Organizations: The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China · The Hong Kong University of Science and Technology, Hong Kong SAR, China
Neural operators (NOs) provide fast, resolution-invariant surrogates for mapping input fields to PDE solution fields, but their predictions can exhibit significant epistemic uncertainty due to finite data, imperfect optimization, and distribution shift. For practical deployment in scientific computing, uncertainty quantification (UQ) must be both computationally efficient and spatially faithful, i.e., uncertainty bands should align with the localized residual structures that matter for downstream risk management. We propose a structure-aware epistemic UQ scheme that exploits the modular anatomy common to modern NOs (lifting-propagation-recovering). Instead of applying unstructured weight perturbations (e.g., naive dropout) across the entire network, we restrict Monte Carlo sampling to a module-aligned subspace by injecting stochasticity only into the lifting module, and treat the learned solver dynamics (propagation and recovery) as deterministic. We instantiate this principle with two 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.
Figures & tables
Method
Training Runs
Inference Runs
Deep Ensembles [ 9 ]
10
1
Laplace Approximation [ 13 ]
1
–
MC Dropout [ 2 ]
1
10 – 1000
Table 1: Comparison of common epistemic UQ methods in neural operators.
Figure 1: Overview of neural operator architectures and our structure-aware uncertainty quantification (UQ) scheme.
Figure 2: Sampling from other parameter groups can lead to uninformative and low-quality samples.
Figure 3: Visualizations of the residual and uncertainty band fields. Compared with other methods, our approach provides a more accurate characterization of the residual fields with reliable coverage.
Figure 4: Comprehensive comparisons with dropout-based baselines. Notably, for each choice of p and T , MC Dropout must be recalibrated since the uncertainty bands are altered by the newly introduced stochasticity, whereas our method A can share a single constant calibration scale across all p and T settings. Note that, to obtain the case-wise Avg. B.W., one should multiply the reported values by the grid size N , which typically ranges from a few hundred to tens of thousands.
Figure 5: Demonstrations of sampling on the Lifting parameters v.s. Sampling on/with other additional choices
Figure 6: Naive MCDropout uncertainty for different dropout probabilities p and sample counts T . Higher p causes more biased predictions and lower coverage of the calibrated uncertainty bands.
Figure 7: Naive MCDropout uncertainty for varying dropout probability p and sample counts T . Relative to the smaller- T results in Figure 6 , increasing T yields a smoother, re-distributed uncertainty-band field.
Figure 8: Naive MCDropout uncertainty for varying dropout probability p with a relatively large sample count T=10,000 . The near-identical predictive means, uncertainty bands, and coverage indicate that further increasing T has a negligible impact on the estimated mean and uncertainty band.
Figure 9: Quantified uncertainty visualizations of the A-implementation of our method for varying dropout probabilities p and sample counts T . With an appropriate dropout rate (i.e., neither extremely small nor large), our approach yields consistently far less biased predictions and higher coverage with calibrated uncertainty bands than the naive MCDropout results in Figure 6 . Notably, the calibration scale k is shared across all p values in our method A, whereas naive MCDropout typically requires case-by-case tuning of k .
Figure 10: Quantified uncertainty visualizations of the A-implementation of our method for varying dropout probabilities p and sample counts T . An appropriate dropout rate (i.e., neither extremely small nor large) provides reliable uncertainty quantification even with small T shown in Figure 9 . Increasing T to 100 or 1000 offers little benefit for normal p but helps reduce under-coverage when the dropout probability is very small ( p=0.01 ).
Figure 11: Quantified uncertainty visualizations of the A-implementation of our method for varying dropout probabilities p and sample counts T . Compared with the previously small T cases, further increasing T to 10000 has fewer benefits on coverage. However, compared with the uncertainty bands in Figure 8 , the A-implementation of our method provides a more faithful estimation of the residual field and achieves consistently much higher coverage.
Figure 12: UQ visualizations of our method B on the 3D ShapeNet Car with C.R.: 0.9162 ( All ) / 0.9248 ( Press ) / 0.9896 ( Velocity ) and Normalized Avg. B.W.: 17.530033 ( All ) / 0.970439 ( Press ) / 67.208816 ( Velocity ).