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Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEs
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Shows that UQ methods for neural operators fail under out-of-domain PDE-parameter shifts while ensembles do well, proposes DiverseNO (multi-head FNO with diversity regularization) and Operator-ProbConserv (applying ProbConserv conservation updates with its uncertainties) to improve OOD accuracy and UQ.
Problem
Neural operators trained on PDE parameters are unreliable out-of-domain: existing uncertainty quantification methods give uncorrelated uncertainty estimates OOD, and ensembles that do work are computationally expensive.
Approach
Use a multi-head last feed-forward layer with a regularizer that maximizes diversity among head weights to mimic ensemble OOD uncertainty cheaply, then plug the resulting variances into the ProbConserv constrained-projection update to enforce conservation and correct OOD predictions.
Key results
- DiverseNO achieves 2x to 70x lower n-MeRCI than BayesianNO, VarianceNO and MC-DropoutNO across PDEs and OOD shifts, comparable to the 10-model EnsembleNO.
- Operator-ProbConserv with DiverseNO improves OOD MSE by 1.4x on the heat equation and fixes oscillations on the Stefan problem where BayesianNO+ProbConserv fails.
- The abstract reports OOD accuracy improvements of up to 34% from Operator-ProbConserv.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use EnsembleNO instead of the other tested UQ methods | FNO on the 1-d heat equation · 1-d heat equation, large OOD shift | n-MeRCI: 0.8 vs 0.05 | Helped |
| Use DiverseNO rather than computationally cheap UQ baselines | FNO for GPME PDEs · GPME, OOD shifts | n-MeRCI: 2× to 70× | Helped |
| Use DiverseNO instead of EnsembleNO at similar FLOPs | FNO on PME with medium OOD shift · PME, medium OOD shift | MSE: 49% to 80% better | Helped |
| Apply ProbConserv using DiverseNO uncertainty estimates | DiverseNO · Heat equation, OOD shifts | MSE: 1.4× | Helped |
| Apply ProbConserv using DiverseNO uncertainty estimates | DiverseNO · Heat equation, OOD shifts | MSE relative to baselines: 1.8× to 5.5× | Helped |
| Apply ProbConserv using VarianceNO uncertainty estimates | VarianceNO · PME and Stefan, small OOD shifts | MSE: 1300% and 31% | Hurt |
| Apply ProbConserv with DiverseNO uncertainty estimates | DiverseNO on medium and hard GPME cases · Medium and hard GPME, large OOD shifts | prediction error: ≈10^{-2} | No clear effect |
| Add diversity regularization over prediction-head weights (ablation) | DiverseNO · 1-d heat equation, OOD shifts | MSE and n-MeRCI | Helped |
| Remove diversity regularization (set λ_diverse = 0) (ablation) | DiverseNO · OOD tasks | MSE and n-MeRCI | Hurt |
| Set the number of prediction heads to M=10 (ablation) | DiverseNO · In-domain and OOD tasks | MSE, n-MeRCI, and FLOPs | Helped |
| Diversify prediction-head outputs or gradients instead of head weights (ablation) | DiverseNO · 1-d heat equation, OOD shifts | MSE and n-MeRCI | Hurt |
| Increase weight-diversity regularization strength (ablation) | DiverseNO · OOD tasks | MSE and n-MeRCI: ≈10 | Helped |
Surprising findings
- Existing UQ methods such as BayesianNO and VarianceNO give narrow, uncorrelated uncertainty bands OOD even when they are well calibrated in-domain.
- ProbConserv applied with uncalibrated uncertainties (VarianceNO) degrades MSE drastically, by 1300% for small OOD shifts on a medium-difficulty PDE.
Limitations
- Only the FNO base architecture is studied; other operators are left to future work.
- Global conservation constraints alone are insufficient for large OOD shifts on medium and hard PDEs.
Experimental setup
Datasets: synthetic PDE solutions for 1-d heat equation, porous medium equation, Stefan equation, linear advection, 2-d Darcy flow
Benchmarks: GPME family (heat, PME, Stefan) linear advection 2-d Darcy flow
Metrics: n-MeRCI MSE conservation error MSE ratio
Baselines: BayesianNO VarianceNO MC-DropoutNO EnsembleNO
Builds on: Fourier Neural Operator ProbConserv DeepEnsembles Bayesian Neural Operator DivDis
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