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

Michael Chin

arXiv:2609.11935v1cs.LG

TL;DR

Neural operators offer accurate PDE surrogates, but rigorous uncertainty quantification remains an open challenge. The paper introduces PI-CP, which embeds PDE residuals into conformal scores, and shows calibrated coverage across physics scenarios while coordinate channels address FNO’s Dirichlet-boundary approximation barrier.

  • Problem

    Neural-operator PDE surrogates lack rigorous uncertainty estimates, while standard deep-learning uncertainty methods lack theoretical coverage guarantees.

  • Method

    PI-CP embeds PDE residuals into split-conformal nonconformity scores, while coordinate-aware FNO theory analyzes and resolves the translation-equivariance barrier for Dirichlet conditions.

  • Results

    Across six physics scenarios, all four Conformal methods achieve 89–91% coverage, and coordinate channels resolve the FNO boundary barrier with up to 63× improvement.

  • Takeaways & Limitations

    PI-CP provides calibrated spatial adaptivity when PDE residuals reliably proxy model error, and coordinate channels should be used for FNOs on bounded Dirichlet domains.

  • Takeaways & Limitations

    PI-CP depends on residual-error correlation and an accurate base model, while FNO fails on shock-forming Burgers and exchangeability may break under distribution shift.

Abstract

from arXiv · show

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.

1 INTRODUCTION

The paper addresses unreliable uncertainty estimates for neural-operator PDE surrogates by embedding PDE residuals into conformal prediction and by resolving an FNO boundary-condition limitation with coordinate channels.

  • 1 INTRODUCTION: Neural operators provide fast, accurate PDE approximations, but safety-critical deployment requires calibrated confidence intervals beyond point predictions.FNO achieves sub-1% relative error on benchmark PDEs such as Darcy flow, Navier-Stokes, and elasticity.
  • 1 INTRODUCTION: PI-CP incorporates PDE residuals into conformal scores, yielding distribution-free calibrated and spatially adaptive intervals when residuals correlate with prediction error.Intervals become tighter where physics is well-satisfied and wider where it is violated.
  • 1 INTRODUCTION: FNO’s translation equivariance creates an approximation barrier for PDEs with Dirichlet boundary conditions.Coordinate channels address this architectural limitation.
  • 1 INTRODUCTION: 63% error on Darcy flow falls to 1% with grid coordinates, a 63× improvement.
  • 1 INTRODUCTION: Across six physics scenarios, all four Conformal methods achieve 89–91% coverage, while MC Dropout and Deep Ensembles lack reliable coverage guarantees.Standard deep-learning uncertainty methods show scenario-dependent coverage ranging from 82% to 100%.

2 RELATED WORK

Related work spans neural operators, conformal prediction, physics-informed learning, and equivariant architectures; PI-CP differs by using PDE residuals as an external physics-based uncertainty signal.

  • 2 RELATED WORK: Neural operators approximate mappings between function spaces but generally provide point predictions without uncertainty estimates.FNO uses spectral convolutions, DeepONet uses branch-trunk decomposition, and Graph Neural Operators support irregular meshes.
  • 2 RELATED WORK: Standard split conformal prediction provides finite-sample distribution-free marginal coverage under exchangeability, while locally adaptive methods normalize scores using estimated local difficulty.
  • 2 RELATED WORK: PI-CP uses PDE residuals rather than model-internal uncertainty to produce spatial adaptivity for neural operators.Well-trained FNO models can have near-uniform dropout variance, whereas residuals capture spatially varying physical information.
  • 2 RELATED WORK: Physics-Informed ML typically embeds PDE constraints during training, whereas PI-CP uses residuals at inference time without retraining.The approaches can be combined by applying PI-CP to a PINN-trained model.
  • 2 RELATED WORK: FNO’s translation equivariance is useful on periodic domains but creates a Dirichlet-boundary approximation barrier that coordinate channels resolve.The paper provides a group-theoretic explanation and 63× empirical validation.

3 BACKGROUND

The background formulates operator learning for PDE solution maps and introduces FNO spectral convolutions alongside split conformal prediction for constructing prediction intervals.

  • 3.1 FOURIER NEURAL OPERATOR: FNO applies learnable spectral weights to Fourier coefficients truncated to the lowest m modes.The architecture performs spectral convolutions in the frequency domain.
  • 3.2 SPLIT CONFORMAL PREDICTION: Split conformal prediction computes calibration nonconformity scores, selects the (1−α) quantile, and returns a symmetric interval around f(x).The interval is C(x) = [f(x) − q̂, f(x) + q̂].
  • 3 BACKGROUND: The PDE framework defines D[u](x) = 0 with boundary conditions and uses R(x) = D[f](x) as the residual of a neural-operator prediction.
  • 3 BACKGROUND: The operator-learning setting maps coefficient fields k to PDE solutions u through a solution operator S approximated by a neural operator fθ.The paper studies uncertainty quantification and the effect of FNO equivariance on approximating solution operators with Dirichlet boundaries.

4 METHOD

PI-CP incorporates PDE residuals into conformal scoring to produce calibrated intervals whose widths adapt to local physics, while coordinate channels remove FNO’s Dirichlet-boundary approximation barrier.

  • Physics-Informed Conformal Prediction: The score s = |y −f(x)| /(1 + λ |R(x)|) normalizes prediction error by the PDE residual, widening intervals where physics is violated.The residual is computed by applying the governing differential operator to the model prediction.
  • Physics-Informed Conformal Prediction: PI-CP retains marginal coverage under exchangeability while producing spatially adaptive intervals when residual magnitude positively correlates with prediction error.The guarantee remains marginal, while conditional-coverage benefits depend on positive residual-error correlation.
  • Interval Width Analysis: PI-CP’s width variation is controlled independently of the global quantile, with CV = std(|R|)/mean(|R|), while score compression reduces the calibration quantile.The resulting redistribution can provide substantial spatial variation without proportional average-width inflation.
  • Coordinate-Aware FNO: The architecture analysis proves that standard FNO is translation-equivariant whereas Dirichlet solution operators generally are not, creating an approximation barrier without coordinates.The mismatch follows from equivariant spectral convolutions and pointwise lifting and projection layers.
  • Coordinate-Aware FNO: Adding coordinate channels breaks the equivariance constraint, enabling position-dependent operators and making PDE residuals informative about local physics rather than architectural error.This transforms systematic boundary error into physics-correlated variation that PI-CP can exploit.

5 EXPERIMENTS

Experiments span six physics scenarios and compare coordinate ablations with conformal uncertainty methods under a 90% nominal-coverage setting.

  • Experimental Setup: The evaluation covers heat conduction, structural mechanics, Darcy flow, and Navier–Stokes across six two- and three-dimensional scenarios using held-out conformal calibration.Experiments use 200 calibration samples and 300 test samples with α=0.1.
  • Coordinate Channel Ablation: Adding coordinate channels reduces FNO errors by 1.2–63× across five scenarios, addressing translation-equivariance failures for Dirichlet boundary conditions.The ablation reports errors up to 76% without coordinates.
  • PI-CP Results on Darcy Flow: PI-CP maintains approximately 90% coverage while achieving width CV=0.221 at λ=1 with only a 4% width increase over standard CP.The comparison is performed on Darcy flow with a model rel_l2 of 1.0%.
  • PI-CP Results on Darcy Flow: At λ=5, PI-CP increases adaptivity to CV=0.643 at the cost of 31% wider intervals, while PI-Normalized CP achieves comparable adaptivity with slightly tighter intervals.The λ parameter therefore provides a smooth adaptivity–width trade-off.

5.4 ARCHITECTURE ABLATION

The architecture study finds that FNO is highly accurate on Darcy flow, with low-frequency modes and coordinate-aware designs supporting strong performance across PDE scenarios.

  • 5.4 ARCHITECTURE ABLATION: < 0.5% rel_l2 is achieved by small Darcy FNO models because the 64×64 problem is dominated by low-frequency content.The error plateaus at m=8, while m=8, width 32, and four layers already achieve this accuracy.
  • 5.4 ARCHITECTURE ABLATION: 0.66% to 0.45% rel_l2 follows from doubling Darcy training samples from 1000 to 2000, consistent with rel_l2 ∝N −0.5.Further hyperparameter changes provide diminishing returns across the tested settings.
  • 5.5 BASELINE COMPARISON: 0.47% error makes FNO outperform CNN by 9.8×, DeepONet by 11.5×, and MGN by 13.7× on Darcy flow.The MGN comparison uses 32×32 resolution, whereas the other models use 64×64; the fairer FNO gaps are against CNN and DeepONet.
  • 5.6 NAVIER-STOKES: 0.31% test error on Taylor-Green vortex shows FNO handling advection-diffusion dynamics across Re ∈[10, 100].Coordinate channels provide spatial anchoring for the predicted vorticity field.

5.7 BURGERS EQUATION: LIMITATION WITH SHOCKS

The Burgers study exposes a sharp limitation of spectral FNOs: smooth dynamics remain tractable, but shock-forming solutions cause catastrophic error because truncated Fourier representations cannot resolve discontinuities.

  • 5.7 BURGERS EQUATION: LIMITATION WITH SHOCKS: > 100% rel_l2 in the shock-forming regime shows FNO completely fails for Burgers solutions at low viscosity.The shock-forming regime uses ν ∈[0.001, 0.02], where discontinuities require infinitely many Fourier modes and truncation causes Gibbs phenomena.
  • 5.7 BURGERS EQUATION: LIMITATION WITH SHOCKS: 5.02% error on smooth Burgers is higher than the 0.03–0.33% errors reported for steady-state PDEs but remains reasonable for time evolution.The smooth regime uses ν ∈[0.05, 0.5].
  • UQ COMPARISON: 90–91% coverage is achieved by all four conformal methods, while MC Dropout spans 82.2–100% and Deep Ensembles 82.1–97.8%.The cross-scenario UQ comparison is reported at the 90% nominal level; PI-CP adaptivity depends on residual-error structure.
  • UQ COMPARISON: PI-CP reaches width CV=0.23–0.49 on Darcy and Navier-Stokes but only CV=0.001 for cantilever elasticity and can over-widen thermal 2D intervals.Thermal 2D has ρ = −0.14, whereas the stronger-adaptivity scenarios have residuals that correlate with model errors.

6 DISCUSSION

The discussion frames PI-CP as useful when PDE residuals track model error, while emphasizing coordinate channels as a prerequisite for reliable FNO predictions on bounded Dirichlet domains.

  • 6 DISCUSSION: PI-CP is most effective when residuals reflect model error and exhibit spatial correlation with actual prediction errors.The recommended diagnostic is ρ = corr(|R(x)| , |y −f(x)|); ρ ≤0 may cause over-widening.
  • 6 DISCUSSION: Darcy and Navier-Stokes achieve width CV=0.23–0.49, whereas cantilever elasticity reaches CV=0.001 because residual-error correlation is poor.Thermal 2D has ρ = −0.14 and correspondingly exhibits over-widening.
  • 6 DISCUSSION: λ ∈[0.5, 2.0] is recommended, with selection by validation-set width minimization subject to the coverage constraint.λ →0 recovers standard conformal prediction, while larger λ increases adaptivity but can widen intervals.
  • 6 DISCUSSION: The main scope limits are Dirichlet-boundary analysis, scenario-dependent residual-error correlation, distribution-shift sensitivity, and failure on shock-forming Burgers.Theorem 1 gives an existence bound whose constant depends on the PDE operator and domain geometry.

7 CONCLUSION

PI-CP provides distribution-free coverage while enabling spatial adaptivity when PDE residuals track model error, and coordinate channels overcome FNO’s Dirichlet-boundary approximation barrier. The approach is validated across six physics scenarios, but broader deployment requires industrial-scale validation beyond the current benchmarks.

  • Across six physics scenarios, all four Conformal methods achieve 89–91% coverage, while MC Dropout and Deep Ensembles show unstable 82–100% coverage without guarantees.PI-CP retains the marginal coverage guarantees of standard conformal prediction.
  • PI-CP achieves spatial adaptivity when PDE residuals correlate with model errors, with minimal width increase in Darcy flow but possible over-widening for thermal problems.Darcy flow shows a +4% width increase, whereas discretization error in thermal problems can break residual-error correlation.
  • The paper recommends computing Pearson ρ(|R|, |y −f|) before applying PI-CP to assess whether residuals are reliable error proxies.
  • Coordinate channels resolve FNO’s translation-equivariance barrier for Dirichlet boundary conditions, producing up to 63× improvement.The result motivates matching neural-operator symmetries to the target PDE’s boundary-condition structure.
  • Current validation uses synthetic, analytical, and controlled FEM benchmarks rather than industrial-scale CAE data with mesh irregularity, multiphysics coupling, and measurement noise.These real-engineering factors may affect residual-error correlation and remain targets for future validation.

A REPRODUCIBILITY DETAILS FOR UQ EXPERIMENTS

The UQ experiments use held-out calibration and test sets under a 90% nominal coverage level, with method-specific configurations for MC Dropout, Deep Ensembles, and CQR. Table 7 provides the full cross-scenario comparison, while this section supplies reproducibility details.

  • Table 7 reports the full UQ cross-scenario comparison, with this section providing additional reproducibility details.
  • Conformal experiments use 200 calibration and 300 test samples per scenario, except CQR with 250 calibration and 250 test samples, at α = 0.1.
  • MC Dropout uses 30 stochastic forward passes with dropout rate 0.1, while Deep Ensembles use three independently trained FNO models.
  • CQR uses an FNO with out_channels=3, pinball loss at quantiles [0.05, 0.5, 0.95], and 200 training epochs.

B IMPLEMENTATION DETAILS

The implementation standardizes experiments around a shared FNO architecture and a common platform, while supporting multiple conformal, stochastic, ensemble, and quantile-based UQ methods. Physics solvers cover Darcy flow, Navier–Stokes, heat, and structural mechanics.

  • All experiments use the same FNO architecture with modes=12, width=32, four layers, and approximately 1.2M parameters.This common architecture supports cross-scenario consistency.
  • Models include two- and three-dimensional FNOs with spectral convolution, implemented in Python 3.10+ with PyTorch.
  • The platform includes Standard, Normalized, PI-CP, PI-Normalized, MC Dropout, Deep Ensemble, and CQR uncertainty methods.
  • Physics solvers cover Darcy flow, analytical Taylor–Green Navier–Stokes, finite-difference heat, and FEM structural problems.
  • Training was performed on an NVIDIA V100 with 16GB memory, and code is planned for release upon publication.
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