Physics-informed neural networks (PINN) is a extremely powerful paradigm used to solve equations encountered in scientific computing applications. An important part of the procedure is the minimization of the equation residual which includes, when the equation is time-dependent, a time sampling. It was argued in the literature that the sampling need not be uniform but should overweight initial time instants, but no rigorous explanation was provided for these choice. In this paper we take some prototypical examples and, under standard hypothesis concerning the neural network convergence, we show that the optimal time sampling follows a truncated exponential distribution. In particular we explain when the time sampling is best to be uniform and when it should not be. The findings are illustrated with numerical examples on linear equation, Burgers' equation and the Lorenz system.

翻译：物理信息神经网络（PINN）是一种极为强大的范式，用于求解科学计算应用中遇到的方程。该过程的一个重要部分是方程残差的最小化，当方程为时间相关时，这涉及时间采样。文献中曾提出，采样不必均匀，而应偏重初始时间点，但并未对此选择提供严格解释。在本文中，我们选取若干典型示例，并在关于神经网络收敛的标准假设下，证明最优时间采样遵循截断指数分布。特别地，我们解释了何时时间采样最佳为均匀分布，以及何时不应均匀分布。这些发现通过线性方程、伯格斯方程和洛伦兹系统的数值算例进行了说明。