Mathematical and Deep Learning Methods in High-Energy Physics: From Temporal Variability in M82 ULXs to Spatial Features in Planck Maps

Modern Astronomy gathers data from a plethora of instruments from the ground and from space, spanning the entire electromagnetic spectrum and even some non-electromagnetic carriers of information, such as neutrinos and gravitational waves. The quality and quantity of data is rapidly increasing over time, but analysis techniques are often lagging behind in terms of fully taking advantage of this complexity. On the one hand, this is expected: popular analysis techniques receive continuous algorithmic development, making the analysis extremely fast. This is the case of the Fast Fourier Transform, which revolutionized frequency analysis. Today, the power spectral density is still routinely modeled using periodograms calculated through an FFT, despite its very well-known limitations in terms of spectral resolution and rather strict assumptions. Renouncing the FFT means increasing the timescales of analysis by many orders of magnitude.One class of analysis tools that have received considerable development is of course Machine Learning, a portmanteau for many different techniques that span from classic algebraic manipulation to algorithms imitating in some way the functioning of the brain (e.g. neural networks). This thesis addresses the challenge of extracting and characterizing faint signals from the wealth of modern astrophysical data, focusing in particular on two major problems in observational astronomy. On one hand, we investigate the timing properties of cosmic X-ray sources, especially ultraluminous X-ray sources (ULXs) in starburst galaxies, aiming to characterize quasi-periodic variability that reflects the physics of accretion onto compact objects. On the other hand, we develop a machine-learning method to detect and classify sources in wide-field infrared survey maps, where instrumental noise and foregrounds often obscure faint signals. In both cases, the goal is to move beyond traditional techniques—such as naive Fourier analysis of evenly sampled data or simple thresholding of images—to apply advanced statistical cleaning, bootstrap resampling, and neural-network algorithms that leverage the high information content of modern observations. By combining rigorous time-series analysis with contemporary deep learning tools, this work seeks to improve the reliability of signal detection in challenging astrophysical datasets.