Control-function instrumental variable estimators need a first-stage residual, not merely a first-stage prediction. High-capacity first stages can interpolate treatment and leave too little residual information for the outcome equation. We study Adaptive Anisotropic Instrumental Heat Flow (A-IHF), a deterministic graph-diffusion residual extractor for flexible control functions. A-IHF treats treatment as a signal on a graph of first-stage features, uses pilot diffusion to detect large treatment jumps, attenuates conductance across those jumps, and computes the generated control with a sparse graph resolvent. Its observational selection rule uses only $(Z,X)$, combining graph generalized cross-validation, roughness, residualized-treatment relevance, and graph-admissibility filtering. The analysis decomposes error into structural leakage, residual attenuation, and residualized treatment variation, yielding finite-sample bounds, graph-admissibility rates under latent piecewise-smooth geometry, and finite-path selection calibration. Across 54 synthetic benchmark cells with tuned graph, kernel, tree, boosting, series, and neural control-function baselines, guarded observational A-IHF has the lowest average structural-response MSE; the A-IHF family beats the best non-A-IHF baseline in 32 cells. Performance is strongest when the graph captures piecewise-smooth first-stage structure.

翻译：控制函数工具变量估计器需要第一阶段的残差，而不仅仅是第一阶段的预测。高容量的第一阶段可能过度拟合处理变量，导致结果方程可用的残差信息不足。我们研究了自适应各向异性工具热流（A-IHF），这是一种用于灵活控制函数的确定性图扩散残差提取器。A-IHF将处理变量视为第一阶段特征图上的信号，利用引导扩散检测处理变量的跳跃，衰减跨跳跃的电导，并通过稀疏图解析器计算生成的控制变量。其观测选择规则仅使用$(Z,X)$，结合图广义交叉验证、粗糙度、残差化处理相关性以及图可容许性过滤。分析将误差分解为结构泄漏、残差衰减和残差化处理变异，从而得到有限样本界、潜在分段光滑几何下的图可容许性速率以及有限路径选择校准。在54个合成基准单元中，与调优的图、核、树、提升、级联和神经网络控制函数基线相比，有监督观测的A-IHF具有最低的平均结构响应均方误差；A-IHF家族在32个单元中优于最佳非A-IHF基线。当图能够捕捉分段光滑的第一阶段结构时，性能最为强劲。