Partial differential equations play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be approximated numerically. The finite element method, for instance, is a usual standard methodology to do so. The recent success of deep neural networks at various approximation tasks has motivated their use in the numerical solution of PDEs. These so-called physics-informed neural networks and their variants have shown to be able to successfully approximate a large range of partial differential equations. So far, physics-informed neural networks and the finite element method have mainly been studied in isolation of each other. In this work, we compare the methodologies in a systematic computational study. Indeed, we employ both methods to numerically solve various linear and nonlinear partial differential equations: Poisson in 1D, 2D, and 3D, Allen-Cahn in 1D, semilinear Schr\"odinger in 1D and 2D. We then compare computational costs and approximation accuracies. In terms of solution time and accuracy, physics-informed neural networks have not been able to outperform the finite element method in our study. In some experiments, they were faster at evaluating the solved PDE.

翻译：偏微分方程在物理、生物及其他科学领域中众多过程和系统的数学建模中扮演着基础性角色。为模拟此类过程和系统，通常需要对偏微分方程的解进行数值近似。例如，有限元方法是实现这一目标的常规标准方法。深度神经网络在各类近似任务中的近期成功，推动了其在偏微分方程数值求解中的应用。这些所谓的物理信息神经网络及其变体已被证明能够成功逼近大范围的偏微分方程。迄今为止，物理信息神经网络与有限元方法主要是在彼此孤立的研究中进行的。本研究通过系统的计算比较，对这两种方法进行了对比。具体地，我们采用这两种方法数值求解多种线性和非线性偏微分方程：一维、二维及三维泊松方程，一维艾伦-卡恩方程，一维及二维半线性薛定谔方程。随后，我们比较了计算成本和近似精度。在求解时间和精度方面，我们的研究中物理信息神经网络未能超越有限元方法。在某些实验中，它们在评估已求解的偏微分方程时速度更快。