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）通过最小化包含残差函数在域内一组配点处评估的目标函数，为求解偏微分方程及其系统提供了一种近似方法。PINNs解的质量取决于多个参数，包括这些配点的数量与分布。本文探讨了若干配点选取策略，并研究了它们对整体方法精度的影响。具体而言，我们认为不存在单一"最优"方法，但展示了多项关键指标如何能够在固定残差评估次数下提升所得结果质量。我们通过两个基准测试问题（Burgers方程与Allen-Cahn方程）对这些方法进行了说明。