Message-Passing Monte Carlo: Generating low-discrepancy point sets via Graph Neural Networks

Significance This article introduces Message-Passing Monte Carlo (MPMC), a machine learning approach for generating low-discrepancy point sets which are essential for efficiently filling space in a uniform manner, and thus play a central role in many problems in science and engineering. To accomplish this, MPMC utilizes tools from Geometric Deep Learning, specifically by employing graph neural networks. MPMC can be extended to a higher-dimensional case which further allows for generating custom-made points. Finally, MPMC point sets significantly outperform previous methods, achieving near-optimal discrepancy in practice for low dimension and small number of points, i.e., for which the optimal discrepancy can be determined. This advancement holds promise for enhancing efficiency in fields like scientific computing, computer vision, machine learning, and simulation.