Neural Operators (NOs) represent a fast solution for mapping input fields to PDE solution fields, but their predictions can exhibit significant uncertainties.

Structure-Aware Uncertainty Quantification

Uncertainty quantification (UQ) must be computationally efficient and spatially faithful. A new structure-aware UQ scheme exploits the modular anatomy common to modern NOs (lifting-propagation-recovering). Instead of applying unstructured weight perturbations, Monte Carlo sampling is restricted to a module-aligned subspace by injecting stochasticity only into the lifting module, and treating the learned solver dynamics (propagation and recovery) as deterministic.

Implementation and Testing

This principle was instantiated with lightweight lifting-level perturbations, including channel-wise multiplicative feature dropout and a Gaussian feature perturbation with matched variance, followed by standard calibration to construct uncertainty bands. Experiments on challenging PDE benchmarks (including discontinuous-coefficient Darcy flow and geometry-shifted 3D car CFD surrogates) demonstrate that the proposed structure-aware design yields more reliable coverage, tighter bands, and improved residual-uncertainty alignment compared with common baselines, while remaining practical in runtime.