Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime

arXiv:2606.08654v1 Announce Type: new Abstract: 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.

Abstract: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.

Machine Learning (cs.LG); Analysis of PDEs (math.AP); Numerical Analysis (math.NA); Applications (stat.AP)

arXiv:2606.08654 [cs.LG]

(or arXiv:2606.08654v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2606.08654 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime

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