# Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime Source: [Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime]() Collection index: [llms.txt]() Data updated: 2026-06-10T05:01:35.776Z Authors: Weinan Wang, Bowen Gang, Hao Deng Published: 2026-06-07T14:49:37.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Conformal Prediction on 2D Navier-Stokes (test)](): Conformal Prediction; 2D Navier-Stokes (test); Coverage: 97.99. ## Original sources and implementation [Original paper]() [PDF]()