Neural Operator: Is data all you need to model the world? An insight into the paradigm of data-driven scientific ML.

Abstract

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 Difference 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, in particular, focus on neural operators, which 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 open problems of machine learning-based approaches. We will note how these new computational approaches can bring immense advantages in tackling many problems in fundamental and applied physics.