This lecture will focus on Scientific Machine Learning (SciML), a fusion of traditional scientific computing and modern machine learning techniques. We will begin by introducing the core concepts of SciML, distinguishing it from conventional numerical analysis and machine learning, and highlighting its unique capabilities in handling scientific and engineering problems. Key methodologies such as Physics-Informed Neural Networks (PINNs), Operator Networks, and spectral coefficient learning will be thoroughly examined, with an emphasis on their theoretical foundations and convergence analysis. The lecture will demonstrate solving fluid dynamics problems with PINNs, finite element operator networks, and spectral operator networks for precise numerical solutions. If time permits, we will explore extensions of SciML, such as inverse design via deep generative models.
