Paper Overview
Field: Machine Learning Authors: Tom Stent, Nicolas Boullé Published: 2026-08-28 arXiv: 2608.28515
Summary
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 \(1-\gamma\) fraction of the evaluation domain, with probability at least \(1-\alpha\) over test and calibration inputs, where \(\alpha,\gamma\in(0,1)\).
Key Points
- The method reduces a normalized residual field to its spatial \((1-\gamma)\)-quantile and computes a scaling factor using a held-out calibration dataset.
- Marginal coverage guarantees are proved for measurable residual fields defined on arbitrary probability spaces, covering both continuum domains and fixed discretizations.
- Under mild assumptions on the data distribution, coverage given the calibration set follows a Beta distribution.
- Numerical experiments on Darcy flow and Navier-Stokes equations show the calibration yields consistently tighter bands than existing corrections while maintaining target coverage.
Original Abstract
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 \(1-\gamma\) fraction of the evaluation domain, with probability at least \(1-\alpha\) over test and calibration inputs, where \(\alpha,\gamma\in(0,1)\). Our method reduces a normalized residual field to its spatial \((1-\gamma)\)-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 ...
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