Title: Neural Low-Discrepancy Sequences

Abstract:

Low-discrepancy points are engineered to distribute themselves uniformly across a space, a characteristic that offers substantial benefits for numerous challenges in science and engineering. These advantages are evident in fields ranging from numerical integration and computer graphics to machine learning, simulation, computer vision, and machine perception. While traditional methods for constructing low-discrepancy sets depend heavily on abstract algebra and number theory, the recently proposed Message-Passing Monte Carlo (MPMC) leverages machine learning to produce point sets with superior uniformity. However, MPMC is restricted to static point sets and cannot generate low-discrepancy sequences (LDS). LDS are critical for many applications because they ensure that every initial segment of the sequence maintains low discrepancy.

To overcome this constraint, we present Neural Low-Discrepancy Sequences (NeuroLDS), a novel machine learning framework designed specifically for generating finite LDS. Inspired by classical LDS methodologies, we utilize a neural network to map indices directly to points, ensuring that the generated sequences achieve minimal discrepancy across all prefixes. This is accomplished through a two-phase training strategy: first, we perform supervised learning to approximate classical constructions, followed by unsupervised fine-tuning aimed at minimizing prefix discrepancies. Our experiments show that NeuroLDS significantly surpasses all existing LDS constructions in terms of discrepancy metrics. Furthermore, we validate the practical utility of NeuroLDS in various domains, including scientific machine learning, robot motion planning, and numerical integration. These findings underscore the potential and widespread impact of Neural Low-Discrepancy Sequences. The source code is available at https://github.com/camail-official/neuro-lds.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC