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Uncertainty Quantification for Forward and Inverse Problems of PDEs via Latent Global Evolution
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Proposes LE-PDE-UQ, which evolves latent state and latent uncertainty vectors in a global latent space with deep ensembles, giving calibrated uncertainty for forward and inverse PDE problems on 2D Navier-Stokes.
Problem
Neural PDE surrogate models provide no uncertainty estimates, and uncertainty accumulates over autoregressive rollouts of time-dependent PDEs, which matters for forward prediction and inverse optimization.
Approach
Extends the LE-PDE latent evolution model with a second latent uncertainty vector that is evolved alongside the state vector in latent space, decoded by a separate uncertainty decoder, and combined with deep ensembles; inverse optimization is done over latent initial vectors rather than the high-dimensional input.
Key results
- On forward Navier-Stokes prediction, LE-PDE-UQ (ensemble, with sigma) reaches MA 0.0142 and RMSCE 0.0160, versus 0.0602 MA for the NoLatent deep ensemble and 0.0445 MA for the best Bayes layer with latent.
- In teacher-forcing ablation, NoLatent ensemble MA drops from 0.0602 to 0.0260, suggesting most of the gap comes from latent uncertainty propagation.
- In inverse optimization, LE-PDE-UQ achieves MA 0.0224 versus 0.0929 for NoLatent ensemble, with L2 0.1863 versus 1.5255.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use latent evolution instead of input-space evolution (ablation) | LE-PDE-UQ · 2D Navier-Stokes forward, autoregressive rollout | MA: 0.0602 -> 0.0142 | Helped |
| Add latent uncertainty propagation with zσ (ablation) | Latent evolution · 2D Navier-Stokes forward | MA: 0.0602 -> 0.0142 | Helped |
| Add single-model σ prediction to the ensemble (ablation) | Latent ensemble · 2D Navier-Stokes forward | MA: 0.1823 -> 0.0142 | Helped |
| Add latent evolution to the Bayes layer (ablation) | Bayes layer · 2D Navier-Stokes forward | MA: 0.2381 -> 0.0445 | Helped |
| Add latent evolution to NoLatent with ensemble and σ (ablation) | Deep Ensembles · 2D Navier-Stokes forward | MA: 0.0602 -> 0.0142 | Helped |
| Use teacher-forcing instead of autoregressive rollout (ablation) | NoLatent ensemble with σ · 2D Navier-Stokes | MA: 0.0602 -> 0.0260 | Helped |
| Add latent evolution under teacher-forcing (ablation) | NoLatent ensemble with σ · 2D Navier-Stokes teacher-forcing | MA: 0.0260 -> 0.0101 | Helped |
| Optimize in latent space and decode instead of optimizing in input space (ablation) | LE-PDE-UQ · 2D Navier-Stokes inverse optimization | L2: 1.5255 -> 0.1863 | Helped |
| Use an ensemble instead of a single latent model (ablation) | Latent model with σ · 2D Navier-Stokes forward | MA: 0.0576 -> 0.0142 | Helped |
| Set Dropout L2 regularization to 10^-5 instead of 0 (ablation) | Dropout · 2D Navier-Stokes forward | MA: 0.1778 -> 0.1924 | Hurt |
| Increase Dropout L2 regularization from 10^-5 to 10^-4 (ablation) | Dropout · 2D Navier-Stokes forward | MA: 0.1924 -> 0.2317 | Hurt |
| Increase Dropout L2 regularization from 10^-4 to 10^-3 (ablation) | Dropout · 2D Navier-Stokes forward | MA: 0.2317 -> 0.3281 | Hurt |
Surprising findings
- Bayes layer with latent achieves better UQ calibration than the proposed model in some settings, though with worse point prediction error.
Limitations
- Evaluated only on a single 2D Navier-Stokes benchmark.
- Uncertainty for inverse optimization relies on deep ensembles rather than a native method.
- Some reported comparisons (e.g., Bayes layer with latent) are close to the proposed method on calibration metrics.
Experimental setup
Datasets: 2D turbulent Navier-Stokes (Re=1e4) simulated trajectories
Benchmarks: 2D Navier-Stokes turbulent flow (Re=1e4)
Metrics: miscalibration area (MA) MACE RMSCE relative L2 MAE
Baselines: Bayes layer MC dropout Deep Ensembles NoLatent ensemble
Builds on: LE-PDE deep ensembles temporal bundling
Closest papers by meaning
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