物理先验的代价其实来自数据划分方式,而非先验本身。
The Cost of a Physics Prior Is Bounded by the Ablation Gap

- 用消融实验对比约束模型与无约束模型,发现代价源于自由特征和验证集划分。
- 在野火严重性预测中,空间屏蔽使代价从0.3470降至0.0473,降幅达7.3倍。
- 提出双拟合筛选法,可提前排除无法识别的实验,避免无效训练。
形状约束与物理信息学习常将精度损失归因于先验本身,我们证明这主要由自由特征和验证划分决定。设P为在特征S上施加形状约束导致的额外风险,D为忽略S的消融模型的风险,则对任意风险函数均有0 ≤ P ≤ D,无需凸性、光滑性或可实现性假设。实证中该界表现为符号检验:约束模型绝不可被其消融版本超越。我们在一个有序野火严重性任务(N=26,681,K=3)上验证,对四个气象驱动变量施加严格单调约束,坐标自由,验证方案从独立同分布重采样到二度空间块化。坐标作为保护屏障:单独使用时,在空间块化下仍能恢复全模型92.9%的宏观F1,使D从0.1288降至0.0427;相同先验在有屏蔽时代价为0.0473,无屏蔽时为0.3470,比值达7.3,物理条件一致。由于D依赖于协议,不具备可迁移性:块粗化从1度到10度使D从0.0942降至0.0050,导致两种配置无法事前区分。通过认证嵌套关系的反向推导,在318次比较中给出自校准下界0.0220宏观F1,低于此值的结果均不可解释,包括我们自身主网格中的四个单元。代价与合规性独立:无约束模型违反先验率达0.48–0.49,而强制遵守仅增加0.0473代价。我们提供双拟合筛选机制,可在训练约束模型前剔除不可识别实验。
原文摘要 · Abstract (English)
Shape-constrained and physics-informed learning reports an accuracy cost of enforcing a prior and treats it as a property of the prior. We show it is mostly a property of the free features and the validation split. Let P be the excess risk of restricting a hypothesis class to functions with a shape constraint on features S, and D the excess risk of the ablated model that ignores S. Because a function constant in x_j is both non-decreasing and non-increasing in x_j, the ablated class is contained in the constrained class, so 0 <= P <= D for every risk functional, with no convexity, smoothness, or realizability assumption. Empirically the bound is a sign test: a constrained model must never be beaten by its own ablation. We instantiate it on an ordinal wildfire-severity task (N = 26,681, K = 3) with hard monotone constraints on four meteorological drivers, coordinates left free, and a validation ladder from i.i.d. resampling to 2-degree spatial blocking. Coordinates act as a shield: alone they recover 92.9% of the full model's macro-F1 under spatial blocking, collapsing D from 0.1288 to 0.0427; the same prior costs 0.0473 shielded and 0.3470 unshielded, a ratio of 7.3 with identical physics. Because D is protocol-dependent it does not transfer: coarsening blocks from 1 to 10 degrees drives D from 0.0942 to 0.0050, leaving two configurations unidentifiable a priori. Inversions of the certified nesting bound the pipeline's additive resolution: over 318 comparisons they give a self-calibrating floor of 0.0220 macro-F1, below which no reported price is interpretable, including four cells in our own headline grid. Cost and compliance are independent: the unconstrained model violates the prior at rate 0.48-0.49 while enforcing it costs 0.0473. We give a two-fit screen that rejects unidentifiable experiments before a constrained model is trained.
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