Physics-Informed Conformal Prediction: Embedding PDE Consistency into Distribution-Free Uncertainty Quantification for Neural Operators

Researchers proposed Physics-Informed Conformal Prediction (PI-CP), a framework that embeds PDE residuals into the nonconformity score of split conformal prediction to give neural operators distribution-free, spatially adaptive uncertainty estimates, according to an arXiv paper (2609.11935v1). Validated across six physics scenarios — heat conduction (2D/3D), structural mechanics (2D/3D), Darcy flow, and Navier-Stokes — PI-CP achieved consistent 89-91% coverage for all four conformal methods, while MC Dropout and Deep Ensembles ranged from 82-100%, and the Fourier Neural Operator outperformed CNN and DeepONet by 10-12x. The authors also proved that FNO's translation equivariance creates a fundamental approximation barrier for PDEs with Dirichlet boundary conditions, which coordinate channels resolve with up to 63x error reduction.

arXiv:2609.11935v1 Announce Type: new Abstract: Neural operators such as the Fourier Neural Operator FNO achieve remarkable accuracy in approximating solutions to partial differential equations PDEs . However, providing rigorous uncertainty estimates remains an open challenge. We propose Physics-Informed Conformal Prediction PI-CP , a framework that embeds PDE residuals into the nonconformity score of split conformal prediction, producing prediction intervals that are i distribution-free with provable coverage guarantees, and ii spatially adaptive when the PDE residual correlates with prediction error -- tighter where physics is well-satisfied, wider where it is violated. Additionally, we prove that FNO's translation equivariance creates a fundamental approximation barrier for PDEs with Dirichlet boundary conditions, and show that coordinate channels resolve this with up to 63x error reduction. We validate PI-CP across six physics scenarios -- heat conduction 2D/3D , structural mechanics 2D/3D , Darcy flow, and Navier-Stokes -- demonstrating consistent 89-91% coverage for all four Conformal methods, while MC Dropout and Deep Ensembles are unstable 82-100% . FNO outperforms CNN and DeepONet by 10-12x.