GenUQ: Predictive Uncertainty Estimates via Generative Hyper-Networks
This work addresses uncertainty quantification for researchers in computational science and engineering, offering a novel approach for surrogate modeling in areas like physics and materials science, though it is incremental as it builds on prior operator learning methods.
The paper tackles the challenge of uncertainty quantification in operator learning for PDEs by introducing GenUQ, a generative hyper-network method that avoids likelihood construction, and demonstrates its superiority over existing methods in recovering operators and modeling stochastic systems with concrete performance gains.
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.