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PI-CP brings provably valid prediction intervals to neural PDE solvers

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Neural operators such as the Fourier Neural Operator (FNO) can approximate solutions to partial differential equations (PDEs) accurately. Giving those predictions rigorous uncertainty estimates, however, has remained an open challenge. A new paper proposes Physics-Informed Conformal Prediction (PI-CP), a framework that embeds the PDE's own residual, how far a prediction deviates from satisfying the governing equation, into the nonconformity score used by split conformal prediction. The resulting prediction intervals are distribution-free with provable coverage guarantees. They are also spatially adaptive when the PDE residual correlates with prediction error: tighter where the PDE residual shows the physics is well satisfied, wider where it is violated.

The authors validate PI-CP across six physics scenarios: heat conduction in 2D and 3D, structural mechanics in 2D and 3D, Darcy flow, and Navier-Stokes fluid dynamics. All four conformal prediction methods they test reach a consistent 89 to 91% coverage across these scenarios. Two widely used non-conformal alternatives, Monte Carlo Dropout and Deep Ensembles, prove unstable by comparison, swinging between 82% and 100% coverage on the same problems. Separately, the underlying FNO architecture itself outperforms CNN and DeepONet by a factor of 10 to 12.

The paper's second contribution concerns FNO's architecture rather than its uncertainty estimates. The authors prove that FNO's translation equivariance creates a fundamental approximation barrier for PDEs with Dirichlet boundary conditions, which fix a specific value at the edge of the domain. They show that adding coordinate channels to FNO resolves this barrier, cutting the resulting error by up to 63 times.

Key facts

  • Proposes Physics-Informed Conformal Prediction (PI-CP), which folds a PDE's own residual into the nonconformity score of split conformal prediction, giving distribution-free prediction intervals with provable coverage guarantees that are also spatially adaptive when the PDE residual correlates with prediction error.
  • Validated across six physics scenarios (2D and 3D heat conduction, 2D and 3D structural mechanics, Darcy flow, and Navier-Stokes), all four conformal methods tested hold a consistent 89 to 91% coverage.
  • Monte Carlo Dropout and Deep Ensembles, the two non-conformal baselines, prove unstable by comparison, swinging between 82% and 100% coverage on the same six scenarios.
  • The underlying Fourier Neural Operator (FNO) itself outperforms CNN and DeepONet baselines by a factor of 10 to 12.
  • Separately proves that FNO's translation equivariance creates a fundamental approximation barrier for PDEs with Dirichlet boundary conditions, and shows that adding coordinate channels resolves it, cutting the error by up to 63 times.

Why it matters

The source frames rigorous uncertainty estimation for neural operators as an open challenge, since these models can approximate PDE solutions accurately but without a trustworthy way to flag where a given prediction might be wrong. PI-CP addresses that by using the PDE residual itself, how far a prediction strays from satisfying the governing equation, as the basis for its confidence intervals. The result combines two properties: distribution-free coverage guarantees that hold regardless of the underlying data, and intervals that adapt spatially when the PDE residual correlates with prediction error, tightening where the physics is well satisfied and widening where it is violated.

Who it affects

The direct audience is researchers and engineers building or evaluating neural-operator surrogates for physics simulation, in the fields the paper tests: heat conduction and structural mechanics, each validated in 2D and 3D, subsurface flow modeled with Darcy's law, and fluid dynamics governed by the Navier-Stokes equations. It also speaks directly to anyone choosing among FNO, CNN and DeepONet as the underlying operator, or among conformal prediction, Monte Carlo Dropout and Deep Ensembles as the uncertainty method sitting on top of it, since the paper compares all of them head to head.

How to use it

PI-CP works as a wrapper placed around an already trained neural operator; conformal methods of this kind calibrate a model's output without retraining it. It takes the operator's prediction, measures how far that prediction deviates from satisfying the governing PDE, and feeds that residual into the nonconformity score that split conformal prediction uses to size its intervals. Applied this way across six different physics scenarios, the approach produced a consistent 89 to 91% coverage using all four conformal methods the authors evaluated.

How solid is it

The paper backs its central claim with breadth: it validates PI-CP across six distinct physics settings, from 2D heat conduction to 3D structural mechanics to Navier-Stokes fluid flow, and reports a narrow, consistent 89 to 91% coverage for all four conformal methods tested across every one of them. The comparison baselines make the contrast concrete: Monte Carlo Dropout and Deep Ensembles, described in the source as unstable, swing between 82% and 100% coverage on the same problems, a much wider range. The FNO approximation barrier is presented as a proven result rather than only an empirical observation: the authors state they prove it mathematically, then confirm empirically that adding coordinate channels fixes it, cutting the error by up to 63 times.

Risks and caveats

The four conformal prediction methods that reach 89 to 91% coverage are not individually named in the source, so it is not possible to tell which specific techniques were tested or whether the result extends to others. The source also gives no comparison of computational or training cost between PI-CP and the baseline uncertainty methods, Monte Carlo Dropout and Deep Ensembles, so any overhead from computing the PDE residual for every prediction is unclear. The demonstrated results are limited to the six scenarios named: heat conduction and structural mechanics in 2D and 3D, plus Darcy flow and Navier-Stokes. The paper is posted on arXiv, a preprint server, meaning it has not necessarily been through formal peer review.