The development of efficient reduced order models (ROMs) from a deep learning perspective enables users to overcome the limitations of traditional approaches. One drawback of the techniques based on convolutional autoencoders is the lack of geometrical consistency when dealing with complex domains defined on unstructured meshes. The present work proposes a framework for nonlinear model order reduction based on Graph Convolutional Autoencoders (GCA) to exploit emergent patterns in different physical problems, including those showing bifurcating behavior, high-dimensional parameter space, slow Kolmogorov-decay, and varying domains [1]. Our methodology extracts the latent space’s evolution while introducing geometric priors, possibly alleviating the learning process through up- and down-sampling operations. Among the advantages, we highlight the high generalizability in the low-data regime and the great speedup. Moreover, we will present a novel graph feedforward network (GFN), extending the GCA approach to exploit multifidelity data, leveraging graph-adaptive weights, enabling large savings, and providing computable error bounds for the predictions [2]. This way, we overcome the limitations of the up- and down-sampling procedures by building a resolution-invariant GFN-ROM strategy capable of training and testing on different mesh sizes, resulting in a more lightweight and flexible architecture. Finally, we will show preliminary results concerning the time-extrapolation regime for dynamical systems in a Deep Operator Networks (DeepONet) framework, integrating the graph-based architectures with core-splitting tensor train decomposition and operator inference to learn the temporal evolution [3]. References [1] Pichi, F., Moya, B. and Hesthaven, J.S. (2024) [2] Morrison, O.M., Pichi, F. and Hesthaven, J.S. (2024) [3] Chen, Y., Pichi, F., Gao, Z., and Rozza, G. (2025)