Numerical Integration on Bounded Domains via Learned Reflected Diffusions

We study score learning for reflected diffusion on bounded domains. Reflection keeps trajectories feasible but does not ensure that the learned score satisfies the boundary behavior implied by the forward process. With implicit score matching, integration by parts leaves a boundary term, and we show that it depends on one scalar at each boundary point: the diffusion- weighted normal component of the score, or conormal trace. The no-flux condition fixes this value while leaving the re- maining boundary components unrestricted; under anisotropic diffusion it generally differs from the ordinary normal score component. On hyperrectangles, our parametrization enforces the required trace without additional trainable parameters or a stochastic boundary estimator and, under regularity assump- tions, can represent the true score, whereas fixing an incorrect value creates an error that more data cannot remove. We ex- tend the construction to simplices and polygonal domains and identify reflection masking: hard reflection can keep samples feasible even when the learned trace is wrong, so post-reflection metrics may hide the error. Experiments show the clearest separation with less frequent reflection, anisotropic diffusion, and mass near intersections of constraints; under full reflection, final sample placement improves inconsistently, illustrating how hard repair can mask boundary-score errors and decouple score accuracy from downstream generation quality.

First order Langevin algorithms for constrained sampling in machine learning, such as projected Langevin Monte Carlo which are based on discretizations of reflected Langevin dynamics, require differentiable log densities that limits their applicability. This paper introduces reflected anchored Langevin dynamics (RALD), a reflected diffusion that converges to non-differentiable targets on constrained domains. The method uses a smooth anchored reference potential and multiplies the drift and noise covariance of its reflected Langevin dynamics by the same state dependent scaling factor. Its Euler-Maruyama discretization with projection gives reflected anchored Langevin Monte Carlo (RALMC) algorithm. We prove explicit convergence bounds and iteration complexity for RALMC in the 2-Wasserstein distance to the target distribution. Numerical experiments are provided to illustrate the theoretical predictions and the empirical performance of the method.

We study the sample complexity of estimating Ornstein--Uhlenbeck diffusion scores in $\mathbb R^d$ using denoising score matching (DSM) with finite-width shallow ReLU networks, and its implications for reverse-process sampling. Under weak differentiability and polynomial-growth conditions on normalized derivatives of the initial density, we combine complementary estimates from the Gaussian transition kernel and the initial density to obtain explicit Sobolev bounds that are uniform in time on each fixed ball. Combining these bounds with existing shallow-network approximation theory gives explicit control of the approximation error and a sufficient path-norm budget. We then use truncation and localized Rademacher complexity to establish high-probability risk bounds for path-norm-constrained empirical DSM minimizers with unbounded inputs and labels. Under additional bounded-density and sub-Gaussian assumptions, suitable choices of network width and path-norm budget yield squared $L^2(P_t)$ score risk $\widetilde O(n^{-(d+3)/(2d+3)})$ at each fixed time $t>0$, where $n$ is the number of independent training samples and $P_t$ is the forward marginal. The nonasymptotic bounds retain explicit dependence on diffusion time and the regularity and tail parameters of the initial distribution. Aggregating the score errors over a time grid and applying existing sampling theory together with an early-stopping bound yields a high-probability total-variation error $\widetilde O(n^{-(d+3)/(4d+6)})$ between the generated and original data distributions under suitable sampling parameters. Here $\widetilde O$ suppresses logarithmic factors for fixed dimension, confidence level, and initial distribution.