局部模型能否外推到更大尺寸，取决于平滑得分的准局部性。
When Do Local Score Models Extrapolate Across Size? A Diagnostic Theory and Benchmark
- 通过Tweedie公式揭示远端扰动如何影响局部得分
- 发现空间混合强时模型可稳定外推，弱时则失败
- 提出FDLF基准，可精确控制响应范围进行诊断
科学生成建模常需规模迁移，即在小系统上训练的模型评估于更大系统。尽管平移不变架构支持此类评估，我们发现仅靠架构局部性不足以保证稳定外推。真正决定因素是高斯平滑得分的准局部性。根据Tweedie公式，远端扰动可通过后验协方差影响局部得分分量，因此局部模型成功依赖其感受野覆盖平滑得分的响应范围。本文形式化该机制，证明了逆扩散下局部边际的尺寸一致比较定理。同时引入有限深度局部流（FDLF），一个具备精确得分、密度和可控响应范围的白盒诊断基准。实验验证了空间混合性、平滑得分准局部性与模型感受野之间的相互作用：当空间混合性强时，平滑得分保持准局部，外推稳定；反之，得分局部性迅速退化，导致外推失败。
原文摘要 · Abstract (English)
Scientific generative modeling often requires size transfer, where models trained on small systems are evaluated on larger ones. While translation-invariant architectures enable this evaluation, we show that architectural locality alone does not guarantee stable size extrapolation. Instead, stable extrapolation is governed by the quasi-locality of the Gaussian-smoothed score. Through Tweedie's formula, far-away perturbations can influence local score components via posterior covariance, meaning a local model succeeds only if its receptive field covers the smoothed score's response range. We formalize this mechanism, proving a size-uniform comparison theorem for local marginals under reverse diffusion. We also introduce Finite-Depth Local Flow (FDLF), a white-box diagnostic benchmark with exact scores, densities, and controllable response ranges. Empirically, we validate the interplay between spatial mixing, smoothed-score quasi-locality, and model receptive fields. Under spatial mixing, the smoothed score remains quasi-local relative to the receptive field, enabling stable extrapolation. Conversely, when spatial mixing weakens, the score's locality rapidly degrades, causing size transfer to fail.
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