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We design and train a neural network (NN) model to efficiently predict the infrared spectra of interstellar polycyclic aromatic hydrocarbons (PAHs) with a computational cost many orders of magnitude lower than what a first-principles calculation would demand. The input to the NN is based on the Morgan fingerprints extracted from the skeletal formulas of the molecules and does not require precise geometrical information such as interatomic distances. The model shows excellent predictive skill for out-of-sample inputs, making it suitable for improving the mixture models currently used for understanding the chemical composition and evolution of the interstellar medium. We also identify the constraints to its applicability caused by the limited diversity of the training data and estimate the prediction errors using a ensemble of NNs trained on subsets of the data. With help from other machine-learning methods like random forests, we dissect the role of different chemical features in this prediction. The power of these topological descriptors is demonstrated by the limited effect of including detailed geometrical information in the form of Coulomb matrix eigenvalues.