Hybrid Machine Learning Model Forecasts Turbulent Flow Dynamics from Sparse Sensor Data

Neural operators such as the Fourier Neural Operator (FNO) have emerged as fast surrogate solvers for computationally expensive partial differential equations (PDEs), but they do not inherently provide calibrated uncertainty estimates for their spatiotemporal predictions. To address this gap, researchers introduced a perturbation-based conformal prediction framework that trains two FNOs on nearly identical datasets—one on original labels and one on labels perturbed by small Gaussian noise—and uses the divergence between their predictions as a local uncertainty scale. This uncertainty proxy is then wrapped with split conformal prediction to produce statistically valid simultaneous coverage bands. The method is specifically designed for the data-scarce regime, where a fixed total label budget makes it costly to allocate data to a separate uncertainty-estimation network. On the standard 2D Navier-Stokes benchmark, the proposed approach yields substantially narrower conformal prediction bands than competing methods under matched data budgets, while still achieving the desired coverage. The results suggest that perturbation sensitivity is a sample-efficient and practical signal for uncertainty quantification in conformalized neural operators. The work sits at the intersection of machine learning, numerical analysis, and applied statistics, with potential implications for scientific computing applications where both speed and reliability are critical.