Space weather is the study of the near-Earth impact of the processes on the Sun. In a this era of technology, space weather emerges as a crucial area of study wherein the state of the near-Earth conditions need to be modeled and forecasted. Crucially, the advent of machine learning provides us with opportunities to study this subject in a rather comprehensive manner. In this chapter, we treat the the processes arising from Sun to the impact on the ground as a set of coupled systems, which can be studied in isolation or in a coupled manner. We define these individual systems in terms of their state vector $\vec{S}$, which provides us with an opportunity to define the physical variables determining the system. This also allows us to consider the systems in terms of their observables, obtained through the action of operators as $\vec{O}[\vec{S}]$. In such a formalism, the machine learning algorithms serve as {\it learned operators}, defined generally as inference ($\mathcal{I}$), propagation ($\mathcal{P}$), and emulation ($\mathcal{E}$). Through such a formalism, we present how machine learning helps advance our understanding of the physical systems under study, while also accelerating the development of operational models for space weather forecasting.
Francesco Marchetti, Sabrina Guastavino, Daniel Collin
High-speed solar wind streams (HSSs) originating from coronal holes (CHs) are an important driver of recurrent geomagnetic activity. Since HSSs are associated to coronal structures visible in extreme-ultraviolet observations, this provides a natural basis for forecasting several days in advance. In this work, we investigate whether the topology of coronal structures observed in extreme ultraviolet (EUV) images contains predictive information on the subsequent occurrence of CH-driven HSSs at Earth. To this end, we consider Topological Data Analysis (TDA), and in particular persistent homology, to classify EUV solar images according to the occurrence of HSSs at Earth four days later. The target is a binary classification of HSS occurrence at a four-day lead time, defined using an HSS interval catalog and solar wind speed thresholds. Several combinations of topological representations and classifiers are evaluated under a common cross-validation framework. The results indicate that the topology of EUV coronal structures alone contains predictive information on subsequent CH-driven HSS occurrence, despite the absence of explicit CH geometric descriptors in the model input. To investigate the origin of this predictive information, we develop a model-level explainability framework based on cross-validation-aggregated feature weights and spatial evidence maps. A fully traced example connects persistence diagrams and persistence-image features to the classifier output and to their spatial localization in the original EUV observation. The proposed framework provides a transparent and computationally lightweight approach for investigating the relationship between coronal topology and subsequent high-speed solar wind streams.
The solar interior and atmosphere constitute a strongly coupled, multiscale dynamical system defined by continuous temporal evolution. Predicting the state of this system requires modeling physical interactions that span disparate spatial and temporal scales, constrained by heterogeneous and sparse data. This chapter formalizes the Sun within the mathematical framework of dynamical systems theory, defining a global state vector ($\vec{S}$) governed by thermodynamical, kinetic, and magnetic properties. Because solar observations represent projected, multi-modal, and partially observable measurements ($\vec{O}[\vec{S}]$) of this hidden state, pure physics-based deduction is fundamentally constrained. We present a unifying thesis: the primary role of machine learning in heliophysics is to act as a structured family of mathematical operators---specifically Inference ($\mathcal{I}$), Emulation ($\mathcal{E}$), Propagation ($\mathcal{P}$), and Control ($\mathcal{C}$)---that reconstruct, simulate, and couple this global state vector across disparate physical layers. By framing the vast, multi-disciplinary literature within this unified state-space taxonomy, we demonstrate how data-driven techniques bridge observational gaps to resolve hidden multiscale couplings and dramatically accelerate the simulation of non-linear processes, establishing a rigorous pathway to predict the global evolution of the Sun.