Operator Sensitivity as an Inductive Prior for Uncertainty Quantification in DeepONets

We investigate whether the input-output sensitivity of a learned operator can provide an inductive prior for the spatial structure of predictive uncertainty. We introduce Jacobian-DeepONet, which parameterizes asymmetric prediction intervals using the normalized pointwise Jacobian norm and two additional trainable scalars, jointly optimized with the operator under mean-squared-error and pinball losses. We also introduce Jacobian Conformal Bands (JCB), which calibrate sensitivity-shaped intervals on a frozen operator without retraining. Both methods exploit the Cartesian DeepONet factorization to compute Jacobian norms exactly through a latent Gram matrix using $K$ reverse-mode vector-Jacobian products, without materializing the full input–output Jacobian. On viscous Burgers, Darcy flow, and compressible Navier-Stokes, the methods achieve competitive interval widths after calibration and stronger alignment with reference physical sensitivities than constant-width and learned-quantile baselines. Monte Carlo perturbations validate the first-order variance approximation, while experiments with noisy inputs show improved coverage retention for JCB relative to conditional quantile bands under the tested perturbations. Our results suggest that operator sensitivity provides useful, problem-dependent structure for prediction intervals in DeepONets, complementing uncertainty models learned from residuals.