Physics-informed neural networks (PINNs) provide a means of obtaining approximate solutions of partial differential equations and systems through the minimisation of an objective function which includes the evaluation of a residual function at a set of collocation points within the domain. The quality of a PINNs solution depends upon numerous parameters, including the number and distribution of these collocation points. In this paper we consider a number of strategies for selecting these points and investigate their impact on the overall accuracy of the method. In particular, we suggest that no single approach is likely to be "optimal" but we show how a number of important metrics can have an impact in improving the quality of the results obtained when using a fixed number of residual evaluations. We illustrate these approaches through the use of two benchmark test problems: Burgers' equation and the Allen-Cahn equation.

翻译：物理信息神经网络（PINNs）通过最小化一个目标函数来获得偏微分方程及方程组的近似解，该目标函数包含在域内一组配置点处对残差函数的评估。PINNs 解的质量取决于众多参数，包括这些配置点的数量与分布。本文探讨了若干选取这些配置点的策略，并研究了它们对方法整体精度的影响。特别地，我们认为单一方法不太可能成为“最优”选择，但我们展示了在使用固定数量残差评估时，若干重要指标如何能够对提升所得结果的质量产生影响。我们通过两个基准测试问题——Burgers 方程和 Allen-Cahn 方程——来阐释这些方法。