Learning Around Analytical Solutions: Risk-Calibrated Corrections for Sparse Graph Sensing

Analytical solutions can provide useful anchors for sparse graph sensing, yet learned flexibility may improve pooled error while worsening local outcomes. We ask when learning should modify or preserve an analytical anchor, whether reconstruction should correct or replace it, and whether pooled gains imply local benefits. We study two settings: a policy that makes one-sensor exchanges around a QR-selected set or retains it, and a spatiotemporal graph network that predicts a residual over Ridge reconstruction. At the smallest river budget, forced swapping improved the whole-period error point estimate but worsened 50.91% of local time-region blocks; allowing no-swap reduced harmful-block frequency by 14.95 percentage points across budgets. Under fixed QR masks, residual prediction reduced pooled unselected-field error by 18.5%–23.0% relative to direct spatiotemporal prediction, yet equal-weight local block gains remained negative on average. A three-domain comparison provided supporting evidence across river, traffic, and ocean graphs. Together, these findings show that intervention choice, correction form, and evaluation scale determine when learning improves on an analytical anchor.