Ordinary and partial differential equations (DE) are used extensively in scientific and mathematical domains to model physical systems. Current literature has focused primarily on deep neural network (DNN) based methods for solving a specific DE or a family of DEs. Research communities with a history of using DE models may view DNN-based differential equation solvers (DNN-DEs) as a faster and transferable alternative to current numerical methods. However, there is a lack of systematic surveys detailing the use of DNN-DE methods across physical application domains and a generalized taxonomy to guide future research. This paper surveys and classifies previous works and provides an educational tutorial for senior practitioners, professionals, and graduate students in engineering and computer science. First, we propose a taxonomy to navigate domains of DE systems studied under the umbrella of DNN-DE. Second, we examine the theory and performance of the Physics Informed Neural Network (PINN) to demonstrate how the influential DNN-DE architecture mathematically solves a system of equations. Third, to reinforce the key ideas of solving and discovery of DEs using DNN, we provide a tutorial using DeepXDE, a Python package for developing PINNs, to develop DNN-DEs for solving and discovering a classic DE, the linear transport equation.

翻译：常微分方程和偏微分方程被广泛应用于科学与数学领域，用于建模物理系统。现有文献主要关注基于深度神经网络（DNN）的方法来求解特定微分方程或一类微分方程。具有使用微分方程模型历史的研究团体可能将基于DNN的微分方程求解器（DNN-DEs）视为当前数值方法的更快且可迁移的替代方案。然而，目前缺乏系统性的综述来详细描述DNN-DE方法在物理应用领域中的使用情况，以及指导未来研究的通用分类体系。本文对先前工作进行综述与分类，并为工程和计算机科学领域的高年级从业者、专业人士及研究生提供教育性指导。首先，我们提出一种分类体系，用于梳理DNN-DE框架下研究的微分方程系统领域。其次，我们考察物理信息神经网络（PINN）的理论与性能，以展示这一具有影响力的DNN-DE架构如何从数学角度求解方程组。第三，为强化使用DNN求解与发现微分方程的核心思想，我们提供使用DeepXDE（一个用于开发PINN的Python包）的教程，以开发DNN-DE来求解并发现经典微分方程——线性输运方程。