Operator learning aims to discover properties of an underlying dynamical system or partial differential equation (PDE) from data. Here, we present a step-by-step guide to operator learning. We explain the types of problems and PDEs amenable to operator learning, discuss various neural network architectures, and explain how to employ numerical PDE solvers effectively. We also give advice on how to create and manage training data and conduct optimization. We offer intuition behind the various neural network architectures employed in operator learning by motivating them from the point-of-view of numerical linear algebra.

翻译：算子学习旨在从数据中发现潜在动力系统或偏微分方程（PDE）的特性。本文提供了一种逐步的算子学习指南。我们阐述了适用于算子学习的问题类型及PDE形式，讨论了多种神经网络架构，并解释了如何有效利用数值PDE求解器。此外，我们还就如何创建与管理训练数据、开展优化提出了建议。通过从数值线性代数的视角进行动机阐释，我们对算子学习中使用的各种神经网络架构背后的直观原理进行了说明。