Research map / Probabilistic and scientific modeling / Uncertainty and Gaussian processes
Enabling Uncertainty Estimation in Iterative Neural Networks
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Proposes using the convergence rate (variance across successive outputs) of iterative networks as a cheap uncertainty proxy, matching Deep Ensembles at roughly 2.7-3x cost, and applies it to road delineation and aerodynamic shape optimization.
Problem
Iterative (recursive) networks that feed their own output back as input boost accuracy but do not provide uncertainty estimates; existing methods like Deep Ensembles are accurate but computationally expensive.
Approach
Treat the sequence of outputs from successive iterations of a recursive network as a signal: the variance across iterations (convergence speed) serves as a per-pixel/node uncertainty proxy, requiring no architectural change and only a single trained model, supervised at every iteration.
Key results
- On RoadTracer, the iterative model reaches F1 24.5 and APLS 77.21 vs 22.1 and 68.81 for Deep Ensembles, with rAULC 69.23 vs 72.19 for Ensembles (comparable uncertainty quality).
- On Massachusetts, the method achieves rAULC 79.27 and uncertainty correlation 87.46, the best among compared methods, at about 2.8x training and 2.7x inference cost versus 5x for Ensembles.
- On airfoil and car OOD detection, ROC-AUC is 0.87 (airfoils) and 0.86 (cars) vs 0.88 and 0.90 for Deep Ensembles, i.e., comparable rather than superior.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a recursive U-Net instead of a standard U-Net | U-Net · RoadTracer | F1: 21.1 -> 24.5 | Helped |
| Use a recursive U-Net instead of Deep Ensembles | U-Net · Massachusetts | APLS: 67.53 -> 78.04 | Helped |
| Estimate uncertainty from successive recursive predictions instead of MC-Dropout | U-Net · Massachusetts | Corr: 32.50 -> 87.46 | Helped |
| Estimate uncertainty from successive recursive predictions instead of Deep Ensembles | U-Net · RoadTracer | Corr: 79.42 -> 74.73 | Hurt |
| Estimate uncertainty from successive recursive predictions instead of Deep Ensembles | U-Net · Massachusetts | rAULC: 78.65 -> 79.27 | Helped |
| Estimate uncertainty from successive recursive predictions instead of Deep Ensembles | U-Net · RoadTracer vs Massachusetts OOD | PR-AUC: 67.85 -> 72.11 | Helped |
| Estimate uncertainty from successive iterative GNN predictions instead of Deep Ensembles | GNN · Airfoils OOD | PR-AUC: 0.86 -> 0.88 | Helped |
| Estimate uncertainty from successive iterative GNN predictions instead of Deep Ensembles | GNN · Cars OOD | ROC-AUC: 0.90 -> 0.86 | Hurt |
| Add gradient-based shape refinement during Bayesian optimization (ablation) | Ours · Airfoil optimization | optimized lift-to-drag ratio | Helped |
| Increase the number of gradient-based shape-refinement steps (ablation) | Ours · Airfoil optimization | optimized lift-to-drag ratio | No clear effect |
| Apply Deep Ensembles to the iterative architecture instead of using the single iterative model (ablation) | Iterative U-Net · RoadTracer | F1: 24.5 -> 24.4 | No clear effect |
| Apply Deep Ensembles to the iterative architecture instead of using the single iterative model (ablation) | Iterative U-Net · Massachusetts | F1: 15.4 -> 14.8 | Hurt |
Surprising findings
- The Ensemble-of-iterative-models variant performs better on some uncertainty metrics than the proposed single iterative model, which the authors note as costing about 14x training time.
- Ensembles beat the proposed method on rAULC and uncertainty correlation for RoadTracer and on OOD ROC-AUC for cars.
Limitations
- Uncertainty quality on some metrics and tasks is comparable to or below Deep Ensembles rather than clearly superior.
- Requires an iterative architecture trained with supervision at every iteration; not applicable to standard pass-through networks without modification.
Experimental setup
Datasets: RoadTracer Massachusetts XFoil-simulated airfoils (NACA) ShapeNet car meshes with OpenFOAM simulations MNIST FashionMNIST
Benchmarks: RoadTracer Massachusetts Airfoils Cars MNIST
Metrics: APLS CCQ (Corr/Comp/Qual) F1 rAULC Pearson correlation ROC-AUC PR-AUC ECE MAE lift-to-drag optimization
Baselines: U-Net MC-Dropout Deep Ensembles Kriging KNN GNN
Builds on: Deep Ensembles MC-Dropout recursive U-Net delineation Bayesian optimization
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