New Framework Quantifies Uncertainty in Neural Operators.

Neural operators are powerful tools for quickly solving Partial Differential Equations (PDEs), acting as fast surrogates. However, their deterministic predictions limit their application in scenarios requiring an understanding of uncertainty, especially when dealing with varying geometries. Existing uncertainty quantification (UQ) methods often focus on network parameters, overlooking the geometry-aware features learned by the operator itself. Researchers propose REEF-GP (Residual on Embedded Features Gaussian Process), a novel post-hoc UQ framework. Instead of learning new features, REEF-GP fits a Gaussian Process to the residuals of a frozen neural operator, using the operator's internal embeddings to define the kernel feature space. This approach directly adapts the operator's learned coordinate-feature representations to construct geometry-aware uncertainties. To ensure stability and scalability, REEF-GP incorporates spectral-normalized projections, heteroscedastic geometry-aware noise, and efficient subset-based training. Across five PDE benchmarks with diverse geometries, REEF-GP maintains predictive accuracy while delivering calibrated uncertainty estimates comparable to deep ensembles but with significantly lower computational cost. It also demonstrates robustness to geometric distribution shifts, concentrating uncertainty in physically meaningful areas like shock fronts.