This paper proposes an approach that combines reduced-order models with machine learning in order to create physics-informed digital twins to predict high-dimensional output quantities of interest, such as neutron flux and power distributions in nuclear reactor cores. The digital twin is designed to solve forward problems given input parameters, as well as to solve inverse problems given some extra measurements. Offline, we use reduced-order modeling, namely, the proper orthogonal decomposition, to assemble physics-based computational models that are accurate enough for the fast predictive digital twin. The machine learning techniques, namely, k-nearest-neighbors and decision trees, are used to formulate the input-parameter-dependent coefficients of the reduced basis, after which the high-fidelity fields are able to be reconstructed. Online, we use the real-time input parameters to rapidly reconstruct the neutron field in the core based on the adapted physics-based digital twin. The effectiveness of the framework is illustrated through a real engineering problem in nuclear reactor physics—reactor core simulation in the life cycle of the HPR1000 governed by the two-group neutron diffusion equations affected by input parameters, i.e., burnup, control rod inserting step, power level, and temperature of the coolant—which shows potential applications for online monitoring purposes.

Machine learning-based modeling of reactor physics problems has attracted increasing interest in recent years. Despite some progress in one-dimensional problems, there is still a paucity of benchmark studies that are easy to solve using traditional numerical methods albeit still challenging using neural networks for a wide range of practical problems. We present two networks, namely the Generalized Inverse Power Method Neural Network (GIPMNN) and Physics-Constrained GIPMNN (PC-GIPIMNN) to solve K-eigenvalue problems in neutron diffusion theory. GIPMNN follows the main idea of the inverse power method and determines the lowest eigenvalue using an iterative method. The PC-GIPMNN additionally enforces conservative interface conditions for the neutron flux. Meanwhile, Deep Ritz Method (DRM) directly solves the smallest eigenvalue by minimizing the eigenvalue in Rayleigh quotient form. A comprehensive study was conducted using GIPMNN, PC-GIPMNN, and DRM to solve problems of complex spatial geometry with variant material domains from the field of nuclear reactor physics. The methods were compared with the standard finite element method. The applicability and accuracy of the methods are reported and indicate that PC-GIPMNN outperforms GIPMNN and DRM.

The aging of operational reactors leads to increased mechanical vibrations in the reactor interior. The vibration of the in-core sensors near their nominal locations is a new problem for neutronic field reconstruction. Current field-reconstruction methods fail to handle spatially moving sensors. In this study, we propose a Voronoi tessellation technique in combination with convolutional neural networks to handle this challenge. Observations from movable in-core sensors were projected onto the same global field structure using Voronoi tessellation, holding the magnitude and location information of the sensors. General convolutional neural networks were used to learn maps from observations to the global field. The proposed method reconstructed multi-physics fields (including fast flux, thermal flux, and power rate) using observations from a single field (such as thermal flux). Numerical tests based on the IAEA benchmark demonstrated the potential of the proposed method in practical engineering applications, particularly within an amplitude of 5 cm around the nominal locations, which led to average relative errors below 5% and 10% in the $$L_2$$L 2 and $$L_{\infty }$$L ∞ norms, respectively.