Deep Learning for Multigroup Cross-Section Representation in Two-Step Core Calculations

Abstract

We investigate using deep learning, a type of machine-learning algorithm employing multiple layers of artificial neurons, for the mathematical representation of multigroup cross sections for use in the Griffin reactor multiphysics code for two-step deterministic neutronics calculations. A three-dimensional fuel element typical of a high-temperature gas reactor as well as a two-dimensional sodium-cooled fast reactor lattice are modeled using the Serpent Monte Carlo code, and multigroup macroscopic cross sections are generated for various state parameters to produce a training data set and a separate validation data set. A fully connected, feedforward neural network is trained using the open-source PyTorch machine-learning framework, and its accuracy is compared against the standard piecewise linear interpolation model.

Additionally, we provide in this work a generic technique for propagating the cross-section model errors up to the k_eff using sensitivity coefficients with the first-order uncertainty propagation rule. Quantifying the eigenvalue error due to the cross-section regression errors is especially practical for appropriately selecting the mathematical representation of the cross sections. We demonstrate that the artificial neural network model produces lower errors and therefore enables better accuracy relative to the piecewise linear model when the cross sections exhibit nonlinear dependencies; especially when a coarse grid is employed, where the errors can be halved by the artificial neural network. However, for linearly dependent multigroup cross sections as found for the sodium-cooled fast reactor case, a simpler linear regression outperforms deeper networks.

Keywords:

• Multigroup cross-section interpolation
• deep learning
• regression
• reactor physics analysis

Acknowledgments

The authors would like to acknowledge Dr. Peter German from the MOOSE framework development team for initially performing the coupling between MOOSE and libtorch that paved the way for this work.

This work was performed and the paper was authored by Battelle Energy Alliance, LLC under contract no. DE-AC07-05ID14517 with the U.S. Department of Energy. The U.S. government retains and the publisher, by accepting the paper for publication, acknowledges that the U.S. government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this paper, or allow others to do so, for U.S. government purposes.

Disclosure Statement

No potential conflict of interest was reported by the authors.

Notes

^a For more information on the activation functions and the type of initializations see http://pytorch.org/docs/stable/nn.init.html.

^b See PyTorch documentation at https://pytorch.org/docs/stable/optim.html for more information.

Additional information

Funding

This work was funded under the Department of Energy’s Nuclear Energy Advanced Modeling and Simulation program. This research made use of Idaho National Laboratory computing resources, which are supported by the Office of Nuclear Energy of the U.S. Department of Energy and the Nuclear Science User Facilities under contract no. DE-AC07-05ID14517.