In recent years, advances in computational power and data availability have propelled machine learning (ML) to the forefront of scientific computing, complementing and enhancing traditional methods. This lecture explores the integration of ML with numerical methods for multi-scale problems, highlighting new perspectives in scientific computing. Key topics include the synergy between ML and numerical analysis, convergence analysis from both fields, and error estimation techniques. Additionally, we will discuss how neural networks contribute to solving complex partial differential equations (PDEs) efficiently. By leveraging these approaches, we can tackle challenging scientific problems with greater accuracy and computational efficiency. If time permits, we will also touch on emerging topics such as foundation models and ML-driven material discovery.