  Numerical approximations of partial differential equations (PDEs) are
routinely employed to formulate the solution of physics, engineering and
mathematical problems involving functions of several variables, such as the
propagation of heat or sound, fluid flow, elasticity, electrostatics,
electrodynamics, and more. While this has led to solving many complex
phenomena, there are some limitations. Conventional approaches such as Finite
Element Methods (FEMs) and Finite Differential Methods (FDMs) require
considerable time and are computationally expensive. In contrast, data driven
machine learning-based methods such as neural networks provide a faster, fairly
accurate alternative, and have certain advantages such as discretization
invariance and resolution invariance. This article aims to provide a
comprehensive insight into how data-driven approaches can complement
conventional techniques to solve engineering and physics problems, while also
noting some of the major pitfalls of machine learning-based approaches.
Furthermore, we highlight, a novel and fast machine learning-based approach
(~1000x) to learning the solution operator of a PDE operator learning. We will
note how these new computational approaches can bring immense advantages in
tackling many problems in fundamental and applied physics.
