Neural Operator: Is data all you need to model the world? An insight into the impact of Physics Informed Machine Learning
This addresses computational inefficiencies in physics and engineering problems, offering a faster alternative, though it appears incremental as it builds on existing machine learning techniques.
The paper tackles the limitations of conventional PDE solvers like FEM and FDM by exploring data-driven machine learning methods, highlighting a novel approach that achieves approximately 1000x speedup in learning PDE solution operators.
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.