@article{moloko_bokov_wu_ivanov_2023, title={Prediction and uncertainty quantification of SAFARI-1 axial neutron flux profiles with neural networks}, volume={188}, ISSN={["1873-2100"]}, url={https://doi.org/10.1016/j.anucene.2023.109813}, DOI={10.1016/j.anucene.2023.109813}, abstractNote={Artificial Neural Networks (ANNs) have been successfully used in various nuclear engineering applications, such as predicting reactor physics parameters within reasonable time and with a high level of accuracy. Despite this success, they cannot provide information about the model prediction uncertainties, making it difficult to assess ANN prediction credibility, especially in extrapolated domains. In this study, Deep Neural Networks (DNNs) are used to predict the assembly axial neutron flux profiles in the SAFARI-1 research reactor, with quantified uncertainties in the ANN predictions and extrapolation to cycles not used in the training process. The training dataset consists of copper-wire activation measurements, the axial measurement locations and the measured control bank positions obtained from the reactor's historical cycles. Uncertainty Quantification of the regular DNN models' predictions is performed using Monte Carlo Dropout (MCD) and Bayesian Neural Networks solved by Variational Inference (BNN VI). The regular DNNs, DNNs solved with MCD and BNN VI results agree very well among each other as well as with the new measured dataset not used in the training process, thus indicating good prediction and generalization capability. The uncertainty bands produced by MCD and BNN VI agree very well, and in general, they can fully envelop the noisy measurement data points. The developed ANNs are useful in supporting the experimental measurements campaign and neutronics code Verification and Validation (V&V).}, journal={ANNALS OF NUCLEAR ENERGY}, author={Moloko, Lesego E. and Bokov, Pavel M. and Wu, Xu and Ivanov, Kostadin N.}, year={2023}, month={Aug} }
 @article{bokov_botes_prinsloo_tomasevic_2022, title={A Multigroup Homogeneous Flux Reconstruction Method Based on the ANOVA-HDMR Decomposition}, ISSN={["1943-748X"]}, DOI={10.1080/00295639.2022.2108654}, abstractNote={Abstract In this paper, we formulate a new flux reconstruction method based on the High Dimensional Model Representation (HDMR) of the nodal flux shape. The method requires the conventional nodal parameters obtained as a result of the transverse-integration nodal solution procedure as well as corner values calculated by a combination of finite-volume and finite-difference approximations. The proposed flux reconstruction procedure is applicable to rectangular nodes and more than two energy groups. The flux reconstruction method is applied to a number of homogeneous and heterogeneous test problems in order to evaluate its accuracy.}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Bokov, Pavel M. and Botes, Danniell and Prinsloo, Rian H. and Tomasevic, Djordje I}, year={2022}, month={Sep} }
 @article{bokov_botes_groenewald_2021, title={DUAL NUMBER AUTOMATIC DIFFERENTIATION AS APPLIED TO TWO-GROUP CROSS-SECTION UNCERTAINTY PROPAGATION}, volume={36}, ISSN={["1452-8185"]}, DOI={10.2298/NTRP3602107B}, abstractNote={This work addresses the problem of propagating uncertainty from group-wise
   neutron cross-sections to the results of neutronics diffusion calculations.
   Automatic differentiation based on dual number arithmetic was applied to
   uncertainty propagation in the framework of local sensitivity analysis. As
   an illustration, we consider a two-group diffusion problem in an infinite
   medium, which has a solution in a closed form. We employ automatic
   differentiation in conjunction with the sandwich formula for uncertainty
   propagation in three ways. Firstly, by evaluating the analytical expression
   for the multiplication factor using dual number arithmetic. Then, by solving
   the diffusion problem with the power iteration algorithm and the algebra of
   dual matrices. Finally, automatic differentiation is used to calculate the
   partial derivatives of the production and loss operators in the perturbation
   formula from the adjoint-weighted technique. The numerical solution of the
   diffusion problem is verified against the analytical formulas and the
   results of the uncertainty calculations are compared with those from the
   global sensitivity analysis approach. The uncertainty values obtained in
   this work differ from values given in the literature by less than 1?10?5.}, number={2}, journal={NUCLEAR TECHNOLOGY & RADIATION PROTECTION}, author={Bokov, Pavel M. and Botes, Danniell and Groenewald, Suzanne A.}, year={2021}, month={Jun}, pages={107–115} }
 @article{mertyurek_jessee_betzler_2021, title={Lattice physics calculations using the embedded self-shielding method in polaris, Part II: Benchmark assessment}, volume={150}, ISSN={["0306-4549"]}, DOI={10.1016/j.anucene.2020.107925}, abstractNote={• Overview of the dual number (DN) arithmetic and DN automatic differentiation (DNAD). • Dual number arithmetic is applied in solving infinite homogeneous diffusion problem. • Two methods based on the DNAD are used for uncertainty propagation. • Utilising dual number arithmetic yielded results for k-inf correct to 3 pcm. • DNAD is a viable additional method for sensitivity analysis in nuclear engineering. Automatic differentiation (AD) is a set of techniques which allows the numeric evaluation of derivatives of functions calculated by a computer program. In recent years, interest in AD has grown significantly in many disciplines, especially in the context of gradient-based optimization algorithms. Sensitivity analysis is another natural application area for AD methods. However, despite the large body of sensitivity and uncertainty (S/U) analysis publications produced in the field of nuclear reactor science and engineering in the last decade, the use of AD by the community has been very limited. The purpose of the present paper is to fill this gap and to demonstrate how AD can be employed in conjunction with some traditionally used sensitivity analysis and uncertainty propagation techniques. Specifically, the forward mode of AD based on dual number arithmetic was considered in the study. We provide a short overview of dual number algebra and dual number automatic differentiation (DNAD) methods, as well as of the tools available for the practical implementation of DNAD, followed by a discussion of its application to S/U analysis. As illustration, we solve a simplistic example of an infinite, homogeneous diffusion problem using parameters that correspond to a plate-type, Material Testing Reactor fuel assembly. Homogenized cross-sections and uncertainty (covariance) data for the test problem are generated with the SCALE code in six energy groups. The diffusion problem is solved through the power iteration algorithm with the algebra of dual matrices, which yields sensitivity information for use in the sandwich formula. DNAD is also used to calculate partial derivatives of the production and loss operators in the perturbation formula in the context of the adjoint-weighted technique. Both of these methods yield uncertainty values for the multiplication factor that are within three pcm of the reference value. Automatic differentiation can, therefore, be useful for uncertainty propagation in the framework of local sensitivity analysis in addition to traditionally employed sampling methods or in conjunction with the perturbation method.}, journal={ANNALS OF NUCLEAR ENERGY}, author={Mertyurek, Ugur and Jessee, Matthew A. and Betzler, Benjamin R.}, year={2021}, month={Jan} }