Scientific machine learning

Abstract Scientific Machine Learning (SciML) is an emerging interdisciplinary field that integrates the strengths of scientific computing and machine learning to address complex modeling, simulation, and inference tasks in the natural and engineering sciences. This paper provides a concise overview of the foundational paradigms underlying SciML, where we highlight the complementary roles of physics-based models and data-driven methods. We discuss the core challenges, including data scarcity, physical consistency, high dimensionality, and computational cost and then introduce key methodological advances such as physics-informed neural networks, operator learning, hybrid modeling, and probabilistic approaches, each designed to address specific limitations of traditional methods. The techniques are illustrated through applications in fluid dynamics and turbulence modeling, automation and control, energy management, and sustainable mobility, demonstrating how SciML enables more accurate, efficient, and interpretable solutions. Finally, we outline open problems and future directions, and emphasize the need for theoretical understanding, scalable algorithms, and interdisciplinary collaboration.