Neural Low-Discrepancy Sequences

Low-discrepancy points are designed to efficiently fill the space in a uniform manner. This uniformity is highly advantageous in many problems in science and engineering, including in numerical integration, computer vision, machine perception, computer graphics, machine learning, and simulation. Whereas most previous low-discrepancy constructions rely on abstract algebra and number theory, Message-Passing Monte Carlo (MPMC) was recently introduced to exploit machine learning methods for generating point sets with lower discrepancy than previously possible. However, MPMC is limited to generating point sets and cannot be extended to low-discrepancy sequences (LDS), i.e., sequences of points in which every prefix has low discrepancy, a property essential for many applications. To address this limitation, we introduce Neural Low-Discrepancy Sequences (NeuroLDS), the first machine learning-based framework for generating finite LDS. Drawing inspiration from classical LDS, we train a neural network to map indices to points such that the resulting sequences exhibit minimal discrepancy across all prefixes. To this end, we deploy a two-stage learning process: supervised approximation of classical constructions followed by unsupervised fine-tuning to minimize prefix discrepancies. We demonstrate that NeuroLDS outperforms all previous LDS constructions by a significant margin with respect to discrepancy measures. Moreover, we demonstrate the effectiveness of NeuroLDS across diverse applications, including numerical integration, robot motion planning, and scientific machine learning. These results highlight the promise and broad significance of Neural Low-Discrepancy Sequences.

Many problems in science, engineering, and computing require sampling points in space as efficiently and evenly as possible. Whether estimating complex integrals numerically, planning robot movements, or training machine learning models, the way points are distributed can have a major impact on accuracy and performance. Sequences designed to spread points as uniformly as possible, known as low-discrepancy sequences, are widely used for this purpose. A key property of these sequences is that not only the full set, but also every initial portion of the sequence, remains well-spread. This makes them suitable for applications where points are added one at a time and results need to be reliable at any stage. Until recently, low-discrepancy sequences have been constructed using classical techniques from mathematics, particularly abstract algebra and number theory. While effective, these approaches have inherent limitations in how uniform they can be. A machine learning method called Message-Passing Monte Carlo showed that neural networks could generate point sets with better uniformity than classical methods, but it was restricted to fixed sets of points and could not produce sequences with the progressive uniformity property needed in practice. In this work, we introduce Neural Low-Discrepancy Sequences (NeuroLDS), the first machine learning framework capable of generating low-discrepancy sequences. Our approach trains a neural network to assign each position in the sequence a point in space such that all prefixes of the sequence remain as uniform as possible. We achieve this through a two-stage training process: the network first learns to approximate well-known classical sequences, then is further refined to directly minimise uniformity error across prefixes. NeuroLDS outperforms all existing low-discrepancy sequence constructions by a substantial margin. We validate its practical value across several applications, including numerical integration, robot motion planning, and scientific machine learning, demonstrating that this approach has broad potential across diverse fields.