Modern weather forecasting: mathematical foundations, tools and current state of the art.

Modern weather forecasting relies on the integration of observational systems, numerical modeling, data assimilation, high-performance computing, and increasingly artificial intelligence techniques. This paper reviews the scientific and technological foundations of contemporary weather prediction, with particular emphasis on the mathematical and computational tools that support operational forecasting. Weather prediction is based on the numerical solution of the nonlinear partial differential equations governing atmospheric dynamics. The accuracy of these forecasts critically depends on the availability of observations collected from heterogeneous sources, including ground-based networks, weather radars, satellites, aircraft, and other remote sensing platforms. These data are combined with model forecasts through advanced data assimilation techniques, which provide dynamically consistent estimates of the atmospheric state and improve forecast quality. Special attention is devoted to nowcasting, one of the most challenging areas of modern meteorology. At lead times of a few minutes to several hours, rapidly evolving phenomena such as severe thunderstorms, hailstorms, and flash floods require the integration of high-frequency observations, radar extrapolation methods, ensemble prediction techniques, and machine learning approaches. Artificial intelligence is increasingly used to enhance the detection, tracking, and short-term prediction of convective systems, complementing traditional physics-based methodologies. The paper also discusses the requirements of a modern national forecasting system and presents the high-resolution ICON-2I modeling framework operational at the ItaliaMeteo Agency. Particular emphasis is placed on the role of high-performance computing infrastructures, ensemble forecasting systems, and kilometer-scale data assimilationmethods. Finally, current challenges and future developments are examined, including convective-scale prediction, uncertainty quantification, exascale computing, digital twins of the atmosphere, and the growing integration of artificial intelligence with numerical weather prediction.