Generating quasirandom points with high uniformity is a fundamental task in many fields. Existing number-theoretic approaches produce evenly distributed points in ${[0,1]}^d$ in asymptotic sense but may not yield a good distribution for a given set size. It is also difficult to extend those techniques to other geometries like a disk or a manifold. In this paper, we present a novel physics-informed framework to transform a given set of points into a distribution with better uniformity. We model each point as a particle and assign the system with a potential energy. Upon minimizing the energy, the uniformity of distribution can be improved correspondingly. Two kinds of schemes are introduced: one based on molecular dynamics and another based on deep neural networks. The new physics-informed framework serves as a black-box transformer that is able to improve given distributions and can be easily extended to other geometries such as disks, spheres, complex manifolds, etc. Various experiments with different geometries are provided to demonstrate that the new framework is able to transform poorly distributed input into one with superior uniformity.

生成具有高一致性的拟随机点是许多领域的一项基本任务。现有的数论方法产生均匀分布的点${[0,1]}^d$在渐近意义上，但对于给定的集合大小可能不会产生良好的分布。也很难将这些技术扩展到其他几何形状，如圆盘或歧管。在本文中，我们提出了一种新颖的基于物理的框架，可以将给定的一组点转换为具有更好均匀性的分布。我们将每个点建模为一个粒子，并为系统分配一个势能。在最小化能量的情况下，分布的均匀性可以相应提高。介绍了两种方案：一种基于分子动力学，另一种基于深度神经网络。新的基于物理的框架充当了一个黑盒转换器，能够改善给定的分布，并且可以轻松扩展到其他几何形状，例如圆盘、球体、复杂流形等。