Numerical Integration on Bounded Domains via Learned Reflected Diffusions
Abstract
We study high-dimensional integration on bounded convex domains. Given a non-negative integrand r, the goal is to estimate Z=∫Dr(x) dx on D⊂ℝd. Standard Quasi-Monte Carlo does not adapt its evaluation points to the integrand, while effective importance sampling requires a proposal that captures its concentration. Learned diffusion samplers can supply such a proposal, but existing methods are formulated on ℝd and do not respect the domain constraint x∈ D. We instead build the proposal from reflected dynamics that stay in D by construction. A reflected diffusion progressively transforms the target distribution toward the uniform distribution on D, and a neural score model is trained to reverse this process. A change-of-measure formula for reflected diffusions yields a training objective without additional boundary terms and an unbiased importance-sampling estimator of Z. For non-smooth integrands, where reparameterization gradients may be undefined or uninformative, we derive a REINFORCE estimator of the training gradient using Gaussian proposal densities before reflection, without evaluating reflected transition densities or differentiating r. We evaluate on numerous synthetic targets and real-world problems, demonstrating strong performance of the proposed method, as well as favorable scaling with the dimension of the problem and the number of points used for the Monte Carlo estimate.
Related
- Monte Carlo Approximations of Time-Nonlocal Diffusions in Bounded Domains
- Time-changed reflected diffusions: subdiffusion, superdiffusion and explosion
- Limit theorems for one-dimensional diffusions: reflected, skew, sticky, skew-sticky, and snapping-out limits
- Schauder estimates for the linearized very fast diffusion equations in bounded domains
- Absence of bounded-orbit wandering domains
BibTeX
@misc{korolev2026numerical,
title = {{Numerical Integration on Bounded Domains via Learned Reflected Diffusions}},
author = {Kirill Korolev and Artur Goldman and Timofei Gritsaev and Tigran Ramazyan and Nikita Morozov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40499},
eprint = {2610.40499},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40499},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Numerical Integration on Bounded Domains via Learned Reflected Diffusions
AU - Kirill Korolev
AU - Artur Goldman
AU - Timofei Gritsaev
AU - Tigran Ramazyan
AU - Nikita Morozov
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40499
UR - https://arxiv.org/abs/2610.40499
AB - We study high-dimensional integration on bounded convex domains. Given a non-negative integrand r, the goal is to estimate Z=∫_Dr(x) dx on D⊂ℝ^d. Standard Quasi-Monte Carlo does not adapt its evaluation points to the integrand, while effective importance sampling requires a proposal that captures its concentration. Learned diffusion samplers can supply such a proposal, but existing methods are formulated on ℝ^d and do not respect the domain constraint x∈ D. We instead build the proposal from reflected dynamics that stay in D by construction. A reflected diffusion progressively transforms the target distribution toward the uniform distribution on D, and a neural score model is trained to reverse this process. A change-of-measure formula for reflected diffusions yields a training objective without additional boundary terms and an unbiased importance-sampling estimator of Z. For non-smooth integrands, where reparameterization gradients may be undefined or uninformative, we derive a REINFORCE estimator of the training gradient using Gaussian proposal densities before reflection, without evaluating reflected transition densities or differentiating r. We evaluate on numerous synthetic targets and real-world problems, demonstrating strong performance of the proposed method, as well as favorable scaling with the dimension of the problem and the number of points used for the Monte Carlo estimate.
ER -