SIAM Review, Volume 63, Issue 1, Page 208-228, January 2021.
Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, from an implementation point of view, PINNs solve inverse problems as easily as forward problems. We propose a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs. For pedagogical reasons, we compare the PINN algorithm to a standard finite element method. We also present a Python library for PINNs, DeepXDE, which is designed to serve both as an educational tool to be used in the classroom as well as a research tool for solving problems in computational science and engineering. Specifically, DeepXDE can solve forward problems given initial and boundary conditions, as well as inverse problems given some extra measurements. DeepXDE supports complex-geometry domains based on the technique of constructive solid geometry and enables the user code to be compact, resembling closely the mathematical formulation. We introduce the usage of DeepXDE and its customizability, and we also demonstrate the capability of PINNs and the user-friendliness of DeepXDE for five different examples. More broadly, DeepXDE contributes to the more rapid development of the emerging scientific machine learning field.

中文翻译：

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DeepXDE：用于求解微分方程的深度学习库

SIAM 评论，第 63 卷，第 1 期，第 208-228 页，2021 年 1 月。
深度学习在各种应用中取得了显着的成功；然而，它在求解偏微分方程 (PDE) 中的应用直到最近才出现。在这里，我们概述了物理信息神经网络 (PINN)，它使用自动微分将 PDE 嵌入到神经网络的损失中。PINN 算法简单，可以应用于不同类型的偏微分方程，包括积分微分方程、分数阶偏微分方程和随机偏微分方程。此外，从实现的角度来看，PINNs 解决逆问题和正问题一样容易。我们提出了一种新的基于残差的自适应细化 (RAR) 方法来提高 PINN 的训练效率。出于教学原因，我们将 PINN 算法与标准有限元方法进行了比较。我们还提供了一个用于 PINN、DeepXDE、它旨在既用作在课堂上使用的教育工具，也用作解决计算科学和工程问题的研究工具。具体来说，DeepXDE 可以解决给定初始和边界条件的正向问题，以及给定一些额外测量的逆向问题。DeepXDE 支持基于构造实体几何技术的复杂几何域，并使用户代码紧凑，与数学公式非常相似。我们介绍了 DeepXDE 的用法及其可定制性，并通过五个不同的示例展示了 PINNs 的能力和 DeepXDE 的用户友好性。更广泛地说，DeepXDE 有助于新兴科学机器学习领域的更快发展。