Neutron Transport Equation

The neutron transport equation delineates the distribution of neutron flux across a cross-sectional plane within a heterogeneous fission medium [61]. In principle, this equation can yield more precise solutions. Nonetheless, the high-fidelity data obtained from solving the transport equation require extensive computational time and substantial computational resources. These limitations often render iterative core design impractical for commercial applications. Conversely, the advantages and disadvantages of grid-based methods are increasingly acknowledged in real-world applications. The accuracy of these methods is fundamentally constrained by the granularity of the grid, quantified by the number of nodes, and the geometric complexity of the problem domain [15, 72].

Neutron Diffusion Equation

Solutions derived from the neutron diffusion equation can be extended to a variety of industrial applications, including the simulation and design of nuclear reactors. Advanced reactor simulations often incorporate multiphysics modeling and feature intricate geometries that require more sophisticated analytical capabilities than those provided by conventional grid-based methods.

Point Kinetic Equations

The point reactor kinetics (PRK) model is derived from point kinetic equations (PKEs). This model assumes that the flux shape function follows the fundamental mode, which is the solution to the external neutron-source-free balance equation. This approximation remains valid primarily when the system is near criticality, where fission sources dominate the external sources [79].

Compared to traditional numerical methods, the PINN method does not require the time-consuming construction of complex grids and has a short computational time; therefore, it can be applied more efficiently to solve problems in irregular and high-dimensional domains. Considering the advantages of PINN, Prantikos et al. [39] developed a PINN model for solving PKEs and obtained results that were consistently compared to standard numerical methods. Schiassi et al. [66] proposed a new method for solving temperature feedback PKEs based on the PINN method, which is flexible and problem-adaptable. Gatchalian et al. [79] explored the role of the ML method in utilizing measurable quantities from reactor operations and experiments as predictors of k_eff and kinetic/subcritical parameters.

Transfer learning physics-informed neural network (TL-PINN), the process of pre-training a neural network on data similar to those used for training to enhance performance on a new task, is currently less used in transient reactor physics problems. Konstantinos et al. [47] achieved an improvement in reactor transient (RTs) prediction performance with the help of a TL-PINN method, which reduces the number of model iterations. Specifically, they used a PRK model with six neutron precursor populations, constructed using the experimental parameters of the Purdue University Reactor-1 (PUR-1) research reactor, to generate different reactor RTs with a range of experimentally relevant variables, while the similarity was characterized using the Hausdorff and Fréchet distances. The model yielded accelerations of up to two orders of magnitude. The average error in the prediction of neutron densities by the conventional PINN and TL-PINN models was less than 1%.

Multiplication Factor

The multiplication factor, an important quantity in reactor physics, is a measurement of the change in the number of fission neutrons from one neutron generation to the next and is a direct reflection of the ability of a nuclear reactor to maintain conditions for a self-sustaining chain reaction. It is vital to achieve an accurate prediction of the multiplication factor, which determines the critical conditions of the reactor, and accurate values that enable reactor operators to make informed decisions regarding fuel management and overall safety.

The original application of ML methods focused on predicting the multiplication factor, and at the same time, the PPF. Kim et al. [80] developed a fast core parameter prediction system based on a pressurized water reactor (PWR) using back propagation neural networks (BPNs) to model the multiplication factor and the maximum power, which resulted in a large speed improvement with errors within a few percent. Ortiz et al. [19] introduced a method based on a multi-state recurrent neural network to optimize the nuclear fuel lattice system of a BWR to predict the local PPF and the infinite multiplication factor (k_inf). Trontl et al. [22] used the support vector regression (SVR) algorithm for the global k_eff at the cycle begin and end, respectively, as well as the PPF as the target parameter, respectively, to build an ML model and optimize the parameters by genetic algorithms (GA), and obtained a good result with the root mean square error (RMSE) controlled in the order of 10^-2.

The application of ML methods in reactor physics is maturing and has achieved better results, particularly in terms of accuracy improvement in recent studies. Berry et al. [30] trained an ANN and random forest (RF) classifier to determine whether a given 20-group energy structure enabled a multi-group collision probability model to compute an accurate neutron multiplication factor in LWR lattice simulations, and the trained model achieved classification with 95.3% and 95.5% accuracy, respectively. Ren et al. [81] explored the potential of utilizing Long Short-Term Memory (LSTM) neural networks to predict the multiplication factor in nuclear reactor physics, modeled the multiplication factor with the first cycle of loading the BEAVRS core for 0–300 days of full-power operation, and adjusted the appropriate parameters to make the predicted k_eff within 2 pcm absolute error.

Oktavian et al. [72] first developed a DNN-based error correction model with high-fidelity data for a 2*2 BWR corelset, generated by the Monte Carlo method. They utilized multiple ML methods to generate high-fidelity data from the error correction of low fidelity data. Compared with traditional numerical methods, such as Monte Carlo simulations, the results of the DNN and XGBoost methods are better. The error of k_eff for PARCS assisted with DNN and XGBoost correction models can be reduced from 200–300 pcm to approximately 50 pcm or lower on average. Bei et al. [82] developed a BP neural network-based surrogate model for the fast prediction of the multiplication factor in the core, based on data modeled and simulated using the Monte Carlo method for the different operating states of the MSR experiment. The maximum absolute error (MAE) of the k_eff prediction surrogate model was reduced to approximately 70 pcm, significantly improving the accuracy of the core calculations compared to traditional physics-based models, where most errors were within ± 50 pcm, with a maximum absolute error of 70 pcm, demonstrating the reliability of the k_eff surrogate model.

Neutron Cross-section

In the two-step approach, which is currently the most commonly used in nuclear reactor simulations, the first step is the generation of uniform few-group cross-sections [83]. Taking high-temperature gas-cooled reactors (HTGR) as an example, to improve the accuracy of the reactor physics module of the HTGR Engineering Simulation System (HTR-ESS), more accurate computational models of the cross-section are generally built in real-time simulations. However, owing to the complex nonlinear relationship between the cross-section parameters and other state parameters, there is no generalized model [84, 85].

Ravichandran et al. [86] trained DNN to predict the value of the cross-section of 2D pin cell model and 2D lattice model nodes of a PWR and confirmed the feasibility of this method in PWR. They also mentioned the optimization of the algorithm with the help of transfer learning (TL) to reduce the dependence on a large dataset.

Vicente-Valdez et al. [84] proposed to support complex tasks of nuclear reactor evaluators with the help of ML technology. They developed decision tree and KNN methods to fit nuclear data and infer the induced reaction cross-sections for neutrons. The predicted values based on the ML methods matched the new measurements accurately. In their work, they explored proof-of-concept models that rely on learning the underlying patterns of cross-sectional data from other radionuclides, demonstrating that ML models can assist traditional physics-guided models and play a role in nuclear data evaluation. They [84] also pointed out that incorporating the ML model into the nuclear data module can enable evaluators to make faster, unbiased decisions in areas of uncertainty and better inform data measurement activities in the most sensitive areas of future Experimental Nuclear Reaction Data (EXFOR). Finally, Ivanov [87] explored the use of ML methods to address the problems faced when approximating neutron cross-sections using NNs.

For industrial applications, especially in the absence of data, Sun et al. [88] investigated the (n, 2n) reaction cross-section of fission products using ANN and XGBoost models to predict these reaction cross-sections. ANN predictions with MAPE deviations of less than 10% accounted for more than 85% of the test set. To meet the real-time requirements in ESS, Li et al. [89] proposed the use of a DNN to consider the complex nonlinear relationships among the reactor variables and used tree regression (TR) to obtain an accurate cross-section and optimize the cross-section calculation among them.

ML methods must be used with caution for different problems. Martin et al. [83] investigated a DL framework for the mathematical representation of multi-group cross-sections in the Griffin reactor multi-physics code for two-step deterministic neutronic calculations. They provided a technique for propagating cross-sectional model errors to k_eff using sensitivity coefficients with first-order uncertainty propagation rules. They also showed that the MLP achieves an accuracy from 1765 to 800 pcm in cross-sections with nonlinear dependence, particularly in the case of rough networks. For linearly dependent multigroup cross-sections in sodium-cooled fast reactors (SFR), simple linear regression is more suitable.

Cross-section prediction is beneficial for high-precision fuel depletion calculations. Leniau et al. [90] discussed the characterization of the spatial distribution of Pu in nuclear fuel as well as the establishment and validation of burnup models. They obtained the spatial distribution of Pu using the Latin Hypercube Sampling (LHS) method and proposed a method based on NNs, which can be used to build a fuel loading model for PWR loaded with MOX fuel and a cross-section predictor. The average cross-section predictor can complete the fuel-consumption calculations with high accuracy in a short time.

Related studies have also explored the internal structure of the cross-section in terms of its internal structure. To determine the resonant structure of the cross-sections, Nobre et al. [91] applied ML methods to automatically assign the angular momentum quantum numbers of the resonances.

Neutron Flux

ML techniques have been used in neutron flux flattening problems since 2002, when Sadighi et al. [92] proposed combining a continuous Hopfield NN with a simulated annealing algorithm (SA) to guarantee an optimal solution with reduced computational effort. A good result was obtained in solving the neutron flux flattening problem at the Bushehr nuclear power plant.

To address the problem of calculating the continuous neutron flux distribution, Xie et al. [34] proposed two boundary processing methods based on ANN and compared their accuracy and efficiency by analyzing benchmark tests. The experimental results show that the latter method is more accurate. They further proposed a trial function construction method to generalize its application.

Zhang et al. [71] proposed a DL-based agent model to replace the traditional diffusion equation solver and thus predict the flux and power distributions of reactor cores. They designed a CNN inspired by a fully convolutional network (FCN) and obtained positive results.

By combining reduced-order modeling and ML methods, Gong et al. [10] proposed a data-enabled physics-informed machine learning (PIML) digital twin model to predict high-dimensional outputs, such as neutron flux distributions and power distributions in reactor cores. Using KNN and decision trees, they attempted to demonstrate the effectiveness of HPR-1000 for online monitoring. This study also explores how the model can be used to quickly reconstruct the neutron field for real-time input parameters.

Furthermore, Bei et al. [82] developed a BPNN-based proxy model for the fast prediction of the channel-by-channel neutron flux distribution in the 3D channel of an MSR core. They generated dataset samples by simulating different operating states of the MSR using Monte Carlo methods and trained and optimized a proxy model. In addition, Chen et al. [35] analyzed the neutron flux in the help of multiple regression methods. As mentioned previously, Elhareef et al. [43] applied the PINN method to a neutron diffusion model to solve a loosely coupled reactor model and used it to optimize the NN hyperparameters, which resulted in a significant reduction in the predicted neutron flux error.

ML techniques have also been used for data enhancement, with Berry et al. [93] exploring the use of CNN to improve the resolution of neutron flux statistics in Monte Carlo simulations.

k-eigenvalue Problems

The k-eigenvalue problems based on diffusion theory have several applications in ML techniques. One of the earliest applications of ML in the prediction of the neutron diffusion equation dates back to 1999. In this study, a query-based adaptive retraining approach was developed, which involved reconstructing a two hidden-layer neural network for the rapid prediction of the eigenvalues of the fundamental mode of the neutron diffusion equation, specifically the effective multiplication factor (k_eff) [94].

Based on their earlier work [77], Elhareef et al. [78] investigated solving the k-eigenvalue diffusion equations in a multiregional configuration subject to a set of Robin boundary conditions, and the relative percentage errors of the two example k-eigenvalues were approximately 0.77% and 1.2%, respectively. They further investigated the forward PINN method, which was applied to the solution of the neutron diffusion equation for single and multi-energy groups, by using a freely learnable parameter to approximate the eigenvalues and a new regularization technique to exclude zero solutions from the PINN frameworks [43]. The computed results are compared with the FEM solutions of the neutron fluxes and power iteration solutions of k_eff. The errors in the fluxes ranged from 0.63% for a simple fixed-source problem to 15% for a two-set k-eigenvalue problem. The predicted k eigenvalue deviations from the power iteration solver ranged from 0.13% to 0.92%. To solve the problem of solution infinity when calculating the neutron diffusion equation using deterministic methods, Li et al. [95] used the power calculation equation as a supplementary equation to form a system of differential-integral equations with the neutron diffusion equation and designed a coupled numerical and machine learning (ML) algorithm based on the finite-volume method of gradient updating for the system of equations, which was applied to calculate the neutron flux densities and volumetric heat rates of the energy groups at specified powers.

In a recent study, Yang et al. [7] proposed a data-enabled PINN (DEPINN) for solving the neutron diffusion eigenvalue problem. They [96] also performed uncertainty analysis and further numerical studies, and the proposed interval loss function to deal with noisy a priori data items significantly improved the robustness of DEPINN to noise. The proposed method is expected to solve practical nuclear engineering problems in the field of nuclear reactor physics in the presence of observed data (with noise). Furthermore, they also studied the Generalized Inverse Power Method Neural Network (GIPMNN), Physics-Constrained GIPMNN (PC-GIPMNN), and Deep Ritz Method (DRM) to solve the k-eigenvalue problem in neutron diffusion theory, among which the PC-GIPMNN method performed the best, and the accuracy of the ML method was very good in all the cases, the ML method accuracy outperformed FEM and was not affected by meshing [48].

Reactor Performance Prediction

As described in the introduction, small reactors are currently being developed as the fourth generation of advanced reactors [4] focusing on reactors such as TFHR. These small reactors are usually located in remote areas for energy supply [27]. To achieve autonomous control of reactor systems with minimal intervention, thereby enabling long-term, stable, and efficient operation of nuclear reactors while reducing costs, Zeng et al. [27] investigated the autonomous control of small reactors. Their study focused on predicting the performance of the system in a steady state. They developed an ML-based performance prediction model that integrates a reactor physics model and a thermo-hydraulic model trained using the SVR method. This model was designed to predict the reactor core behavior and its future evolution. The system comprised two main components: a prediction module and a decision-making module, which worked together to accurately and efficiently assess the future operational conditions of the reactor.

Temperature Field Prediction

Temperature prediction also serves as a key to the optimization of nuclear reactors, and with the help of accurate prediction, we can achieve temperature regulation. In the field of nuclear power generation, reactors with heavy water coolants and moderators are widely used, and monitoring the moderator state is crucial to ensure the proper operation of the plant [26]. Starkov et al. [26] focused on the prediction of the temperature field of the nuclear reactor moderator under steady-state conditions and established a prediction system for the temperature of the heavy water moderator. They predicted the temperature distribution in the core cross-section based on the cellular neural network architecture. Based on this, continuous optimization of the moderator can be achieved in the future.

Additionally, related studies have focused on predicting critical conditions, particularly the temperature. Park et al. [54] employed a ML model to predict wall temperature at critical heat flux (CHF), a crucial parameter in nuclear reactor heat transport. The CHF marks the transition from nucleate boiling to transition boiling and serves as a key indicator of the reactor performance under steady-state operating conditions in terms of heat transport efficiency.

Power Prediction

In the operation of a nuclear power plant, ensuring a homogeneous radial power distribution in the reactor core is essential to avoid local overheating and ensure safe operation. The study [97] used the group data processing method (GMDH) to predict the axial power and axial offset of the reactor core, as well as the parameters for the fuel assembly. Using the GMDH algorithm, the researchers were able to reconstruct the axial power offset across the reactor core from the in-heap detector signals and determine the optimal relationship between the detector signals, core axial power, and axial offset [97].

The performance of the reactor can also be effectively assessed by predicting the power distribution in the steady state. This is crucial because the power distribution directly influences the temperature distribution and the transport and migration of fission products within the reactor. In Ref. [98], lattice physics calculations were used to generate data to find analytical expressions for the radial power distribution in fuel cores, with the help of ML techniques to find representative expressions. Daniell et al. [99] designed and compared several Multi-Stage DNN (MSDNN) architectures for predicting the final steady state power of a research nuclear reactor after power variations. The results show that MSDNN models tend to perform better than standard feed-forward neural networks in terms of accuracy and generalization.

The use of DNN continues to be a viable alternative to standard physics modeling in nuclear systems and engineering. Alam et al. [32] attempted to develop a multi-stage prediction model consisting of 2 feed-forward DNNs to determine the final steady-state power at each transient in a real reactor facility. They separately output a resolution representing the large frequency band power output, which is used as an input to realize the prediction. Physical information is collected from the initial conditions of the transient and the final steady state [32]. The final combined model achieved a classification accuracy of 96% and an absolute prediction accuracy of 92%.

Furthermore, Wan et al. [100] proposed a neural network-based variable universe fuzzy control method for regulating power and axial power distribution in large PWRs. By establishing fuzzy inference rules based on a two-node reactor dynamics model and adopting a variable universe approach to design the input scaling factor, optimized online via a neural network, the method achieves precise control of reactor power and spatial distribution. The simulation results indicate that, under transient conditions, the control performance of this method is significantly superior to that of conventional fuzzy controllers.

Power Peak Factor Estimation

The problem of computing the PPF is directly related to the thermal stress and potential local risks of nuclear fuel rods. Under steady-state conditions, it reflects the ratio of the highest local power density to the average power of the core reached by the nuclear reactor after a long period of operation. To address the problem of accurate estimation of the PPF, Bae et al. [21] developed a model to process the signal and predict it accurately using the measured signals from the reactor cooling system. The model is based on the SVR method and determines the NN weights by solving quadratic programming problems.

In practice, this approach demonstrates superior generalization ability compared to traditional ANN when addressing nonlinear problems, resulting in lower prediction error. In the context of solving the fuel loading problem, Trontl et al. [22] and Hedayat et al. [101] focused on optimizing the PPF, which facilitated precise regulation of core fuel loading. Tayefi et al. [23] further optimized fuel management strategies to enhance operational efficiency and safety, ensuring that the nuclear reactor operates under optimal conditions over long-term use. Additionally, they utilized the Hopfield neural network (HNNA) to identify the optimal distribution of axial enrichment variations, which effectively reduced the peak neutron flux and power peaking factors [102].

Core Loading Pattern Optimization

The prediction of a core loading pattern or reload strategy is a typical reactor core design task that involves optimizing the material composition, geometrical arrangement, and thermal-hydraulic parameters of the core. This optimization process must also account for constraints related to economic considerations, safety standards, and nuclear safeguards. These strategies aim to reduce fuel cycle costs while ensuring safe and efficient reactor operation. Conventional optimization usually consists of nuclear physics computational codes that rely on engineering judgments, heuristic rules, optimization algorithms, and evaluations, which are typically computationally intensive processes. To ameliorate this problem, Trontl et al. [22] quickly evaluated the core loading pattern using SVR methods, thus significantly reducing the computational time.

Hedayat et al. [101] explored the idea of using ANN to develop a rapid evaluation system for assessing and optimizing core parameters during reactor fuel reloading, which are essential to ensure optimal performance and compliance with safety standards during long-term operation.

Doppler Reactivity

The Doppler reactivity coefficient is a key safety parameter of the core, which is determined during the design phase alongside the moderator reactivity coefficients. In 3D reactor core behavior analysis, these reactivity coefficients are predicted but are not required by solvers. The solvers consider local neutronic and thermal-hydraulic feedback through nodal or homogenized cross-sections provided to them. Only point kinetics models (PKM) require an explicit input of reactivity coefficients. However, PKM is inherently fast, and therefore, there appears to be no need to replace it with an AI-based model to enhance its speed. Choi et al. [105] developed a comprehensive reactor dynamics model incorporating point kinetics equations (PKE), a thermodynamics model for hot water (TH), and a xenon (Xe) dynamics model to facilitate the automatic control of small modular reactors (SMRs). In a related study, Dorde et al. [33] proposed a method for calculating the Doppler reactivity in conjunction with the PRK model for Sodium Fast Reactors (SFRs), which was based on an ANN and yielded similar or even better performance.

Furthermore, the startup transients of the Purdue University Reactor-1 (PUR-1) were monitored using a reactivity feedback scheme. This involved the use of experimental parameters derived from a neutron source, providing valuable insights into the behavior of the reactor during the initial startup conditions. The integration of advanced modeling techniques and real-time monitoring is essential for enhancing the safety and operational efficiency of modern nuclear reactors. These studies underscore the importance of accurately calculating the Doppler reactivity and the potential benefits of incorporating machine learning methodologies in reactor dynamics.

Behavior Prediction

The behavior of reactors undergoing rapid changes or transient events must be predicted. Traditional numerical methods are time-consuming and susceptible to accumulated rounding errors or floating-point overflow in the transient state. Therefore, Pirouzmand et al. [73] proposed a cellular neural network-based simulation of complex nuclear reactor behaviors. Specifically, the method simulates the time-space response of the flux distribution in a two-dimensional Cartesian geometry, which solves the problem of behavior prediction. The model is well stabilized and can be accurately analyzed to provide fast and accurate solutions for complex and rigid models.

Park et al. [106] proposed an evolutionary algorithm, the GMDH algorithm, to process and analyze real-time data obtained from detectors inside the reactor to reconstruct and monitor the axial power distribution of a nuclear reactor. The algorithm is robust to different power distributions and can cover the entire reaction cycle, thus monitoring and evaluating the behavior of nuclear reactors in real time.

Utilizing extensive data from the Multi-Application Small LWR (MASLWR) integrated test facility at Oregon State University, Gomez Fernandez et al. [25] developed an NN topology designed to simulate and predict the behavior of a nuclear reactor during a loss-of-feedwater transient (LOFW). This NN model effectively captures the dynamic changes in various physical parameters within the reactor during the LOFW event, providing a robust tool for analyzing the reactor response under such critical conditions.

In the previous subsection on the application of ML methods in the steady state, Zeng et al. [27] investigated the prediction of steady-state properties for small reactors. Similarly, for the transient problem of microreactors, Mendoza et al. [107] proposed a power transient module that reliably identifies reactor operation as steady state, power rise, or power fall, using data from a large-scale simulator. They also explored various data-driven methods to accurately assess power transients, namely, principal component analysis, SVM, DNN, and CNN [107].

To identify the state of a nuclear reactor, Zubair et al. [108] developed a model that includes the primary and auxiliary systems of a nuclear power plant. In addition, it includes models of the control system and instrumentation used to monitor and regulate the reactor as an important component of data extraction and transient modeling as an integral component of data extraction and transient modeling. Based on the data generated by the Western Services Cooperation’s General purpose PWR (GPWR) simulator as well as MATLAB tools, they [108] simulated nine different transient events under twelve different initial conditions to create a dataset with 72,000 observations. Nine types of classification models (a total of 33 predefined models) were trained and validated using a classification learner application. The Neural Network Classifier had the highest average accuracy of 90% in the validation of the ML-based approach. The study maintained a low execution and computation time while achieving high accuracy.

In a further study, to optimize the model, the study [109] integrated feature selection using the minimum redundancy maximum relevance (mRMR) algorithm. As a result, the KNN model achieved 97% of the highest accuracy in the primary system, while the effective linear discriminant and logistic regression models obtained 99% of the highest average accuracy in the secondary system [109].

Forward and Inverse Problems

The real-time estimation of PPF is also important under transient conditions. This is needed to capture the rapid changes in PPF due to changes in operational variables, including control rod movement, to accurately detect and respond to special safety issues. Bae et al. [21] investigated the capture of transient PPF changes based on the SVR method. Focusing on the VVER-1000 reactor core power distribution problem, Pirouzmand et al. [24] estimated the relative power distribution and PPF of the reactor core using ANN. They first monitored the radial RPD of the core and screened the axial relative power of the fuel assembly to detect the PPF of the core.

To achieve real-time power prediction, Gong et al. [38] proposed a reactor physics digital twin model that simultaneously solves the positive and negative problems of predicting power-given parameters and additional power field measurements to solve the input parameters of operation. They [38] developed a non-intrusive forward model utilizing a Singular Value Decomposition (SVD) Autoencoder (AE) downscaling model, in conjunction with machine learning methods such as KNN and decision tree. The test results demonstrated that this approach outperforms traditional reduced-order methods, such as the Proper Orthogonal Decomposition (POD). Regarding the inverse problem, they investigated a generalized latent data assimilation method to achieve real-time construction of the inverse model characteristics by evaluating the forward model multiple times [38]. Furthermore, Gong et al. [10] rapidly reconstructed the reactor’s high-fidelity neutron field with real-time input parameters based on the online situation. They simulated different scenarios, such as burnup, control rod insertion step, power level, and coolant temperature, over the life cycle of the HPR1000, demonstrating the role of transient problem-solving in online monitoring.

Response Decision

Anil et al. [110] predicted future reactor states under possible control strategies, particularly for reactor systems under transient conditions, to evaluate and determine the optimal control strategy for the autonomous control of nuclear reactors. The prediction model, based on the initial and boundary conditions, can predict the fuel centerline temperature with predictive uncertainty and predict the future reactor state for all possible control strategies to select the optimal control strategy in the face of contingencies.

Konstantinos et al. [47] used a point kinetic model as a basis for predicting the neutron density in transients with the help of transfer learning PINN (TL-PINN). This is an alternative to the traditional reactor transient modeling. Transfer learning achieves a significant performance improvement compared to PINN, greatly reducing the number of iterations required for model training. They generated transients with different ranges of experimentally relevant variables using the PUR-1 experimental data.

Because the extreme stress of operators in most nuclear accident scenarios is prone to operational errors, ML models can provide real-time support and advice and act as expert systems to assist in accident handling and decision-making. Based on this idea, Nguyen et al. [111] developed a multivariate time-series ML meta-model to predict the transient response of a nuclear power plant experiencing steam generator tube rupture (SGTR). The model employs a hybrid CNN-LSTM model of recurrent neural networks (RNNs). They also implemented Bayesian Neural Networks (BNNs) to address the uncertainty in the prediction. The final model successfully captured the latent features of the data, and the BNN-LSTM approach provided additional monitoring of the level of uncertainty associated with heap prediction [111].

The successful application of LSTM extends beyond this point. Bae et al. [112] proposed a data-driven prediction model based on a multi-step prediction strategy and ANN. By testing MLP, RNN, and LSTM, they finalized a multi-input multiple-output strategy and LSTM, which successfully addressed multivariate problems under multiple emergencies of incident handling. The model not only achieves prediction accuracy but also guarantees accident transient prediction capability, with a prediction time of only 0.06 s.

Thermo-hydraulic coupling

The nuclear reactor is a complex system with a wide range of physical fields such as neutron density field, flow field, temperature field, etc. widely present in the reactor system [113]. These physical fields interact with each other; therefore, there is a strong coupling relationship between them. These coupling relations are not negligible during reactor operation, particularly during transient processes. Additionally, for neutronics, the neutron fluence rate determines the distribution of fission power inside the core, which affects the temperature field of the fuel rods in addition to that of water, and ultimately changes the entire temperature field. In thermal hydraulics, the thermodynamic parameters strongly affect the macroscopic cross-section. The interaction of these physical fields forms the basis for the coupling of neutronics and thermo-hydraulics [114]. Therefore, we will broadly introduce machine learning applications in the scenario of reactor thermal hydrodynamics coupled with reactor physics.

In recent years, the application of ML techniques in nuclear reactor thermo-hydrodynamic coupling and related areas has achieved remarkable progress, offering innovative solutions for the real-time prediction, optimization, and control of complex systems. For instance, Zhang et al. [115] proposed a neural network-based rapid prediction method for multi-physics field coupling in heat pipe reactors. By replacing traditional numerical modules with neural networks, this method achieves high-precision coupling predictions for multiple physical domains (e.g., neutronics, thermal, and mechanical) while reducing computation time from several hours to less than four minutes—without compromising key parameter accuracy (e.g., maximal stress difference <2 MPa, average fuel temperature difference <3 K). Moreover, the approach requires relatively modest training data and permits dynamic model correction and optimization based on prediction performance, thereby offering significant support for early accident warning and on-site deployment.

Traditional approaches often decouple these phenomena, for instance, by using one-way coupling, where neutronics and thermal-hydraulics provide boundary conditions for fuel performance analysis. However, this simplification may overlook essential core-wide interactions. To address this, Che et al. [116] developed a machine learning-assisted surrogate model that fully integrates coupled neutronics, thermal-hydraulics, and fuel thermo-mechanics. By combining look-up tables with advanced ML algorithms, including rule-based techniques for effective feature extraction, the surrogate model accelerates the simulation time by up to four orders of magnitude relative to conventional codes (e.g., FRAPCON) while preserving key accuracy metrics. This approach enables more realistic full-core predictions and facilitates the tighter integration of fuel performance into core design optimization.

In addition, Aghili Nasr et al. [117] tackled a core challenge in coupled simulations: dynamically updating neutron cross sections as a function of the evolving state variables. Their method employs Gaussian Process Regression (GPR) to substitute traditional logarithmic correlations for temperature-dependent cross section updates. Implemented within a multiphysics framework based on OpenFOAM, the GPR approach was validated for benchmark problems and a 2D axisymmetric Molten Salt Reactor configuration. The results demonstrate that, particularly for isotopes with prominent absorption resonances (e.g., ²³²Th and ²³⁸U), the method significantly influences the effective multiplication factor (k_eff) and transient peak power predictions, highlighting the importance of robust coupling for reactor safety analysis.

Overall, these studies underscore the transformative potential of advanced surrogate modeling and ML techniques in capturing the nuanced interplay between reactor physics and thermal hydraulics. These approaches not only substantially reduce computational burdens and enhance predictive fidelity, but also offer flexible and efficient strategies for regulating reactor operations. The vast application potential of ML in improving multiphysics field coupling and thermo-hydrodynamic integration is pivotal for achieving the safe, economical, and intelligent operation of nuclear power plants. For a comprehensive review of ML applications in thermal hydraulics, readers are encouraged to consult [118].

Optimization of Burnup Calculation

To simplify and accelerate the process of pressurized water reactor burnup calculations, Leniau et al. [90] proposed a method to generate a precalculated set of fuel burnup calculations. They developed an NN-based approach. Their method [90] was applied to the dynamic fuel cycle simulation tool CLASS to predict the Pu content and average cross-section in the fuel using a neural network, which significantly improved the computational efficiency and accuracy. The method was able to complete burnup calculations in less than a minute and had an average error of only 0.37% in predicting Pu content, demonstrating its potential for nuclear fuel cycle management and burnup prediction.

On this basis, Courtin et al. [123] investigated the MOX extended fuel nuclide prediction. Their proposed fuel loading model, based on infinite multiplication factor (k_inf) calculations, predicts the amount of Pu in the fuel required to reach the target burnup, as well as the enrichment of U in specific cases, with an accuracy of 270 pcm and a deviation of less than 4% for the primary nucleus at the end of the cycle (EOC).

To perform safeguards, the spent nuclear fuel needs to be checked for the correctness of the fuel assembly declarations and the completeness of the relevant declarations for each fuel assembly, i.e., it is determined that the spent fuel assembly does contain nuclear material and that no diversion of a part of the assembly has taken place [124]. Based on simulated pressurized water reactor fuel data, Grape et al. [124] used random forest regression to explore the predictive power of fuel parameters for the initial enrichment, burnup, and cooling time.

Burnup Distribution Prediction

Accurate prediction of the burnup distribution is also crucial for fuel reloading design and optimization, which is essential for the safety and economy of the reactor. However, owing to the heterogeneous flow distribution in the reactor core and the manufacturing deviation of each component, there is a deviation between the actual and the simulation results, leading to an error in the power distribution that is subsequently transferred to the burnup distribution [121]. Guo et al. [121] proposed a data assimilation method. They first combined the measured values of the power distribution with the results of the numerical simulation, established the relationship between the burnup distribution and power distribution using ANN, and finally calibrated the burnup distribution using the 3D variational method. The method was validated on the CNP1000 PWR, and the error in the burnup distribution was significantly reduced.

Nuclide Density Prediction

A related study developed and validated a DL-based model for predicting the burnup nuclide densities. To predict the nuclide densities of ²³⁵U, ²³⁹Pu, ²⁴¹Pu, ¹³⁷Cs, and ¹⁵⁴Nd, Lei et al. [122] used the mean square error (MSE) of the DNN as a loss function by comparing it with the relative error of a multilayer perceptron model. They found that DNN overcame the problem of excessive prediction errors of traditional ML algorithms in the low-burnup region and performed better in the middle and high-burnup regions, proving the feasibility of ML techniques for nuclide density prediction.

Other Fields in Burnup

To analyze the equilibrium burnup state of molten salt reactors (MSRs), Chen et al. [35] generated equilibrium and non-equilibrium state neutron information of fuel assemblies with different geometrical parameters as a dataset, and quickly filtered the models for the equilibrium burnup state and predicted the neutron fluxes and multiplication ratios using 15 different ML regression algorithms. The LGBM model performed the best.

Soto et al. [125]found that with the help of ML methods, it is possible to achieve a significant improvement in accuracy over a short fuel cooling time, increased robustness to spectral compression, and competitive burnup prediction even when only background signals are used, which is difficult to achieve with conventional methods.

The burnup of nuclear reactors is also affected by the interaction between the fuel and cladding, which is currently the subject of more frontier studies [126]. The literature indicates that the ML method is expected to be used in conjunction with energy dispersive x-ray spectroscopy (EDS) analysis of scanning electron microscopy collected in this study for the analysis of high-magnification back-scattered electron (BSE) datasets to facilitate the rapid and consistent extraction of statistical information on fuel-cladding chemical interaction (FCCI).

Observable Measurement Descriptions

Accurate definitions of nuclear physics observables become difficult since the information from differential nuclear physics experiments and theories is usually subject to large uncertainties, thus making the precise definition of nuclear physics observables challenging. However, the overall experimental data representing the applications of these observables are usually more precise. Simultaneoursly, there is too much experimental data to identify problems in the observables through human expert analysis alone [55].

To describe these quantities, Neudecker et al. [55] proposed an analytical approach to identify groups of observable measurements in nuclear physics that may lead to biases in integral experimental simulations using the Random Forest algorithm and SHAP metric.

In a practical application, the study characterizes the fission observable measurements of ²⁴¹Pu. They [55] combined differential experimental data and theoretical modeling to determine that the ²⁴¹Pu fission cross-section may be the main cause of the simulation bias. The high-energy interval data for ²³⁹Pu, which could not be accurately described, were still able to narrow down the range of nuclear data that could be problematic. This demonstrates the successful application of ML technology in the description of observed quantities and its wide range of prospects.

Radiation Transport Problems

The radiation transport problem encompasses various types of radiation, including neutron transport in reactor physics. This describes the radiation flux through a heterogeneous medium. For radiative transport problem, traditional ML methods are difficult to apply because of the large computational cost of creating data through simulations. Huhn et al. [53] investigated the PINN method to solve the radiative transport problem in the absence of data. They also proposed the application of Fourier features in the algorithm, which yielded good performance for heterogeneous problems. For sampling, they investigated heuristic-based methods. In a practical set of one-dimensional radiative transport problems, PINN produces consistent results with traditional method solutions under fine grids [53].

Model Correction Problem

Although high-fidelity transport calculations are certainly accurate, their large number of calculations makes them difficult to apply, especially in the implementation of forecasting, where it is almost impossible to obtain direct access to high-fidelity model/data in real time. Therefore, this study considers error correction for low-fidelity models/data.

For operational simulation and real-time prediction in boiling water reactor (BWR), Oktavian et al. [127] investigated machine learning (ML) methods. Specifically, they generated initial results using a standard two-step simulation [112] process and then used an ML model to correct the errors between the high- and low-fidelity data. To correct the errors between the nuclear reactor mechanical model and the actual reactor, Wang et al. [128] proposed a hybrid mechanism and NN approach for nuclear reactor modeling to eliminate or minimize these deviations. Specifically, based on simulation data from RELAP5, they built a point reactor dynamics model as a model of the core and calibrated it using a genetic algorithm (GA) to optimize the BPNN reactor reactivity feedback.

Nuclear Data Validation Analysis

Nuclear data validation analysis focuses on identifying possible problems in nuclear data using ML models. This process focuses on the overall quality and predictive power of the nuclear data. As mentioned in the literature [56], ML methods assess the reliability of a large number of nuclear data features and corresponding experimental results by analyzing them in the simulation of critical nuclear experiments. These models are based on the measurements and simulations of selected key components from the International Criticality Safety Benchmark Evaluation Project (ICSBEP). The k_eff, along with its sensitivity to nuclear data and baseline features, was used as an input to the Random Forest (RF) regression model. The RF model is employed to encode the complex interdependencies between factors such as sensitivity profiles, descriptive measurements of the nuclear data, and both simulated and experimental k_eff values. This approach aims to identify the nuclear data features that are most influential in predicting deviations, thereby enhancing our understanding of the underlying relationships.

A related study [84] also focused on the potential role of ML in supporting cross-section evaluation and proposed a data enhancement model for neutron-induced response cross-sections. Specifically, they fitted nuclear data from the EXFOR database based on decision trees and KNN to predict data for important nuclides, and the final predictions matched the measurements very accurately.

Data Synthesis Problem

Before building a nuclear power plant, the proliferation potential of nuclear reactors must be understood. A digital twin is a solution for identifying the behavior of nuclear proliferation reactors. Based on the parameters of the AGN-201 reactor of Idaho State University, Palmer et al. [129] discussed data synthesis techniques to generate data for training the digital twin model. Additionally, they trained the LSTM on synthetic data to further validate the application of these methods in the implementation of digital twins. The benefit of this approach is that it effectively captures the aging and sensor drift of equipment caused by environmental factors and other influences. By monitoring the error between the actual and synthesized data, the need for reprocessing the synthesized data at the appropriate time can be signaled, ensuring more accurate and reliable monitoring and analysis over the equipment’s lifespan [129].

Noise Diagnostic Problems

Nuclear power plants are large and complex systems equipped with numerous monitoring sensors. Within the reactor core, it is critical to detect anomalies as early as possible to prevent irreversible effects. Although the core can accommodate only a limited number of sensors for data acquisition, neutron detectors remain highly effective in capturing perturbations, even those occurring at significant distances from the sensors. This capability is due to the intrinsic nature of fission and scattering reactions, as well as the transport of particles within the core, which allows for the detection of changes throughout the reactor system [130].

Based on the capture of anomalous states, Kollias et al. [130] used an ML method for noise capture and anomaly localization. Specifically, they performed semantic segmentation, classification, and localization for multiple simultaneous perturbations using DL models. They further developed a domain-adaptive approach and applied it to real reactor simulations.

Data Fidelity Mapping

To reduce the cumulative error of long-term prediction while minimizing computational costs, Zhou et al. [131] introduced a novel prediction model known as the multi-scale physics-constrained neural network (MSPCNN). This model utilizes a multi-fidelity convolutional autoencoder (CAE) to map data of varying fidelity into a uniformly shared potential space. This approach serves two key purposes. First, it enables low-fidelity data to complement high fidelity data during the training phase. Second, it allows the enforcement of physical constraints within the low-fidelity domain rather than at the high-fidelity level. This strategy significantly reduces the costs associated with offline data acquisition and preprocessing while ensuring model accuracy. This advancement represents a notable improvement over the traditional PINN. Additionally, the MSPCNN model is robust to noise, providing reliable predictions even under suboptimal conditions.

Applications in Operational Simulation

Neutron simulations of reaction processes are essential for the design and operation of nuclear reactors, serving as both a validation of safety and operational capability. Efficient simulations not only provide operators with real-time data and critical metrics to optimize power distribution and burnup and extend equipment life, but also hold the potential to significantly accelerate the design of next-generation advanced reactors [132]. A considerable body of literature focuses on the application of ML methods in reactor simulations, which will be systematically reviewed and discussed in this section.

Applications in Online Monitoring

The safe operation and effective maintenance of nuclear reactors are critical. The development of economically beneficial and safe operations requires a more accurate, comprehensive, and real-time analysis of neutron parameters [138]. A real-time monitoring system is required to confirm that the relevant safety requirements are not violated during reactor operation [139]. In the current literature related to online supervision, ML techniques have been mainly applied in real-time monitoring, autonomous control, condition detection, parameter verification, signal monitoring, and fault monitoring and diagnosis of reactors. This section focuses on the ML approaches used in these applications and the practical benefits achieved.

Applications in Safety Design

Core optimization aims to adjust the key state parameters of a reactor core, such as the power distribution, temperature field, and reactivity coefficients, to ensure safe operation and optimal performance. Typically, two major classes of optimization methods are employed: gradient descent-based methods and evolutionary algorithms (EA).

However, because most existing studies rely on external neutron simulation programs[19] to compute these state parameters, the resulting objective functions are often non-differentiable in nature. This non-differentiability can lead to issues such as vanishing gradients, which hampers the effectiveness of gradient descent–based optimization.

Method-based Challenges

The frontier ML prediction methodology is represented in reactor physics using DL. The DNN architecture derived from DL is also used as a fundamental architecture in many applications.

It is well known that in ML methods, especially supervised or semi-supervised learning, the quality of the data has a significant impact on the results. Therefore, one of the challenges is the selection of a particular sampling method. In reactor physics, which usually involves a high-dimensional parameter space, traditional sampling methods are inefficient. During the research process, suitable sampling algorithms can facilitate the training process and provide better approximations [150], which is one of the future research directions.

The core of an NN is the selection of the loss function and the structure of the data grid, which can be further investigated in these two aspects. For example, during the research of MSPCNN for seamless encoding of multi-fidelity data [131], the adaptability and computational efficiency are high, but for this class of networks, it is easy to amplify the error in scenes with limited spatial correlations. Therefore, customizing the loss function to balance the fidelity and reduce the error will be an extension of this problem. In addition, in real-world applications, it is often necessary to model unstructured or even adaptive grids, where the number and arrangement of grids can be dynamically varied to better capture the phenomena or optimize the computational resources [131].

Breakthroughs can also be achieved in terms of the scale of the problem. The acquisition of high-resolution, high-fidelity data is a significant challenge for larger and more complex problems. The generation of high-fidelity data, such as those obtained using the statistical Monte Carlo method, is particularly resource intensive. The current mainstream approach consists of fitting high-fidelity data from low-fidelity data with error predictions; for example, the literature [127] shows how to augment the running simulation of a boiling water reactor (BWR) with the help of ML in combination with traditional simulation techniques. This complementary approach is only in its infancy and is limited to small reactor problems presented in the literature [72]. The enhancement of the simulation of high-dimensional problems with the aid of ML for large-scale problems is still under research [7]. This approach may even be extended to the study of other types of reactors. This is necessary; removing the uncertainty approach to simulation, although for realistic reactors, it is possible to obtain operational data, how the data noise is handled should also be considered; in addition, not every demanded parameter has been pre-measured [7, 72].

Finally, the topic can be considered as an improvement in the algorithm architecture. Specifically, it involves changes in the existing network structure. The DL-based PINN method and its variants are particularly used in the review of the main text, which are known for their efficient performance and low mesh dependency. If the Transformer architecture could be integrated into the PINN architecture [131], or a new auxiliary ML model inspired by the Transformer could be created, completely different results could be obtained.

Model-Based Challenges

One of the important concerns of the industry regarding theoretical models is their interpretability. In some cases, reactor operators may be more interested in the input parameters that led to the observation or state of interest. Therefore, parameter identification (PI) is another intriguing task [10]. Therefore, a previous study [10] established a digital twin based on non-intrusive modeling and ML methods for the inverse problems of PI, achieving good performance in terms of accurate prediction in a short period. Non-intrusive modeling refers to a modeling approach that does not require the modification of the original (forward) model or direct intervention in the physical process. The model can directly utilize the physical model as an input, which retains some physical meaning and improves its interpretability. Another issue is uncertainty quantification [130]. The nuclear reactor itself is a complex system consisting of multi-physics, multi-scale, and for different scales, there may be different parameters, assumptions, and models. To more accurately and comprehensively consider the interpretability of the model, the literature considers multi-scale uncertainty quantification as a future research direction [137]. Stewart et al. [142] investigated the appropriateness of different ML methods for final implementation by assessing the pros and cons of each method in terms of accuracy, access, development, efficiency, and transfer learning approaches. The authors also investigated the explainability and interpretability, assessed the model performance on live data streams, and identified strategies for model verification.

To improve the applicability and reliability of the model, another critical issue is the generalization of the model, which is also known as the robustness of the model. This is also a solution strategy for multiscale problems. In most studies, ML methods have been applied to specific reactors. Therefore, the specificity of ML methods for data is a problem to be considered; in other words, a generalization model needs to be found to reduce the dependence on data or even provide a better model without knowing the data in advance [7, 48].