Physics-Informed Conformal Prediction: Embedding PDE Consistency into Distribution-Free Uncertainty Quantification for Neural Operators
Physics-Informed Conformal Prediction (PI-CP) embeds PDE residuals into split conformal prediction's nonconformity score, producing distribution-free prediction intervals with provable coverage guarantees PI-CP generates spatially adaptive uncertainty estimates: tighter intervals where physics is well-satisfied, wider where PDE residuals indicate violations FNO's translation equivariance creates a fundamental approximation barrier for Dirichlet boundary conditions; adding coordinate channels res
Analysis
TL;DR
- Physics-Informed Conformal Prediction (PI-CP) embeds PDE residuals into split conformal prediction's nonconformity score, producing distribution-free prediction intervals with provable coverage guarantees
- PI-CP generates spatially adaptive uncertainty estimates: tighter intervals where physics is well-satisfied, wider where PDE residuals indicate violations
- FNO's translation equivariance creates a fundamental approximation barrier for Dirichlet boundary conditions; adding coordinate channels resolves this with up to 63x error reduction
- PI-CP achieves consistent 89-91% coverage across six physics scenarios (heat conduction, structural mechanics, Darcy flow, Navier-Stokes), outperforming MC Dropout and Deep Ensembles (82-100% unstable coverage)
- FNO outperforms CNN and DeepONet by 10-12x in accuracy across the tested physics benchmarks
Why It Matters
This work bridges two critical gaps in scientific machine learning: rigorous uncertainty quantification and physics consistency. For AI practitioners deploying neural operators in safety-critical domains like engineering and climate modeling, PI-CP offers statistically guaranteed coverage without distributional assumptions, while simultaneously leveraging physical laws to refine uncertainty estimates spatially.
Technical Details
- PI-CP Framework: Integrates PDE residual norms into the nonconformity score of split conformal prediction, enabling distribution-free uncertainty quantification with finite-sample coverage guarantees of the form P(true solution ∈ prediction set) ≥ 1 - α
- Spatial Adaptivity: When PDE residuals correlate with prediction error, PI-CP automatically produces tighter intervals in regions where the physics is well-satisfied and wider intervals where boundary conditions or physics are violated
- FNO Boundary Condition Analysis: Proves that FNO's translation equivariance inherently prevents accurate approximation of PDEs with Dirichlet boundary conditions; introducing coordinate channels as additional input features resolves this barrier, achieving up to 63x error reduction
- Benchmark Validation: Tested across six physics scenarios — 2D/3D heat conduction, 2D/3D structural mechanics, Darcy flow, and Navier-Stokes — comparing four conformal methods against MC Dropout and Deep Ensembles
- Performance Results: All four conformal methods maintained stable 89-91% coverage; MC Dropout and Deep Ensembles showed unstable coverage ranging from 82% to 100%; FNO exceeded CNN and DeepONet by 10-12x in accuracy
Industry Insight
- The provable coverage guarantees of PI-CP make it a strong candidate for regulatory-compliant deployment of physics ML models in industries like aerospace, energy, and civil engineering where uncertainty quantification is mandatory
- The finding that coordinate channels resolve FNO's Dirichlet boundary limitation should inform architectural choices for any neural operator deployed on bounded domains with fixed boundary values
- The instability of MC Dropout and Deep Ensembles in physics-informed settings suggests practitioners should favor conformal prediction methods over ensemble-based uncertainty estimation when deploying neural operators for scientific computing applications
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