Signal decomposition (SD) techniques are fundamental tools in signal processing, breaking down complex, non-stationary signals into their constituent components. This serves as a vital preliminary step that provides valuable insights into the underlying data and the systems that generated it. SD finds applications in a wide range of fields, such as, e.g., fault diagnosis, structural health monitoring, biomedical signal processing, and time-series analysis. By decomposing signals into their constituent parts, these techniques facilitate tasks like noise and artifacts removal, as well as the extraction of key features. In this talk, first, we review some of the most recent and sophisticated model-based SD approaches proposed for different types of signals, ranging from 1D signals [1, 2] to images [3, 4] up to scalar fields defined on manifolds [5]. These approaches rely on a variational formulation of the SD problem and a suitable choice / design of the ’norms’ enforcing the decomposition of the desired components. Then, a novel machine learning-based SD approach is presented [6], relying on a supervised end-to-end training of a transformer neural network architecture. The comparison between the results achieved by this and the variational approaches allows us to experimentally outline the potential of model-based and data-driven methods for SD. [1] Girometti, L., Lanza, A., and Morigi, S. Fractional Derivative Variational Model for Additive Signal Decomposition. Lecture Notes in Computer Science (2025) Vol. 15668 LNCS: 136-149. [2] Girometti, L., Huska, M., Lanza, A., and Morigi, S. Convex Predictor–Nonconvex Corrector Optimization Strategy with Application to Signal Decomposition. Journal of Optimization Theory and Applications (2024) 202(3): 1286-1325. [3] Girometti, L., Huska, M., Lanza, A., and Morigi, S. Quaternary Image Decomposition with Cross-Correlation-Based Multi-parameter Selection. Lecture Notes in Computer Science (2023) Volume 14009 LNCS: 120-133. [4] Girometti, L., Lanza, A., and Morigi, S. Ternary image decomposition with automatic parameter selection via auto- and cross-correlation. Advances in Computational Mathematics (2023) 49(1): 1. [5] Huska, M., Lanza, A., Morigi, S., and Selesnick, I. A convex-nonconvex variational method for the additive decomposition of functions on surfaces. Inverse Problems (2019) 35(12): 124008. [6] Lanza, A., Morigi, S., and Salti, S. Additive decomposition of 1D signals using transformers. Under revision.