Physics-informed neural networks (PINNs) offer a novel and efficient approach to solving partial differential equations (PDEs). Their success lies in the physics-informed loss, which trains a neural network to satisfy a given PDE at specific points and to approximate the solution. However, the solutions to PDEs are inherently infinite-dimensional, and the distance between the output and the solution is defined by an integral over the domain. Therefore, the physics-informed loss only provides a finite approximation, and selecting appropriate collocation points becomes crucial to suppress the discretization errors, although this aspect has often been overlooked. In this paper, we propose a new technique called good lattice training (GLT) for PINNs, inspired by number theoretic methods for numerical analysis. GLT offers a set of collocation points that are effective even with a small number of points and for multi-dimensional spaces. Our experiments demonstrate that GLT requires 2--20 times fewer collocation points (resulting in lower computational cost) than uniformly random sampling or Latin hypercube sampling, while achieving competitive performance.

翻译：物理信息神经网络（PINNs）提供了一种新颖且高效的方法来求解偏微分方程（PDEs）。其成功关键在于物理信息损失函数，该损失函数通过训练神经网络在特定点上满足给定偏微分方程并逼近解来实现。然而，偏微分方程的解本质上是无限维的，其输出与解之间的距离定义为区域上的一个积分。因此，物理信息损失函数仅提供有限逼近，而选择合适的配置点以抑制离散误差变得至关重要——尽管这一方面经常被忽视。本文提出了一种名为良好格点训练（GLT）的新技术，该技术受数值分析中数论方法的启发。GLT提供了一组配置点，即使在点数较少且面向多维空间时也能保持有效。我们的实验表明，与均匀随机采样或拉丁超立方采样相比，GLT所需的配置点数量减少了2至20倍（从而降低了计算成本），同时实现了具有竞争力的性能。