GenUQ: Predictive Uncertainty Estimates via Generative Hyper-Networks

TL;DR

GenUQ introduces a measure-theoretic, likelihood-free uncertainty quantification framework for operator learning by training a generative hyper-network to sample parameter distributions whose induced output distribution matches observed data. The method minimizes an energy-score discrepancy between predicted and actual joint samples, enabling robust aleatoric UQ while keeping training efficient by generating only a subset of parameters as stochastic. Across three diverse problems, GenUQ outperforms traditional UQ methods in capturing stochastic behavior and delivering calibrated predictive distributions, with evidence of a regularizing effect in some tasks. The work expands the toolkit for SciML by providing a practical approach to quantify uncertainty in neural operators without relying on explicit likelihoods, though it acknowledges limitations in epistemic uncertainty quantification and suggests future extensions.

Abstract

Operator learning is a recently developed generalization of regression to mappings between functions. It promises to drastically reduce expensive numerical integration of PDEs to fast evaluations of mappings between functional states of a system, i.e., surrogate and reduced-order modeling. Operator learning has already found applications in several areas such as modeling sea ice, combustion, and atmospheric physics. Recent approaches towards integrating uncertainty quantification into the operator models have relied on likelihood based methods to infer parameter distributions from noisy data. However, stochastic operators may yield actions from which a likelihood is difficult or impossible to construct. In this paper, we introduce, GenUQ, a measure-theoretic approach to UQ that avoids constructing a likelihood by introducing a generative hyper-network model that produces parameter distributions consistent with observed data. We demonstrate that GenUQ outperforms other UQ methods in three example problems, recovering a manufactured operator, learning the solution operator to a stochastic elliptic PDE, and modeling the failure location of porous steel under tension.

Figures (7)

• Figure 1: (Left) Sample input function in training data. (Right, orange) Sample output function in training data. (Right, black) Actions of resampled operators on same input function.
• Figure 2: Comparison of UQ methods for recovering the 1D operator in \ref{['eq:elu']}. (Left) the predictive mean (orange) and 95% confidence intervals (blue) of the action of each method on a test function. (Right) Actions of resampled operators on same test function. Qualitatively, the learned GenUQ operator better matches the true operator. Energy distances between predictions and data, VI: 0.0265, CoV: 78.4386, NF: 0.1215, Gen: 0.2122, DO: 0.1994, GenUQ: 0.0020.
• Figure 3: Fine grained evaluation of GenUQ in ELU operator learning example. Stochastic operators' actions on test input function in Figure \ref{['fig:elu']}. (blue) Test data. (green) VI. (orange) GenUQ.
• Figure 4: (Top Left) Sample input function from training set. (Top Right) Sample action on input function from training set. (Bottom Right) Trace of sample input function along the red line in the corresponding contour plot. (Bottom Left, orange) Trace of sample output function along the red line. (Bottom Left, black) Action of multiple realizations of solution operator on fixed input function.
• Figure 5: Visual comparison of VI and GenUQ for recovering the stochastic Poisson equation. between predictions and data, VI: 0.2643 and GenUQ: 0.0780