提出频谱诊断法，揭示等变力场模型在分子振动频率上的能力极限。
Diagnosing Spectral Ceilings in Equivariant Neural Force Fields
- 通过注入可控角频率扰动，用轻量网络检测模型保留的频率成分。
- 在阿司匹林上发现模型在l=4处仍可恢复，l=5时性能骤降11.7倍。
- 适用于研究等变神经势能模型的频率响应边界，适合量子化学与机器学习交叉研究者。
我们提出一种频谱注入诊断方法，用于衡量训练好的等变力场主干网络保留的角频率：向分子力场注入受控的角频率扰动，将一个轻量级频谱预测网络（SPN）连接到冻结的主干网络上，读取可恢复的频率。在阿司匹林分子上，一个二次型SPN连接至L=2的NequIP主干，在l=4处仍能恢复边界信号，但在l=5处性能崩溃，预测值下降11.7倍，相关系数p从0.913降至0.078。该边界与高于边界的对比在四个独立训练的主干网络中一致（原始增益差异，分层聚类自举），并得到无分母注入残差度量验证（R²_inj(4)=0.374 vs R²_inj(5)=0.006）。有限度跨度定理校准了该诊断：对于单一方向标记，度为d的多项式在度为L的球谐特征上恰好能表示不超过dL的阶数，且在边界处达到单重饱和（仅限单方向度受限探测，非多原子MPNN函数类上界）。合成C5校准及容量、激活函数和跨架构控制排除了参数量单独解释的可能性。
原文摘要 · Abstract (English)
We introduce a spectral-injection diagnostic for measuring which angular frequencies a trained equivariant force-field backbone preserves: inject a controlled angular-frequency perturbation into a molecular force field, attach a lightweight Spectral Prediction Network (SPN) to the frozen backbone, and read off which frequencies are recoverable. On aspirin, a quadratic SPN attached to an L = 2 NequIP backbone recovers the boundary signal at l = 4 but collapses at l = 5: a 11.7x cliff at the predicted drL boundary, with p dropping from 0.913 to 0.078. The same boundary-vs-above contrast persists across n = 4 independently trained backbones (raw-gain delta contrast, hierarchical cluster bootstrap) and is corroborated by a denominator-free injected-residual metric (R2_inj(4) = 0.374 versus R2_inj(5) = 0.006). A finite-degree span theorem calibrates the diagnostic: for a single marked direction, degree-d polynomials of degree-L spherical-harmonic features span exactly H less than or equal to dL with multiplicity-one saturation at the boundary (scoped to single-direction degree-bounded probes, not a function-class upper bound on multi-atom MPNNs). A synthetic C5 calibration plus capacity, activation, and cross-architecture controls rule out parameter count alone as the explanation.
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