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 still significant limitations. Conventional approaches such as Finite Element Methods (FEMs) and Finite Differential Methods (FDMs) require considerable time and are computationally expensive. In contrast, machine learning-based methods such as neural networks are faster once trained, but tend to be restricted to a specific discretization. This article aims to provide a comprehensive summary of conventional methods and recent machine learning-based methods to approximate PDEs numerically. Furthermore, we highlight several key architectures centered around the neural operator, a novel and fast approach (1000x) to learning the solution operator of a PDE. 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（PDE）的数值逼近常规用于构建涉及多个变量函数的物理、工程和数学问题的解，例如热传导或声波传播、流体流动、弹性力学、静电学、电动力学等。尽管这已成功解决许多复杂现象，但仍存在显著局限性。传统方法如有限元法（FEM）和有限差分法（FDM）需要大量时间且计算成本高昂。相比之下，基于机器学习的方法（如神经网络）在训练完成后速度更快，但通常局限于特定的离散化格式。本文旨在全面综述传统方法和近期基于机器学习的PDE数值逼近方法。此外，我们重点介绍围绕神经算子（一种学习PDE解算子的新型快速方法，速度提升达1000倍）的若干关键架构。我们将阐述这些新型计算方法如何为基础物理和应用物理中的诸多问题带来巨大优势。