Recent years have witnessed the emergence of Scientific Machine Learning (SciML) as a new paradigm for the numerical treatment of differential equations. Complementing traditional discretization techniques, SciML leverages machine learning algorithms to address forward and inverse PDE problems, construct surrogate models, learn solution operators, and accelerate traditional numerical methods. Despite the remarkable success across a broad range of applications, many fundamental questions remain open regarding the mathematical foundations, convergence properties, computational efficiency, and generalization of these methods. Addressing these challenges naturally calls for concepts and techniques from high-order numerical methods. Variational formulations, accurate quadrature, approximation theory, adaptive sampling, high-order discretization schemes and rigorous error analysis offer powerful tools for improving both the theoretical understanding and practical performance of SciML algorithms. This minisymposium aims to bring together researchers working at the intersection of Scientific Machine Learning and Numerical Analysis to highlight open challenges, explore emerging synergies, and stimulate new research directions at the interface of these fields. We welcome contributions spanning theoretical analysis, algorithmic developments, and computational applications.