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.

本文研究了物理信息神经网络在科学计算应用中解决方程时的残差最小化问题，通过标准假设下的神经网络收敛性，证明了最优时间采样符合截断指数分布，并给出了在线性方程、Burgers' 方程和 Lorenz 系统中最佳时间采样的解释和数值实例。

物理信息神经网络中的最佳时间采样

Optimal time sampling in physics-informed neural networks

Training Neural Ordinary Differential Equations (ODEs) is often
computationally expensive. Indeed, computing the forward pass of such models
involves solving an ODE which can become arbitrarily complex during training.
Recent works have shown that regularizing the dynamics of the ODE can partially
alleviate this. In this paper we propose a new regularization technique:
randomly sampling the end time of the ODE during training. The proposed
regularization is simple to implement, has negligible overhead and is effective
across a wide variety of tasks. Further, the technique is orthogonal to several
other methods proposed to regularize the dynamics of ODEs and as such can be
used in conjunction with them. We show through experiments on normalizing
flows, time series models and image recognition that the proposed
regularization can significantly decrease training time and even improve
performance over baseline models.

本文提出一种新的正则化技术：在训练过程中随机抽样 ODE 的结束时间，实验证明该技术能够在多个任务中显著减少训练时间并提高性能