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ABSTRACT: In the past decade, machine learning methods (MLMs) for solving partial differential equations (PDEs) have gained significant attention as a novel numerical approach. Indeed, a tremendous number of research projects have surged that apply MLMs to various applications, ranging from geophysics to biophysics. This surge in interest stems from the ability of MLMs to rapidly predict solutions for complex physical systems, even those involving multi-physics phenomena, uncertainty, and real-world data assimilation. This trend has led many to hopefully think MLMs as a potential game-changer in PDE solving. However, despite the hopeful thinking on MLMs, there are still significant challenges to overcome. These include limitations compared to main numerical approaches, a lack of thorough analytical understanding of its accuracy, and the potentially long training times involved. In this talk, I will first assess the current state of MLMs for solving PDEs. Following this, we will explore the role mathematics can play to make MLMs a main numerical scheme.