Neural operators provide fast surrogate models for approximating operators between function spaces, but their predictions often lack uncertainty quantification. We develop a split conformal framework to guarantee that a calibrated pointwise band around the neural operator output contains the true solution on at least a 1−γ fraction of the evaluation domain, with probability at least 1−α over test and calibration inputs, where α,γ∈(0,1). Our method reduces a normalized residual field to its spatial (1−γ)-quantile and computes a scaling factor using a held-out calibration dataset. We prove marginal coverage guarantees for measurable residual fields defined on arbitrary probability spaces, covering both continuum domains and fixed discretizations. Under mild assumptions on the data distribution, we show that the coverage conditional on the calibration set follows a Beta distribution, which we verify with numerical experiments on Darcy flow and Navier--Stokes equations, where our calibration yields bands consistently tighter than existing corrections while retaining the target coverage.
Figures & tables
Figure 1: 2D Darcy flow dataset. Left: median calibrated scaling factor λ^ with respect to the spatial resolution N over 3000 random disjoint re-splits ( ncal=500 , ntest=1000 ) along with empirical 5 th– 95 th percentiles; smaller λ^ indicates tighter bands. Right: empirical distribution of resampled coverage at resolution N=60 compared with the exact Beta-Binomial law predicted by Theorem 2.5 at the target level 1−α=0.9 . Our distribution aligns with the predicted law, while ( Ma et al., 2024 ) is shifted right with more conservative and wider bands.
Figure 2: Navier–Stokes at N=64 : input vorticity, solution at t=21 , FNO prediction, pointwise error, and estimated error field for a calibrated operator pair. The estimated error tracks the spatial error pattern, enabling tight pointwise bands while maintaining the target containment fraction 1−γ .
Appendix figures & tables7 assets
Supplementary material from the paper’s appendix.
Appendix
Darcy flow
Navier–Stokes
Resolutions N
12,20,30,42,60,84
16,32,64
Fourier modes
10 at N=12 ; 16 otherwise
16,24,32
Hidden channels
32
32
Fourier layers
4
4
Lifting / projection
64 / 64
64 / 64
Optimiser
AdamW
AdamW
Appendix
Table 1: Configuration for both problems. Fourier modes are given per axis and are capped by the grid size at each resolution.
Ours
Ma et al.
N
λ^
B
λ^
B
ratio
12
1.84
9.43×10−4
4.38
2.24×10−3
2.37
20
1.59
6.90×10−4
2.24
9.71×10−4
1.41
30
1.48
5.52×10−4
1.92
7.13×10−4
1.29
42
1.64
4.62×10−4
2.05
5.77×10−4
1.25
60
1.42
5.41×10−4
1.66
6.34×10−4
1.17
Appendix
Table 2: Darcy flow: calibrated factor λ^ (single calibration split), mean bandwidth B , and width ratio between methods at each resolution. Mean bandwidth is B:=ntest1∑i=1ntestM1∑j=1Mλ^E^M(Ai)(xj) .
Figure 3: Darcy flow: distribution of re-sampled coverage by resolution and method, with means and target 1−α . Our distributions are centred near the target at each N , consistent with Theorem 2.5 ; those of Ma et al. (2024) are shifted upward and collapse to coverage 1 at the coarsest resolution.
Figure 4: Darcy flow: coverage on the fine reference grid ( Nmax=420 ) versus calibration resolution. Left: exact spectral interpolation of calibrated coarse-grid bands. Right: zero-shot super-resolution. This evaluates cross-grid transfer, which is outside the scope of Theorem 2.5 (a fixed-grid guarantee). The correction in Ma et al. (2024) increases width (up to 2.4× ) yet reaches the target only at N=84 , and zero-shot coverage is 0 at all resolutions.
Ours
Ma et al.
N
λ^
B
λ^
B
ratio
16
1.92
0.419
3.42
0.747
1.78
32
1.76
0.354
2.34
0.472
1.33
64
1.72
0.354
2.11
0.436
1.23
Appendix
Table 3: Navier-Stokes: calibrated factor λ^ , mean bandwidth B , and the width ratio between the methods, per resolution.
Figure 5: Navier–Stokes. Left: resampled median calibrated scaling factor λ^ versus resolution N , with empirical 5 th– 95 th percentile bars. Right: resampled coverage at N=64 against the Beta-Binomial reference implied by Theorem 2.5 and the target 1−α .
Figure 6: Navier–Stokes: distribution of re-sampled coverage by resolution and method, with means and target 1−α . Our distributions centre near the target, while those of Ma et al. (2024) are shifted upward with smaller spread, indicating conservative validity.
Neural operators such as the Fourier Neural Operator (FNO) have emerged as powerful surrogates for solving partial differential equations (PDEs), achieving speedups of several orders of magnitude over traditional numerical solvers. However, deploying these models in safety-critical engineering applications -- such as thermal management of electronic components and battery systems -- requires not only accurate point predictions but also rigorous uncertainty guarantees. Existing uncertainty quantification (UQ) methods for neural operators, including Monte Carlo Dropout and Deep Ensembles, provide only relative uncertainty estimates without formal coverage guarantees. In this work, we propose the first application of split conformal prediction to neural operator-based physics simulation, providing distribution-free prediction intervals with finite-sample coverage guarantees. We further introduce a normalized conformal prediction scheme that leverages MC Dropout uncertainty to produce adaptive-width intervals, yielding tighter intervals in regions of low uncertainty and wider intervals where the model is less certain. Full-scale experiments (33.7M parameters, 800 training samples, 5 ensemble members, NVIDIA V100) on steady-state heat conduction benchmarks demonstrate that our method achieves 89.1% empirical coverage at the target level of alpha=0.1, while producing spatially adaptive prediction intervals that reflect the underlying physical uncertainty structure. We also provide an uncertainty decomposition framework that separates epistemic uncertainty (68% of total) from aleatoric uncertainty (32% of total), offering actionable guidance for data collection and model improvement. Our method is implemented in an open-source platform with REST API endpoints and interactive 3D visualization.
In this paper, we propose a perturbation-based conformal prediction framework for uncertainty quantification in operator learning, with a focus on the 2D Navier--Stokes equations. While neural operators provide fast surrogates for expensive PDE solvers, they do not by themselves provide calibrated uncertainty for spatiotemporal field predictions. Our approach wraps a trained Fourier Neural Operator (FNO) with split conformal prediction and constructs the local uncertainty scale by comparing the predictions of two operators trained on nearly identical datasets: one on the original labels and one on labels perturbed by small Gaussian noise. We consider this procedure in the data-scarce regime, where the total label budget is fixed and methods that require a separate uncertainty network must divide training data between multiple models. On the 2D Navier--Stokes benchmark, the perturbation-based method produces substantially narrower conformal bands than existing methods under matched total data budgets while maintaining the target simultaneous coverage. These results suggest that perturbation sensitivity is a practical and sample-efficient uncertainty proxy for conformalized neural operators.
Weinan Wang, Bowen Gang, Hao Deng
Department of Mathematics, University of Oklahoma, Norman, OK, USA · Department of Statistics and Data Science, School of Management, Fudan University, Shanghai, China
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.