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Rare event modeling with self-regularized normalizing flows: what can we learn from a single failure?
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Proposes CalNF, a self-regularized conditional normalizing flow that learns posteriors from very few failure observations by training on target subsamples with a calibrated label, and applies it to benchmarks and the 2022 Southwest Airlines crisis.
Problem
Learning generative posterior models of rare failure events from very limited target data, where prior regularization requires a hard-to-tune penalty and bootstrapped ensembles remain data-sensitive.
Approach
Train one conditional normalizing flow on random subsamples of target data plus nominal data, using one-hot labels for subsamples and a zero label for nominal; then freeze the flow and optimize a low-dimensional calibration label on the full target set, with a KL regularizer between candidate target posteriors providing self-regularization.
Key results
- CalNF achieves the best held-out target ELBO on SWI (46.4 ± 0.26 x10^-3 nats/dim) and UAV (7.55 ± 0.60) and ATC (-2.11), while the ensemble is best on the 2D toy (-0.84 vs -0.90).
- Ablations indicate most of the gain comes from training on random target subsamples; removing nominal data drops UAV ELBO to -9.22 ± 10.
- The case study infers aircraft deficits accumulating at LAS, DAL, and PHX over the first four days of the Southwest 2022 disruption, and service-time increases only at weather-affected airports (DEN, MDW, DAL).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CalNF instead of the ensemble baseline | Rare-event posterior inference · 2D, SWI, UAV, and ATC | ELBO (nats/dim): 2D: -0.84_{\pm 0.14} vs -0.90_{\pm 0.10}; SWI: 46.1_{\pm 0.42} vs 46.4_{\pm 0.26}; UAV: 6.65_{\pm 0.98} vs 7.55_{\pm 0.6 | Mixed |
| Add calibration by optimizing c* rather than fixing c (ablation) | CalNF · UAV | ELBO (nats/dim): 6.95_{\pm 1.2} vs 7.86_{\pm 1.0} | Hurt |
| Add calibration by optimizing c* rather than fixing c (ablation) | CalNF · 2D, SWI, and ATC | ELBO (nats/dim): 2D: -0.96_{\pm 0.2} -> -0.90_{\pm 0.1}; SWI: 46.2_{\pm 0.4} -> 46.3_{\pm 0.2}; ATC: -2.02_{\pm 0.1} -> -2.01_{\pm 0.1} | Helped |
| Include nominal data during training (ablation) | CalNF · UAV | ELBO (nats/dim): -9.22_{\pm 10} -> 6.95_{\pm 1.2} | Helped |
| Include nominal data during training (ablation) | CalNF · 2D, SWI, and ATC | ELBO (nats/dim): 2D: -1.12_{\pm 0.2} -> -0.90_{\pm 0.1}; SWI: 46.1_{\pm 0.4} -> 46.3_{\pm 0.2}; ATC: -2.03_{\pm 0.2} -> -2.01_{\pm 0.1} | Helped |
| Use random target-data subsamples during training (ablation) | CalNF · 2D, SWI, UAV, and ATC | ELBO (nats/dim): 2D: -1.03_{\pm 0.2} -> -0.90_{\pm 0.1}; SWI: 43.9_{\pm 2.8} -> 46.3_{\pm 0.2}; UAV: -3.65_{\pm 11} -> 6.95_{\pm 1.2}; AT | Helped |
Surprising findings
- Larger regularization strength beta helps in extremely data-sparse regimes while smaller beta is better when target data is larger.
- Theory shows bootstrapping has no effect on data sensitivity in highly data-constrained settings as K grows to infinity, matching the ensemble to the non-bootstrapped solution.
Limitations
- Training costs K+1 joint likelihood evaluations per step, roughly 1.7-2x slower than the ensemble on the benchmarks.
- Does not estimate probability of failure risk.
- Assumes failure examples share structure with nominal data; may not help for radically different distributions.
Experimental setup
Datasets: Southwest Airlines public arrival and departure data ALFA UAV dataset synthetic seismic waveform inversion data (Deepwave) MNIST CIFAR-10 synthetic 2D toy problem
Benchmarks: 2D toy SWI UAV (ALFA) ATC air traffic MNIST few-shot CIFAR-10 few-shot UAV and SWI anomaly detection
Metrics: ELBO nats/dim on held-out target data AUROC AUCPR negative log-likelihood (bits/dim)
Baselines: KL-regularized normalizing flow W2-regularized RNODE ensemble of normalizing flows supervised anomaly detector
Builds on: neural spline flows prior regularization bootstrapping ensembles Zuko Pyro
Closest papers by meaning
- Enhanced Importance Sampling Through Latent Space Exploration in Normalizing Flows (AAAI 2025 · similarity 0.82)
- PAGER: Accurate Failure Characterization in Deep Regression Models (ICML 2024 · similarity 0.81)
- Scalable, Explainable and Provably Robust Anomaly Detection with One-Step Flow Matching (NeurIPS 2025 · similarity 0.81)
- Harnessing Event Sensory Data for Error Pattern Prediction in Vehicles: A Language Model Approach (AAAI 2025 · similarity 0.81)
- An Error Analysis of Flow Matching for Deep Generative Modeling (ICML 2025 · similarity 0.81)
- Uncertainty Quantification for Deep Regression using Contextualised Normalizing Flows (NeurIPS 2025 · similarity 0.81)
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