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Calibrated Physics-Informed Uncertainty Quantification
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Proposes CP-PRE, a data-free conformal prediction framework that uses PDE residuals (computed with finite-difference convolution kernels) as nonconformity scores to give marginal and joint coverage guarantees for neural PDE surrogates.
Problem
Neural PDE surrogate solvers produce predictions with high confidence but lack statistically guaranteed uncertainty quantification, and existing UQ methods either lack guarantees, need extensive simulation data, or require architectural changes.
Approach
Use the physics residual error (PDE residual evaluated via finite-difference stencils implemented as convolution kernels) as a data-free nonconformity score in conformal prediction, so calibration needs no labelled simulation data and prediction sets are input-independent, with marginal (per-cell) and joint (whole-domain) variants.
Key results
- On the wave equation, CP-PRE achieves 95.52% in-distribution and 95.39% out-of-distribution coverage at 95% target with 32 s evaluation, versus 2022 s for CP-AER.
- On Navier-Stokes, CP-PRE reaches 95.44% (in-dist.) and 95.57% (OOD) coverage, while MC Dropout, Deep Ensemble, BNN and SWA-G fail to reach coverage on OOD data (e.g. Deep Ensemble 30.74%).
- CP-PRE is data-free and evaluates in 134 s on Navier-Stokes versus 20026 s for CP-AER, which requires simulation data for calibration.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use physics residual errors for conformal calibration instead of labelled-data absolute errors | FNO on Wave Equation · Wave Equation | evaluation time (s): 2022 -> 32 | Helped |
| Use physics residual errors for conformal calibration instead of labelled-data absolute errors | FNO on Wave Equation · Wave Equation in-distribution | coverage (95%): 95.70 ± 0.21 -> 95.52 ± 0.21 | No clear effect |
| Use physics residual errors for conformal calibration instead of labelled-data absolute errors | FNO on Navier-Stokes · Navier-Stokes | evaluation time (s): 20026 -> 134 | Helped |
| Use physics residual errors for conformal calibration instead of labelled-data absolute errors | FNO on Navier-Stokes · Navier-Stokes in-distribution | coverage (95%): 95.56 ± 0.40 -> 95.44 ± 0.22 | No clear effect |
| Use physics residual errors for conformal calibration instead of labelled-data absolute errors | FNO on Magnetohydrodynamics · Magnetohydrodynamics | evaluation time (s): 40042 -> 482 | Helped |
| Use physics residual errors for conformal calibration instead of labelled-data absolute errors | FNO on Magnetohydrodynamics · Magnetohydrodynamics out-of-distribution | coverage (95%): 95.60 ± 0.27 -> 95.67 ± 0.22 | No clear effect |
Surprising findings
- Bayesian UQ methods (MC Dropout, Deep Ensemble, BNN, SWA-G) fail to attain the target coverage on out-of-distribution data for several PDEs.
Limitations
- Coverage is in PDE residual space rather than physical variable space; transforming to physical space needs approximations or Monte Carlo sampling.
- Finite-difference residual estimation introduces Taylor truncation errors and the method is limited to regular grids with fixed spacing.
- Does not distinguish aleatoric from epistemic uncertainty.
Experimental setup
Datasets: simulated wave equation data (spectral solver) simulated Navier-Stokes data (spectral solver) simulated MHD Orszag-Tang data (finite volume) JOREK reduced-MHD tokamak plasma data 1D advection and Burgers simulations
Benchmarks: 2D wave equation 2D Navier-Stokes 2D MHD Orszag-Tang tokamak reduced-MHD temperature equation Grad-Shafranov equilibrium
Metrics: empirical coverage at 95% MSE train time evaluation time
Baselines: MC Dropout Deep Ensemble BNN SWA-G CP-AER
Builds on: Fourier Neural Operator inductive conformal prediction physics-informed neural networks
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