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.

翻译：偏微分方程（PDEs）的数值近似通常用于表述涉及多变量函数的物理、工程和数学问题的解，例如热或声的传播、流体流动、弹性、静电学、电动力学等。尽管这有助于解决许多复杂现象，但仍存在一些局限性。传统方法如有限元法（FEMs）和有限差分法（FDMs）需要大量时间且计算成本高昂。相比之下，基于数据驱动的机器学习方法（如神经网络）提供了一种更快且相当精确的替代方案，并具有离散化不变性和分辨率不变性等优势。本文旨在全面洞察数据驱动方法如何补充传统技术以解决工程和物理问题，同时指出基于机器学习方法的一些主要缺陷。此外，我们重点介绍一种新颖且快速的基于机器学习的方法（加速约1000倍），用于学习偏微分方程算子解算子。我们将说明这些新的计算方法如何在解决基础和应用物理中的许多问题时带来巨大优势。