CTRL-FM: Inference-Time Structural Control for Flow-Based Virtual Sensing under Long-Horizon Node Outages

Boshra Khalili; Andrew Smyth

cs.LGOct 21, 2026

CTRL-FM: Inference-Time Structural Control for Flow-Based Virtual Sensing under Long-Horizon Node Outages

Authors: Boshra Khalili, Andrew Smyth

Organizations: Department of Civil Engineering and Engineering Mechanics Columbia University New York, NY 10027, USA

Abstract

Long-horizon node outages are harder than conventional missing-value imputation because complete sensor trajectories remain unobserved for extended periods, while the usefulness of spatial and temporal dependencies varies across outage locations and operating regimes. Existing graph, diffusion, and flow-based methods typically impose a fixed structural bias, which can oversmooth abrupt changes or propagate errors from unreliable neighbors. We introduce CTRL-FM, an observation-preserving flow-matching framework that treats structural guidance as an outage-conditioned inference-time decision. A missing-only flow-matching backbone learns a conditional velocity field while keeping observed measurements exactly fixed. For each realized outage, CTRL-FM constructs self-supervised proxy outages from surviving measurements and uses their known values to select among uncontrolled, graph-dominant, temporal-dominant, and balanced control policies. The selected policy is locally modulated by graph and temporal reliability estimated exclusively from surviving observations, while boundary-anchored trends provide structural guidance. Control is applied only to missing entries and is rejected when proxy evidence does not sufficiently improve over uncontrolled sampling. Across PeMS08, METR-LA, Air-36, and AQI, CTRL-FM achieves the strongest overall performance under 30% and 60% node outages. Stress tests with up to 75% sensor loss further show graceful degradation and consistent gains over training-time and fixed structural control. These results demonstrate that validating, selecting, and locally adapting structural guidance for each realized outage is more effective than imposing a single global structural prior.

Figures & tables

Figure 1: Overview of CTRL-FM. A missing-only flow backbone is trained while preserving observations. At inference, local proxy outages select an outage-specific policy, observed-only reliability modulates structural guidance, and the controlled velocity reconstructs only missing entries while maintaining exact observation consistency.

Figure 2: Qualitative PeMS08 reconstruction under a 60% long-horizon node outage. Rows (a)–(d) show CSDI, FM-TrainStruct, CDSTE, and CTRL-FM, respectively; columns show flow, occupancy, and speed at two failed sensors. Blue points denote ground truth, red and green curves denote the predictive mean and median, and shading denotes the 90% predictive interval.

Dataset	Valid proxy (%) $\uparrow$	$\boldsymbol{\rho}\uparrow$	Oracle match (%) $\uparrow$
PeMS08	98.7	0.82	88.4
METR-LA	99.1	0.79	84.8
Air-36	87.4	0.68	78.6
AQI	98.9	0.76	81.6

Table 1: Validity of proxy-based policy selection at 60% node outage. Valid proxy measures calibration coverage, Spearman’s

\rho

measures agreement with the post-hoc true policy ranking, and oracle match reports selection of the true best policy. Higher values are better.

Variant	PeMS08 Speed	METR-LA Speed	Air-36 PM 2.5	AQI PM 2.5	Time $\downarrow$	Mem. $\downarrow$
FM-Base	3.853 / 0.393	3.960 / 0.417	28.548 / 3.709	25.927 / 3.813	1.00 $\times$	1.00 $\times$
Fixed-CTRL	2.938 / 0.295	3.226 / 0.384	17.350 / 2.244	14.250 / 2.310	1.08 $\times$	1.03 $\times$
CTRL-FM w/o reliability	2.796 / 0.252	2.818 / 0.299	16.758 / 2.195	13.210 / 2.223	1.15 $\times$	1.04 $\times$
CTRL-FM w/o abstention	2.830 / 0.271	2.936 / 0.312	16.720 / 2.232	13.666 / 2.265	1.16 $\times$	1.05 $\times$
CTRL-FM	2.559 / 0.249	2.512 / 0.294	16.381 / 2.187	12.490 / 2.153	1.16 $\times$	1.06 $\times$

Table 3: Component ablation under 60% long-horizon node outage. Dataset entries report MAE / Peak-

\Delta

MAE, with PeMS08 evaluated on speed. Runtime and memory are normalized to FM-Base. Lower values are better, and the best reconstruction value for each dataset and metric is shown in bold .

Policy	$\boldsymbol{\alpha}$	$\boldsymbol{\lambda_{g}}$	$\boldsymbol{\lambda_{t}}$
$h_{0}$ : uncontrolled	0.00	0.00	0.00
$h_{1}$ : weak balanced	0.50	0.50	0.50
$h_{2}$ : graph-dominant	0.75	0.75	0.25
$h_{3}$ : temporal-dominant	0.75	0.25	0.75
$h_{4}$ : strong balanced	1.00	0.50	0.50

Table 4: Candidate structural-control policies. The parameter

\alpha

controls the total correction strength, while

\lambda_{g}

and

\lambda_{t}

determine the graph–temporal balance.

Dataset	Domain	Nodes	Steps	Resolution	Variables
PeMS08	Traffic	170	17,856	5 min	Flow, occupancy, speed
METR-LA	Traffic	207	34,272	5 min	Speed
Air-36	Air quality	36	8,760	1 hour	PM 2.5
AQI	Air quality	437	8,760	1 hour	PM 2.5

Table 5: Summary of the datasets used in the experiments.

Dataset	$\boldsymbol{T}$	Duration	Train stride	Test windows	Masks/window	Main ratios
PeMS08	288	24 hours	12	12	10	30%, 60%
METR-LA	288	24 hours	12	23	10	30%, 60%
Air-36	168	7 days	24	10	10	30%, 60%
AQI	168	7 days	24	10	10	30%, 60%

Table 6: Window and outage configuration. Each outage spans the complete evaluation window, and ten failed-node masks are evaluated per test window and ratio.

Quantity	Value or validation candidates
Hidden dimension	$128\;(64,128,256)$
Graph-attention groups	$4\;(2,4,8)$
Optimizer / learning rate	AdamW / $1\times 10^{-3}$
Weight decay	$1\times 10^{-4}$
Batch size / maximum epochs	$32/200$
Early-stopping patience	$20$ epochs

Table 7: Training and final-sampling configuration. Parentheses contain validation candidates, with the selected value listed first.

Quantity	Value or validation candidates
Proxy budget $B$	$4\;(1,2,4,8)$
Graph radius $r_{p}$	$2\;(1,2,3,\mathrm{All})$ hops
Air-36 proxy search	3 hops, expanded to all stations if needed
Probe length $L_{\mathrm{probe}}$	$12\;(6,12,24)$ steps
Proxy-node budget $m_{p}$	4 nodes per proxy
Candidate policies $\|\mathcal{H}\|$	$5$

Table 8: Test-time calibration configuration. Parentheses contain validation candidates, with the selected default listed first.

Flow
Method	MAE $\downarrow$	RMSE $\downarrow$	CRPS $\downarrow$	Peak- $\Delta$ MAE $\downarrow$
Historical Average	48.550	74.851	–	7.498
Graph Neighbor	39.950	60.851	–	6.498
SAITS	35.960	54.855	–	6.196
IGNNK	33.451	52.958	–	6.298
GRIN	30.965	47.552	–	5.840

Table 9: Reconstruction performance on PeMS08 under 30% long-horizon node outage, with flow, occupancy, and speed reported in vertically stacked panels. Results are mean

\pm