用学习到的反射扩散模型在有界域上做高维数值积分
Numerical Integration on Bounded Domains via Learned Reflected Diffusions
给需要算高维积分的人：用反射扩散造提议分布，天然满足域约束，还能无偏估计，比 QMC 灵活。
论文研究有界凸域 D 上的高维积分，目标是用蒙特卡洛估计 Z=∫_D r(x)dx。现有学习型扩散采样器只能在 R^d 上构造提议分布，无法满足 x∈D 的约束，作者改用反射动力学，使采样过程天然保持在域内。方法通过反射扩散把目标分布逐步变换为 D 上的均匀分布，并训练神经分数模型逆转该过程，借助测度变换公式得到无偏重要性采样估计器。对不可微的被积函数，作者用反射前的 Gaussian 提议密度推导出 REINFORCE 梯度估计器，无需对 r 求导。实验覆盖多个合成目标和真实问题，性能随维度和蒙特卡洛点数扩展表现良好。
Numerical Integration on Bounded Domains via Learned Reflected Diffusions
We study high-dimensional integration on bounded convex domains. Given a non-negative integrand $r$, the goal is to estimate $Z=\int_{D}r(x)\,\mathrm dx$ on $D\subset\mathbb R^d$. 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 $\mathbb R^d$ and do not respect the domain constraint $x\in D$. 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.