Xiaoping Du

Xiaoping Du

ML models have errors when used for predictions. The errors are unknown but can be quantified by model uncertainty. When multiple ML models are trained using the same training points, their model uncertainties may be statistically dependent. In reality, model inputs are also random with input uncertainty. The effects of these types of uncertainty must be considered in decision-making and design. This study develops a theoretical framework that generates the joint distribution of multiple ML predictions given the joint distribution of model uncertainties and the joint distribution of model inputs. The strategy is to decouple the coupling between the two types of uncertainty and transform them as independent random variables. The framework lays a foundation for numerical algorithm development for various specific applications.

Amirreza Tootchi, Xiaoping Du

Machine learning surrogates are increasingly employed to replace expensive computational models for physics-based reliability analysis. However, their use introduces epistemic uncertainty from model approximation errors, which couples with aleatory uncertainty in model inputs, potentially compromising the accuracy of reliability predictions. This study proposes a Gauss-Hermite quadrature approach to decouple these nested uncertainties and enable more accurate reliability analysis. The method evaluates conditional failure probabilities under aleatory uncertainty using First and Second Order Reliability Methods and then integrates these probabilities across realizations of epistemic uncertainty. Three examples demonstrate that the proposed approach maintains computational efficiency while yielding more trustworthy predictions than traditional methods that ignore model uncertainty.

Xiaoping Du

Machine learning (ML) surrogate models are increasingly used in engineering analysis and design to replace computationally expensive simulation models, significantly reducing computational cost and accelerating decision-making processes. However, ML predictions contain inherent errors, often estimated as model uncertainty, which is coupled with variability in model inputs. Accurately quantifying and propagating these combined uncertainties is essential for generating reliable engineering predictions. This paper presents a robust framework based on Polynomial Chaos Expansion (PCE) to handle joint input and model uncertainty propagation. While the approach applies broadly to general ML surrogates, we focus on Gaussian Process regression models, which provide explicit predictive distributions for model uncertainty. By transforming all random inputs into a unified standard space, a PCE surrogate model is constructed, allowing efficient and accurate calculation of the mean and standard deviation of the output. The proposed methodology also offers a mechanism for global sensitivity analysis, enabling the accurate quantification of the individual contributions of input variables and ML model uncertainty to the overall output variability. This approach provides a computationally efficient and interpretable framework for comprehensive uncertainty quantification, supporting trustworthy ML predictions in downstream engineering applications.