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

2026-06-07

AI Summary

This work addresses the challenge of achieving both accurate predictions and well-calibrated spatiotemporal uncertainty quantification for solutions to partial differential equations under data-scarce conditions, where existing neural operators often fall short. The authors propose a perturbation-based conformal prediction framework that constructs local uncertainty scales by comparing the outputs of two Fourier Neural Operators (FNOs)—one trained on original labels and the other on Gaussian-noise-perturbed labels—and integrates these scales into split conformal prediction for efficient uncertainty quantification. Notably, the method avoids training a separate uncertainty network and, under a fixed label budget, yields substantially narrower prediction bands while rigorously maintaining the desired coverage level. Experiments on the 2D incompressible Navier–Stokes equations demonstrate that the approach produces tighter and better-calibrated conformal prediction intervals compared to current methods at the same data budget.

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.

Problem

• operator learning
• uncertainty quantification
• Navier-Stokes equations
• data-scarce regime
• conformal prediction

Innovation

• conformal prediction
• operator learning
• Fourier Neural Operator
• uncertainty quantification
• data-scarce regime