Inverse UQ

Inverse Uncertainty Quantification (UQ)

Uncertainty Quantification (UQ) is an essential step in computational model validation because assessment of the model accuracy requires a concrete, quantifiable measure of uncertainty in the model predictions. The concept of UQ in the nuclear community generally means forward UQ, in which the information flow is from the inputs to the outputs. Inverse UQ, in which the information flow is from the model outputs and experimental data to the inputs, is an equally important component of UQ but has been significantly underrated until recently. Forward UQ requires knowledge in the computer model input uncertainties, such as the statistical moments, probability density functions, upper and lower bounds, which are not always available. Historically, expert opinion or user self-evaluation have been predominantly used to specify such information in previous Verification, Validation and Uncertainty Quantification (VVUQ) studies. Such ad-hoc specifications are subjective, lack mathematical rigor, and can sometimes lead to inconsistencies. Inverse UQ is defined as the process to inversely quantify the input uncertainties based on experimental data. It seeks statistical descriptions of the uncertain input parameters that are consistent with the observation data.

Inverse UQ methods can be broadly categorized by three main groups: frequentist (deterministic), Bayesian (probabilistic), and empirical (design-of-experiments). Our group has been developing Bayesian inverse UQ methods, which assume that the uncertain input parameters have true but unknown values, and use probabilistic treatment of these parameters with uncertain distributions, because it is impossible to quantify the exact values given limited available information. The Bayesian inverse UQ methods are built upon the Bayes’ rule as a procedure to update information after observing experimental data. Knowledge about the physical model uncertainties is first characterized as prior distributions, which is updated to posterior distributions based on a comparison of model and data. When sampling-based approaches are used to explore the posterior distributions, such as Markov Chain Monte Carlo (MCMC) sampling, the computational cost can be tremendous because typically tens of thousands of samples are needed. In this case, surrogate models based on machine learning algorithms (e.g. Gaussian Processes, deep neural networks, etc.) are usually employed.

We have developed an innovative inverse UQ method called the modular Bayesian approach (MBA). It has been successfully demonstrated on two cases with different levels of complexity: system thermal-hydraulics code TRACE with a relatively large amount of void fraction data from the BFBT benchmark, and fuel performance code Bison with very limited time series fission gas release measurement data. Compared to traditional Bayesian calibration, modularization was introduced to separate various modules in Bayesian inverse UQ to prevent suspect information belonging to one part from overly influencing another part. The resulting modular Bayesian-based inverse UQ process has reduced complexity from reasonable simplification and better convergence for MCMC sampling. The most significant characteristic of the MBA method is the simultaneous consideration of all major sources of quantifiable uncertainties in modeling & simulation (M&S), i.e., uncertainties from parameter, experiment, model and code. The resulting posterior distributions can effectively represent the input uncertainties that are consistent with the experimental data. Our current work focuses on improving the MBA method by resolving several important existing issues in inverse UQ, including a more rigorous ML-based representation of the model uncertainty, extrapolation of the model uncertainty term to generalized domains and test source allocation when the experimental data is limited.

Relevant publications:

1. Xie, Z., Wang, C., and Wu, X. (2025). Hierarchical Bayesian Modeling for Inverse Uncertainty Quantification of System Thermal-Hydraulics Code using Critical Flow Experimental Data. International Journal of Heat and Mass Transfer, 239:126489.
   https://doi.org/10.1016/j.ijheatmasstransfer.2024.126489
2. Xie, Z., Yaseen, M., and Wu, X. (2024). Functional PCA and Deep Neural Networks-based Bayesian Inverse Uncertainty Quantification with Transient Experimental Data. Computer Methods in Applied Mechanics and Engineering, 420:116721
   https://doi.org/10.1016/j.cma.2023.116721
3. Wang, C., Wu, X., Xie, Z., and Kozlowski, T. (2023). Scalable Inverse Uncertainty Quantification by Hierarchical Bayesian Modeling and Variational Inference. Energies, 16(22):7664
   https://www.mdpi.com/1996-1073/16/22/7664
4. Xie, Z., Jiang,W., Wang, C., and Wu, X. (2022). Bayesian inverse uncertainty quantification of a MOOSE-based melt pool model for additive manufacturing using experimental data. Annals of Nuclear Energy, 165:108782
   https://www.sciencedirect.com/science/article/pii/S0306454921006599
5. Wu, X., Xie, Z., Alsafadi, F., and Kozlowski, T. (2021). A comprehensive survey of inverse uncertainty quantification of physical model parameters in nuclear system thermal–hydraulics codes. Nuclear Engineering and Design, 384:111460
   https://www.sciencedirect.com/science/article/pii/S002954932100412X
6. Che, Y., Wu, X., Pastore, G., Li, W., and Shirvan, K. (2021). Application of Kriging and Variational Bayesian Monte Carlo method for improved prediction of doped UO2 fission gas release. Annals of Nuclear Energy, 153:108046
   https://www.sciencedirect.com/science/article/pii/S0306454920307428
7. Lu, C., Wu, Z., and Wu, X. (2021). Enhancing the one-dimensional sfr thermal stratification model via advanced inverse uncertainty quantification methods. Nuclear Technology, 207(5):692–710
   https://www.tandfonline.com/doi/abs/10.1080/00295450.2020.1805259
8. Wu, X., Shirvan, K., and Kozlowski, T. (2019). Demonstration of the Relationship between Sensitivity and Identifiability for Inverse Uncertainty Quantification. Journal of Computational Physics, 396:12–30
   https://www.sciencedirect.com/science/article/pii/S0021999119304401
9. Wang, C., Wu, X., and Kozlowski, T. (2019). Gaussian process–based inverse uncertainty quantification for trace physical model parameters using steady-state PSBT benchmark. Nuclear Science and Engineering, 193(1-2):100-114
   https://www.tandfonline.com/doi/abs/10.1080/00295639.2018.1499279
10. Wu, X., Kozlowski, T., Meidani, H., and Shirvan, K. (2018). Inverse uncertainty quantification using the modular Bayesian approach based on Gaussian Process, Part 2: Application to TRACE. Nuclear Engineering and Design, 335:417–431
    https://www.sciencedirect.com/science/article/pii/S0029549318306411
11. Wu, X., Kozlowski, T., Meidani, H., and Shirvan, K. (2018). Inverse uncertainty quantification using the modular Bayesian approach based on Gaussian process, part 1: theory. Nuclear Engineering and Design, 335:339–355
    https://www.sciencedirect.com/science/article/pii/S0029549318306423
12. Wu, X., Kozlowski, T., and Meidani, H. (2018). Kriging-based Inverse Uncertainty Quantification of Nuclear Fuel Performance Code BISON Fission Gas Release Model using Time Series Measurement Data. Reliability Engineering & System Safety, 169:422–436
    https://www.sciencedirect.com/science/article/pii/S095183201730532X
13. Wu, X., Mui, T., Hu, G., Meidani, H., and Kozlowski, T. (2017). Inverse uncertainty quantification of TRACE physical model parameters using sparse gird stochastic collocation surrogate model. Nuclear Engineering and Design, 319:185–200
    https://www.sciencedirect.com/science/article/pii/S0029549317302406
14. Wu, X. and Kozlowski, T. (2017). Inverse uncertainty quantification of reactor simulations under the Bayesian framework using surrogate models constructed by polynomial chaos expansion. Nuclear Engineering and Design, 313:29–52
    https://www.sciencedirect.com/science/article/pii/S0029549316304824