NeuroLDS: Neural Low-Discrepancy Sequences

• NeuroLDS is a neural network approach that generates sequences with minimal prefix discrepancy, ensuring uniform coverage in the unit hypercube.
• It uses a two-stage learning process—supervised pre-training from classical methods and unsupervised fine-tuning with a differentiable loss—to optimize sequence uniformity.
• NeuroLDS enhances quasi-Monte Carlo integration, motion planning, and scientific machine learning by adapting to arbitrary sequence lengths and high-dimensional spaces.

Neural Low-Discrepancy Sequences (NeuroLDS) are neural network–based constructions devised to generate sequences of points in the unit hypercube $[0,1]^d$ such that every prefix of the sequence exhibits minimal discrepancy. Discrepancy quantifies a sequence’s deviation from perfect uniformity in filling the space; low-discrepancy sequences (LDS) are essential in quasi-Monte Carlo (QMC) integration, simulation, motion planning, and scientific machine learning. Classical LDS are synthesized from number-theoretic and combinatorial principles, such as Sobol’ and Halton constructions. NeuroLDS surpasses these by leveraging neural networks trained to minimize prefix discrepancies, demonstrating improved sequence quality and adaptability to modern computational pipelines (Huffel et al., 4 Oct 2025).

1. Motivation and Conceptual Framework

NeuroLDS arises from limitations in existing LDS constructions. Traditional point set generators (e.g., Sobol’, Halton, (t,s)-sequences) only guarantee low discrepancy for sets of fixed size and dimension, offer limited adaptivity, and struggle to generalize to arbitrary-length sequences necessary in sequential or online algorithms. Message-Passing Monte Carlo (MPMC) provided smaller discrepancy for fixed point sets by machine learning techniques, but cannot produce sequence-valued outputs where every prefix maintains low discrepancy. NeuroLDS addresses these two major deficiencies—sequence extensibility and prefix uniformity—by treating index $\rightarrow$ point mapping as a supervised and unsupervised learning problem for neural networks (Huffel et al., 4 Oct 2025).

2. Methodology and Loss Function Design

NeuroLDS employs a two-stage learning approach:

(a) Supervised Pre-training:

A neural network (typically an index-conditioned multilayer perceptron) is trained to reproduce a classical LDS, such as Sobol’. The training objective minimizes the mean-squared error between the output point $\mathbf{x}_i$ for input index $i$ and the corresponding classical sequence point, imparting the network with number-theoretic regularity as an inductive bias.

(b) Unsupervised Fine-tuning:

Fine-tuning adjusts the network parameters $\theta$ to minimize an explicit differentiable loss based on prefix discrepancy. The loss is formally given by:

$$\mathcal{L}(\theta) = \sum_{p=2}^N w_p \, D_2\big(\{ X_i \}_{i=1}^{p} \big)^2$$

where $X_i = f_\theta(i)$ is the output point for index $i$, $w_p$ are weighting coefficients, and $D_2$ denotes an $L_2$ discrepancy metric, such as the kernel discrepancy with

$k(x, y) = \prod_{j=1}^{d} (1 - \max(x_j, y_j))$

The summation over all prefixes ensures uniformity at every scale. Optimization proceeds via stochastic gradient descent or variants. The final neural architecture is capable of mapping arbitrary indices to points in $[0,1]^d$, extending to any desired sequence length.

3. Discrepancy Measures and Evaluation

Various discrepancy measures, including star, symmetric, and centered $L_2$ discrepancies, are utilized for both the loss function and evaluation. Each can be computed efficiently in a reproducing kernel Hilbert space (RKHS) framework. For a sequence $\{X_i\}_{i=1}^N$, the $L_2$ star discrepancy may be expressed as:

$$D_N^* = \sup_{a \in [0,1]^d} \bigg| \frac{1}{N} \sum_{i=1}^{N} \mathbf{1}_{[0,a)}(X_i) - \lambda([0,a)) \bigg|$$

with $\lambda$ the Lebesgue measure. Kernel-based discrepancies admit closed-form evaluation and differentiability, facilitating neural training.

NeuroLDS demonstrates empirically lower discrepancy than classical and scrambled sequences across a range of sequence lengths and dimensions. Every initial segment (prefix) of the sequence retains superior uniformity, which is quantifiable via the selected $L_2$ discrepancy metrics (Huffel et al., 4 Oct 2025).

4. Applications Across Computational Domains

NeuroLDS supports a broad spectrum of applications:

• Numerical Integration:

Quasi-Monte Carlo (QMC) integration benefits directly from lower discrepancy, as the integration error $|I_N - \int f| \leq V(f) D_N^*$ tightens with reduced $D_N^*$. For benchmark functions, such as in the 8-dimensional Borehole model, NeuroLDS sequences yield smaller errors than Sobol’ or Halton.

• Robot Motion Planning:

Rapidly-exploring Random Tree (RRT) methods require sequential generation of sample points in configuration space. NeuroLDS ensures uniform space coverage and expeditious exploration of challenging areas (e.g., narrow passages) by maintaining low discrepancy throughout all prefixes.

• Scientific Machine Learning:

Training neural surrogates for parametric PDEs or ODEs, NeuroLDS sequences can supplant random mini-batch selection, improving generalization, faster convergence, and reduced sample requirements. Theoretical generalization gap bounds follow from the Koksma-Hlawka inequality and empirical studies show enhanced surrogate accuracy.

5. Comparisons, Advantages, and Adaptability

NeuroLDS advances several attributes beyond classical LDS methods:

• Prefix Uniformity:

Unlike most classical constructions that focus on fixed-size sets, every initial segment of a NeuroLDS sequence exhibits low discrepancy due to prefix-wise optimization.

• Flexibility and Extensibility:

Neural mapping from index to point is easily extendable to arbitrary $N$ and $d$, supporting dynamic and online sampling without loss of uniformity.

• Customizable Discrepancy Objective:

By selecting alternate kernels or discrepancy metrics (e.g., Stein discrepancies for non-uniform distributions), NeuroLDS can be adapted for specific integration or simulation requirements.

• Integration with Modern ML Pipelines:

The neural framework supports integration with advanced architectures (autoregressive, graph neural networks), domain-specific regularizers, and sensitivity-weighted objectives for coordinates of decaying relevance.

6. Future Research Directions

Potential research avenues include:

• Alternative Discrepancy Metrics:

Optimization for discrepancy measures other than $L_2$ or star discrepancy to accommodate irregular integration domains or distributions.

• Architectural Innovation:

Incorporation of domain-aware neural architectures, exploration of autoregressive or attention-based models, and hybridization with symbolic approaches (e.g., automata-theoretic constructs, spectral regularization).

• High-Dimensional Scalability:

Development of capacity and generalization guarantees for very high $d$; paper of weighted discrepancy and coordinate prioritization.

• Industrial and Simulation Pipelines:

Deployment in simulation, design optimization, uncertainty quantification, and real-time robotic exploration where uniformly covering growing parameter spaces is essential.

7. Mathematical Foundations and Summary Table

Key mathematical constructs used in NeuroLDS are summarized below:

Construct	Formula / Description	Role
$L_2$ star discrepancy	$$D_N<sup>*</sup> = \sup_{a}</td> <td>\frac{1}{N} \sum_{i=1}<sup>{N}</sup> \mathbf{1}_{[0,a)}(X_i) - \lambda([0,a))</td> </tr> <tr> <td>Kernel discrepancy</td> <td>$$k(x, y) = \prod_{j=1}^{d} (1 - \max(x_j, y_j))$$</td> <td>Differentiable loss</td> </tr> <tr> <td>Loss over prefixes</td> <td>$$\mathcal{L}(\theta) = \sum_{p=2}^{N} w_p D_2(\{X_i\}_{i=1}^{p})^2$$</td> <td>Training objective</td> </tr> <tr> <td>Neural mapping</td> <td>$$X_i = f_\theta(i)$$</td> <td>Sequence generation</td> </tr> <tr> <td>Koksma–Hlawka error bound</td> <td>$$|I_N - \int f| \leq V(f) D_N^*$	Generalization guarantee

These foundations enable NeuroLDS to provide guarantees and empirically demonstrated improvements over classical low-discrepancy sequences.

NeuroLDS has been developed and evaluated in (Huffel et al., 4 Oct 2025). The approach addresses both theoretical and practical shortcomings of classical LDS, synthesizing machine learning–based sequence generation with discrepancy minimization to achieve superior performance in diverse computational domains. Direct code and models are available at https://github.com/camail-official/neuro-lds.