In the present work, we propose a neural-network-based data-inversion approach to reduce structured contamination sources, with a particular focus on the mapmaking for Planck High Frequency Instrument data and the removal of large-scale systematic effects within the produced sky maps. The removal of contamination sources is made possible by the structured nature of these sources, which is characterized by local spatiotemporal interactions producing couplings between different spatiotemporal scales. We focus on exploring neural networks as a means of exploiting these couplings to learn optimal low-dimensional representations, which are optimized with respect to the contamination-source-removal and mapmaking objectives, to achieve robust and effective data inversion. We develop multiple variants of the proposed approach, and consider the inclusion of physics-informed constraints and transfer-learning techniques. Additionally, we focus on exploiting data-augmentation techniques to integrate expert knowledge into an otherwise unsupervised network-training approach. We validate the proposed method on Planck High Frequency Instrument 545 GHz Far Side Lobe simulation data, considering ideal and nonideal cases involving partial, gap-filled, and inconsistent datasets, and demonstrate the potential of the neural-network-based dimensionality reduction to accurately model and remove large-scale systematic effects. We also present an application to real Planck High Frequency Instrument 857 GHz data, which illustrates the relevance of the proposed method to accurately model and capture structured contamination sources, with reported gains of up to one order of magnitude in terms of performance in contamination removal. Importantly, the methods developed in this work are to be integrated in a new version of the SRoll algorithm (SRoll3), and here we describe SRoll3 857 GHz detector maps that were released to the community.