Advanced computational methods and machine learning (ML) techniques are playing a crucial role in solving complex problems across various scientific and engineering disciplines. This minisymposium will explore the mathematical and computational foundations of modern numerical approaches to scientific computing, highlighting advanced algorithms, reduced-order modeling techniques, and hybrid methods that integrate data-driven learning with physics-based models. The goal is to demonstrate how these approaches enhance computational efficiency, improve predictive accuracy, and expand the applicability of simulation techniques in scientific research. The session will focus on the synergy between traditional mathematical models and ML techniques, showcasing how their combination leads to more efficient, scalable, and precise simulations, ultimately addressing real-world scientific challenges. Topics of interest include, but are not limited to: • Sparse and low-rank approximations in data-driven modeling. • Hybrid approaches combining multiscale mathematical models with data integration. • Mathematical foundations of scientific ML in computational science and numerical simulations. • ML techniques for physics-informed modeling, uncertainty quantification, and data assimilation. • Advanced numerical methods for solving high-dimensional PDEs and optimization problems. The session will illustrate how these advanced techniques are driving progress in various fields, including computational biology, environmental sciences, and beyond.