This thesis is at the intersection of Information Theory (IT), solar physics, signal processing and Machine Learning (ML). It discusses the evolution of astronomy, highlighting the influence of technological advancements that, together with statistics, and the rise of Big Data in modern astronomy, enable efficient space exploration. It emphasizes the challenges and opportunities presented by ML in analyzing vast astronomical datasets, particularly focusing on solar data from NASA’s IRIS satellite. The thesis explores IT and ML applications for several detection problems, classification in decentralized systems, and forecasting multivariate data, such as solar activity. It addresses the challenges of astronomical data, with the design of sufficient statistics and information bottleneck formulations. The thesis emphasizes the importance of these formulations to enhance the scientific value learned on large collected datasets that are often multidimensional, sparse, weekly annotated, and processed with limited dedicated hardware as, for instance, in space-based instruments.