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

Boshra Khalili, Andrew Smyth

arXiv:2610.30362v1cs.LG

TL;DR

Long-horizon node outages hide complete sensor trajectories, while the usefulness of spatial and temporal dependencies varies across outages and operating regimes. CTRL-FM uses a missing-only flow backbone and proxy-selected, locally reliability-modulated structural control at inference. Across four datasets, it achieves the strongest overall reconstruction performance under 30% and 60% outages, with graceful degradation in stress tests up to 75% sensor loss.

  • Problem

    Long-horizon node outages leave complete sensor trajectories unavailable, while spatial and temporal dependencies can vary across outage locations and operating regimes.

    Source: §1 · p. 1, §3.1 · p. 3

  • Method

    CTRL-FM preserves observed measurements with a missing-only flow backbone and uses self-supervised proxy outages to select and locally adapt inference-time structural control.

    Source: §1 · p. 2

  • Results

    Across four traffic and air-quality datasets, CTRL-FM achieves the strongest overall reconstruction performance under 30% and 60% node outages and degrades gracefully in stress tests up to 75% loss.

    Source: §1 · p. 2, §5 · p. 10

  • Takeaways & Limitations

    The results support validating and adapting structural guidance for each realized outage rather than imposing a single global structural prior.

    Source: §5 · p. 10

    CTRL-FM relies on sufficiently informative surviving observations and meaningful sensor-graph dependencies; performance may decline under extremely sparse or highly clustered failures.

    Source: §4.5 · p. 8

Abstract

from arXiv · show

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.

1 Introduction

Long-horizon node outages remove complete sensor trajectories, making reconstruction difficult when spatial and temporal dependencies vary across outages and regimes. CTRL-FM adapts structural guidance at inference using surviving observations and local proxy outages.

1 Introduction

  • Long-horizon outages leave complete sensor trajectories unavailable, with distant temporal anchors and multiple trajectories compatible with surviving observations.The usefulness of graph and temporal assumptions also varies by outage and operating regime.Source: §1 · pp. 1–2
  • Existing methods commonly use a global structural choice and lack a direct way to validate or reject it for the realized outage.A prior helpful in one outage may oversmooth another or propagate errors from unrepresentative neighbors.Source: §1 · p. 2
  • CTRL-FM separates general reconstruction learning from outage-specific inference-time control, adapting a frozen missing-only flow backbone without retraining.Observed measurements remain fixed throughout training and sampling.Source: §1 · p. 2
  • Local self-supervised proxy outages use known surviving values to compare uncontrolled, graph-dominant, temporal-dominant, and balanced policies.Structural intervention is applied to the true outage only when the selected policy sufficiently improves over uncontrolled sampling.Source: §1 · p. 2
  • Observed-only reliability determines where selected graph and temporal guidance should be trusted, while boundary-anchored trends provide structural guidance.Reliability is estimated exclusively from surviving observations.Source: §1 · p. 2
  • Evaluation covers PeMS08, METR-LA, Air-36, and AQI under 30% and 60% node outages, with additional stress tests up to 75% sensor loss.The experiments also assess reconstruction accuracy, proxy validity, and robustness.Source: §1 · p. 2

2 Related Work

Prior spatio-temporal imputation methods use temporal, graph, or probabilistic structure, but generally apply a single reconstruction strategy or fixed structural conditioning across outages. CTRL-FM instead validates and adapts guidance for each realized outage.

2 Related Work

  • Classical methods use low-rank, tensor, statistical, or kernel assumptions, while neural methods learn richer dependencies from data.BRITS Cao et al. [2018] and SAITS Du et al. [2023] model temporal and cross-variable relationships; IGNNK Wu et al. [2021a], KCN Appleby et al. [2020], GRIN Cini et al. [2022], SPIN Marisca et al. [2022], and OPCR Deng et al. [2024] exploit graph structure.Source: §2 · p. 2
  • Diffusion methods model conditional distributions, but their spatial, temporal, or geographic conditioning is typically fixed during training and reused across test outages.Examples include CSDI Tashiro et al. [2021], PriSTI Liu et al. [2023], CDSTE Lei et al. [2024], Diffputer Zhang et al. [2025a], and SSD-TS Gao et al. [2025].Source: §2 · p. 3
  • Flow matching offers a continuous-time alternative, but graph-informed and training-free guidance methods commonly use predefined structural objectives or strengths.GiFlow Zhang et al. [2026] applies graph-informed flow matching; these choices may be unreliable when dependencies vary across outages.Source: §2 · p. 3
  • CTRL-FM uses proxy outages to select among graph, temporal, balanced, and uncontrolled policies, then modulates guidance with survivor-only reliability and abstains when unsupported.It validates both when and where structural guidance should be applied for each realized outage.Source: §2 · p. 3

3 Method

CTRL-FM separates a missing-only observation-preserving flow backbone from outage-specific structural control at inference. Proxy evidence selects a policy, observed-only reliability localizes its guidance, and projection and clamping preserve observed values.

3 Method

  • Training restricts noise and transport to synthetically missing entries, keeping observed measurements fixed while learning a reusable conditional velocity field.The backbone is optimized only where reconstruction is required and does not embed one fixed graph–temporal control policy.Source: §3.1 · p. 3
  • Local proxy outages temporarily hide surviving measurements, score candidate policies using their known values, and support abstention when control does not sufficiently improve over uncontrolled sampling.The proxy objective combines pointwise reconstruction accuracy with temporal-change recovery to discourage oversmoothing abrupt changes.Source: §3.1 · pp. 4–5
  • Boundary-anchored guidance uses valid observed changes as trend anchors and measures graph and temporal compatibility of the missing-region velocity.The measurement-supported trend term is used for structural comparison; observed measurements are not modified.Source: §3.1 · p. 5
  • Reliability features are recomputed under the active mask from surviving observations, with support gates preventing guidance where valid observational support is absent.The features summarize observed support, neighborhood agreement, change magnitude, graph distance, and proximity to observed temporal boundaries.Source: §3.1 · p. 6
  • The central controlled velocity selects outage-specific structural corrections, modulates them by graph and temporal reliability, and projects all corrections onto missing entries.With the uncontrolled policy, the correction vanishes and the method reduces to the missing-only flow backbone.Source: §3.1 · p. 6
  • After each numerical solver stage, re-clamping prevents the backbone or controller from altering observed measurements; one frozen backbone thus adapts without retraining or test-time parameter updates.The method does not access true failed values.Source: §3.1 · p. 6

4 Experiments

Experiments evaluate CTRL-FM across traffic and air-quality datasets, test whether proxy outages support policy selection, and assess reconstruction, component contributions, and severe-outage behavior. Results favor outage-specific control overall, with performance boundaries under sparse surviving support or weak graph dependencies.

4.1 Experimental Setup

  • Evaluation uses four traffic and air-quality datasets, simulated 30% and 60% long-horizon node outages, shared test conditions, and failed-region metrics.Comparisons include deterministic and probabilistic imputers, GiFlow, training-regularized FM-TrainStruct, and fixed-policy Fixed-CTRL; metrics are MAE, CRPS, and Peak-∆MAE.Source: §4.1 · pp. 6–7

4.2 Validity of Proxy-Based Policy Selection

  • Proxy rankings correlate 0.68–0.82 with post-hoc oracle rankings, and CTRL-FM selects the true best candidate in 78.6–88.4% of valid cases.Valid proxies are available for 98.7–99.1% of outages on PeMS08, METR-LA, and AQI, versus 87.4% on Air-36.Source: §4.2 · p. 7, Table 1 · p. 8

4.3 Reconstruction under Long-Horizon Node Outages

  • CTRL-FM delivers the strongest overall results across datasets and outage levels, with dataset- and metric-specific exceptions at 30% outage.At 60% outage, it ranks first across all reported metrics on both air-quality datasets; Table 2 reports reconstruction results for 30% and 60% outages.Source: §4.3 · p. 8, Table 2 · p. 9

4.4 Ablation and Inference Efficiency

  • Removing reliability modulation or abstention consistently degrades MAE and Peak-∆MAE, while complete CTRL-FM achieves the best reconstruction in every setting.Runtime and memory are 1.16× and 1.06× FM-Base, respectively; Table 3 reports the component ablation and normalized costs.Source: §4.4 · p. 8, Table 3 · p. 9

4.5 Extreme-Outage and Qualitative Analysis

  • CTRL-FM remains the strongest evaluated controlled flow-matching variant on PeMS08, METR-LA, and AQI at 65% and 75% outages; valid-proxy coverage remains 84.8–89.3% at 75%.The qualitative PeMS08 reconstruction shows CTRL-FM most consistently following ground truth across flow, occupancy, and speed, with intervals expanding around abrupt transitions.Source: Figure 2 · p. 7, §4.5 · p. 8
  • CTRL-FM relies on sufficient surviving observations for informative proxies and a sensor graph with meaningful dependencies; performance may decline under extremely sparse or highly clustered failures.The stated future directions include real outage events, larger networks, and adaptive graph structures.Source: §4.5 · p. 8

5 Conclusion

CTRL-FM separates reconstruction learning from outage-specific structural control, using proxy outages and surviving observations to select and adapt guidance while preserving observed measurements. Experiments report the strongest overall reconstruction performance across four datasets at 30% and 60% outages, with graceful degradation in stress tests up to 75%.

5 Conclusion

  • CTRL-FM treats structural guidance as an outage-conditioned inference-time decision rather than a fixed modeling assumption.Its frozen flow-matching backbone is adapted without retraining, while proxy outages, observed-only reliability, boundary-anchored guidance, and conservative abstention inform control.Source: §5 · p. 10
  • Across four traffic and air-quality datasets, CTRL-FM achieves the strongest overall reconstruction performance at 30% and 60% outages, with graceful degradation up to 75%.The conclusion also reports modest computational overhead relative to the uncontrolled flow-matching backbone.Source: §5 · p. 10
  • Effectiveness may be limited by insufficient surviving observations or weak spatial relationships, especially under extremely sparse or spatially clustered failures.Future directions include adaptive proxy construction, dynamically evolving graphs, and validation on real-world sensor outage events.Source: §5 · p. 10

1 Additional Method Details

CTRL-FM evaluates a fixed library of structural-control policies separately for each outage using proxy outages formed from surviving measurements. Proxy scoring combines reconstruction and temporal-change criteria, while calibration defaults to uncontrolled sampling when evidence is insufficient.

1.1 Policy Library and Proxy Calibration

  • The candidate policies are fixed before test evaluation, but policy selection is repeated locally for each realized outage using self-supervised proxy outages.The library includes uncontrolled, weak balanced, graph-dominant, temporal-dominant, and strong balanced control; its parameters are selected on validation outages.Source: §1.1 · p. 13, §1.2 · p. 13

1.2 Candidate Policy Library

  • The control parameters encode total structural-correction strength and its graph–temporal balance, while the uncontrolled policy applies no correction.Candidate policies control the same frozen flow-matching backbone rather than defining separately trained reconstruction models.Source: §1.2 · p. 13

1.3 Outage-Conditioned Proxy Construction

  • Each proxy temporarily hides nearby surviving measurements, creating a local self-supervised evaluation set without accessing values inside the true outage.Eligible nodes are fully observed over the proxy interval and close to the true outage; each proxy hides at most four nodes and covers up to the probe interval length.Source: §1.3 · pp. 13–14
  • Proxy inputs, boundary trends, guidance, context features, and reliability maps are recomputed after hiding proxy values, preventing those targets from contributing to their own reconstruction or reliability estimates.Proxy values are used afterward to score candidate policies.Source: §1.3 · p. 14

1.3.1 Proxy Calibration Objective

  • Policies are compared using identical Gaussian initialization and reduced solver grids, and proxy scoring combines value error with a temporal-dynamic term that discourages oversmoothing transitions.Only temporarily hidden proxy entries are evaluated; temporal differences are scored only when they touch a proxy entry and both endpoints were originally observed.Source: §1.3.1 · pp. 14–15
  • Ties favor smaller guidance strength, with uncontrolled sampling preferred on a tie; CTRL-FM defaults to uncontrolled sampling when valid proxies are absent or improvement is insufficient.No value inside the true outage is used for proxy construction, reliability estimation, scoring, or policy selection.Source: §1.3.1 · p. 15

1.4 Boundary-Anchored Structural Guidance

CTRL-FM uses valid trends from surviving measurements as boundary anchors for structural guidance into missing regions. Its comparison field combines the frozen backbone velocity on unavailable entries with measurement-supported trends on observed entries, without changing observations.

1.4 Boundary-Anchored Structural Guidance

  • Boundary-anchored guidance compares missing-region flow velocities with valid trends extracted exclusively from surviving measurements.Anchors adjacent to the observed–missing boundary influence unavailable nodes, while observed measurements remain fixed.Source: §1.4 · p. 15

1.4.1 Boundary-Anchored Comparison Field

  • A temporal change is valid only when both measurements used to compute it are available under the active mask.With one neighboring measurement, the observed trend uses a one-sided difference; with both, it reduces to a centered difference.Source: §1.4.1 · p. 16
  • The comparison field uses detached backbone velocity on unavailable entries and inserts measurement-supported trends on observed entries.Observed components support graph and temporal compatibility evaluation but never update observed measurements.Source: §1.4.1 · p. 16

1.5 Graph and Temporal Guidance Fields

Graph and temporal guidance fields measure compatibility against valid observed trends, and their corrections affect only unavailable entries. Observed-only reliability estimates locally modulate those corrections, with reliability modules trained using synthetic outages.

1.5 Graph and Temporal Guidance Fields

  • The channel-averaged discrepancy is differentiable, nonnegative, zero for equal vectors, and has bounded derivatives; graph-edge affinity depends on normalized graph cost and temperature.Each undirected edge is included once in the graph compatibility objective.Source: §1.5 · p. 16
  • Temporal compatibility measures weighted disagreement between connected sensors’ velocity vectors, encouraging coherent transport rather than directly smoothing signals.The guidance fields are negative gradients of graph and temporal compatibility energies, evaluated with the detached comparison field.Source: §1.5 · p. 17
  • Only unavailable-region components of the graph and temporal guidance fields affect reconstruction, with their local contributions modulated by observed-only reliability maps.Outage-level strengths come from the selected policy; local reliability maps determine where each guidance field is trusted.Source: §1.5 · p. 17

1.6.1 Observed Node Context

  • Reliability uses surviving-observation context, including support, neighborhood agreement, change magnitude, graph distance, and distance to an observed temporal boundary.Temporal support weakens deeper inside a long outage, and context quantities depend only on the active mask and surviving measurements.Source: §1.6 · p. 17, §1.6.1 · p. 18

1.6.2 Compact Reliability Features

  • Compact graph and temporal feature vectors summarize observed support, neighborhood agreement or disagreement, local change, and graph or temporal distance without unavailable endpoint values.A temporal difference is valid only when both endpoints are observed.Source: §1.6.1 · p. 17, §1.6.2 · p. 18

1.6.3 Reliability Weights and Maps

  • Graph and temporal reliability modules map compact feature vectors to support-gated values in (0, 1), which directly modulate their corresponding guidance fields.Reliability becomes zero when no supporting observation is available, and proxy features and maps are recomputed under the proxy mask.Source: §1.6.3 · p. 19

1.6.4 Self-Supervised Training

  • Reliability modules are trained self-supervisedly on synthetic outages, using masked-signal features and complete training signals to form reliability targets.Training relations must intersect the synthetic missing region, have valid complete-data targets, and retain positive observed support.Source: §1.6.4 · pp. 19–20

1.7 Observation Preservation and Final Sampling

After proxy calibration, CTRL-FM restores the original test masks and samples with the selected outage-specific policy. Policy strengths set the graph–temporal balance, while reliability maps determine where each guidance field is trusted.

1.7 Observation Preservation and Final Sampling

  • After policy calibration, CTRL-FM discards temporary proxy masks and restores the original test masks for final sampling.The initial generated component is sampled from Gaussian noise on unavailable entries.Source: §1.7 · p. 20
  • The selected policy determines outage-level correction strength and graph–temporal balance, while reliability maps control where each guidance field is trusted.When the uncontrolled policy is selected, the structural correction vanishes and sampling reduces exactly to the missing-only backbone.Source: §1.7 · p. 20
  • Every candidate policy preserves surviving observations exactly by projecting the complete state and re-clamping observed entries after solver stages.The final reconstruction combines measured values on observed entries with generated values on unavailable entries.Source: §1.7 · p. 20

1.8 Local Policy Calibration Algorithm

CTRL-FM calibrates a policy for each outage using temporary proxy outages and short controlled-flow probes. It selects a controlled policy only when its weighted proxy improvement passes the abstention threshold; otherwise it returns the uncontrolled policy.

1.8 Local Policy Calibration Algorithm

  • The calibration algorithm creates proxy masks from eligible surviving nodes and intervals, then recomputes boundary trends and observed-only reliability under each proxy mask.Proxies without a valid missing region or dynamic-score mask are skipped.Source: §1.8 · p. 21
  • For each valid proxy and candidate policy, CTRL-FM probes controlled flow using shared proxy noise, reduced Heun steps, and re-clamping at both solver stages.The resulting proxy reconstruction is scored against the temporarily hidden values.Source: §1.8 · p. 21
  • CTRL-FM selects the policy with the best weighted proxy score, breaking ties toward smaller correction strength and the uncontrolled policy.It accepts the selected control only if relative proxy improvement reaches the abstention threshold; otherwise it returns the uncontrolled policy.Source: §1.8 · p. 21
  • Each evaluated outage spans the complete evaluation window, with ten failed-node masks per test window and outage ratio.The table caption states the outage and mask-count configuration.Source: Table 6 · p. 22

2 Additional Experimental Details

This section documents CTRL-FM’s reconstruction benchmarks on traffic and air-quality datasets under long-horizon node outages, including matched evaluation masks and missing-region metrics. It also covers higher-outage robustness and proxy-calibrated policy selection across outages, including selected-policy frequencies.

2.1 Datasets, Preprocessing, and Outage Protocol

  • The protocol uses complete-window node outages at 30% and 60% for main tests and 65% and 75% for stress tests, with shared test outages and observation-preserving sampling.The four datasets cover PeMS08 and METR-LA traffic plus Air-36 and AQI air quality; CTRL-FM uses 20 Heun solver steps and 20 samples per outage.Source: §2.1 · pp. 22–23, §2.2 · p. 23

2.3 Test-Time Calibration Settings

  • CTRL-FM selects among five validation-fixed policies using up to four nearby proxy outages per test failure, accepts control only with at least 2% improvement over uncontrolled sampling, and otherwise abstains.Its variants distinguish training-time structural regularization (FM-TrainStruct), one fixed inference policy (Fixed-CTRL), and outage-specific control; Peak-∆MAE measures large-change reconstruction, while CRPS assesses probabilistic quality.Source: §2.3 · p. 24, §2.4 · p. 26, §2.5 · p. 26

2.6.1 METR-LA Results

  • On METR-LA, CTRL-FM has the lowest MAE, RMSE, and Peak-∆MAE at both outage levels, while CDSTE leads CRPS; CTRL-FM consistently improves over Fixed-CTRL.The comparison supports outage-specific policy selection over applying one fixed structural policy to every outage.Source: §2.6.1 · p. 28

2.6.2 Air-36 Results

  • On Air-36, OPCR leads MAE and RMSE at 30% outage, but CTRL-FM leads CRPS and Peak-∆MAE then and all four metrics at 60%.At 60%, CTRL-FM clearly outperforms Fixed-CTRL.Source: §2.6.2 · p. 28

2.6.3 AQI Results

  • On AQI, GiFlow leads MAE and RMSE at 30% outage, while CTRL-FM leads CRPS and Peak-∆MAE then and all four metrics at 60%.At 60%, CTRL-FM substantially improves over Fixed-CTRL.Source: §2.6.3 · p. 28

2.7 Extreme-Outage Robustness

  • At 65% and 75% outage, CTRL-FM has the lowest MAE and Peak-∆MAE across PeMS08, METR-LA, and AQI; valid-proxy coverage remains above 84% at 75%.The results indicate graceful degradation and continued calibration reliability under extreme sensor loss.Source: §2.7 · p. 28

2.8.1 Selected-Policy Frequencies

  • Weak- and strong-balanced control are selected most often, while graph- and temporal-dominant policies remain useful for particular failures and uncontrolled sampling is chosen only occasionally.The pattern indicates that structural guidance is generally useful but should adapt to each outage.Source: §2.8.1 · p. 28

2.6 Complete Dataset-Level Results

  • At 60% outage, CTRL-FM ranks first across every PeMS08 channel and metric; at 30%, it leads all flow and speed metrics and occupancy CRPS and Peak-∆MAE, while OPCR leads occupancy MAE and RMSE.The larger advantage at 60% indicates greater effectiveness when fewer observations remain available.Source: §2.6 · p. 28, Table 9 · p. 29, Table 10 · p. 30

2.8.1 Selected-Policy Frequencies

  • Tables 11–13 report reconstruction performance at 30% and 60% outages for METR-LA, Air-36 PM2.5, and 437-station AQI PM2.5.They report mean ± standard deviation; MAE, RMSE, and Peak-∆MAE use original physical units, while CRPS uses standardized space.Source: Table 11 · p. 31, Table 12 · p. 31, Table 13 · p. 32
  • Table 14 evaluates robustness at 65% and 75% node outages for traffic speed and AQI PM2.5, including valid-proxy coverage where applicable.It defines Peak as Peak-∆MAE; lower reconstruction errors and higher valid-proxy coverage are better.Source: Table 14 · p. 32
  • Table 15 reports the frequency of selected policies at 60% node outage, conditioned on valid-proxy outages only.Source: Table 15 · p. 32
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