在反射扩散中是否应学习边界项？余法迹与反射掩蔽

中文摘要 AI 辅助

我们研究有界域上反射扩散的分数学习问题。反射操作可保持轨迹的可行性，但无法确保学习到的分数满足前向过程隐含的边界行为。使用隐式分数匹配时，分部积分会产生一个边界项，我们证明该边界项依赖于每个边界点处的一个标量：即分数的扩散加权法向分量，或称余法迹。无通量条件确定了该值，同时使其余边界分量不受限制；在各向异性扩散下，该值通常与普通法向分数分量不同。对于超矩形，我们的参数化方法无需额外可训练参数或随机边界估计器即可强制执行所需的迹，且在正则性假设下可表示真实分数；而固定错误值会产生无法通过更多数据消除的误差。我们将该构造扩展到单纯形和多边形域，并识别出反射掩蔽现象：即使学习到的迹有误，硬反射仍可保持样本的可行性，因此反射后指标可能会掩盖误差。实验表明，在反射频率较低、各向异性扩散以及质量靠近约束交点时，误差的分离最为明显；在完全反射下，最终样本位置的改善不一致，说明硬修复可掩盖边界分数误差，使分数准确性与下游生成质量解耦。

英文摘要

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.

发表机构

• Institute of Science Tokyo（东京科学大学）

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