Deep learning has emerged as the predominant approach in machine learning and has achieved remarkable success in various domains such as computer vision and natural language processing. Its influence has progressively extended to numerous research areas within the fields of science and engineering. In this presentation, I will discuss our recent endeavors, which involve the integration of notions from scientific computing, such as reduced order modeling, with concepts from machine learning, such as meta-learning. The objective of our work is to develop data-driven solvers for parametric PDEs and explore their applications in inverse problems and design optimization.