human brain networks vary as a function of a continuous trait, but flexible and efficient procedures to accomplish this goal are limited. We develop a Bayesian semiparametric model, which combines low-rank factorizations and flexible Gaussian process priors to learn changes in the conditional expectation of a network-valued random variable across the values of a continuous predictor, while including subject-specific random effects.The formulation leads to a general framework for inference on changes in brain network structures across human traits, facilitating borrowing of information and coherently characterizing uncertainty. We provide an efficient Gibbs sampler for posterior computation along with simple procedures for inference, prediction and goodness-of-fit assessments. The model is applied to learn how human brain networks vary across individuals with different intelligence scores. Results provide interesting insights on the association between intelligence and brain connectivity, while demonstrating good predictive performance.Source code implemented in R and data are available at https://github.com/wangronglu/BNRR.email@example.com.Supplementary data are available at Bioinformatics online.