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Joint Initialization of Flux Networks and Effective Multiplication Factor for Physics-Informed Neural Networks Solving Neutron Diffusion Problems
Qin Hang, Yangdi Yi, Jiayi Li, Xu Wang, Heng Zhang
TL;DR
Determining keff and neutron flux efficiently remains challenging because PINN training can depend strongly on random network initialization and empirical keff settings. JI-PINN uses a low-resolution approximate solution to jointly initialize the flux network and keff, then optimizes both under physical constraints. Across four cases, it reduced total computational time while retaining comparable accuracy and reducing anomalous keff results.
Problem
PINN training for neutron-diffusion K-eigenvalue problems can be slow or difficult to converge when flux parameters are randomly initialized and keff is specified empirically.
Method
JI-PINN uses a low-resolution approximate K-eigenvalue solution to jointly initialize flux-network parameters and keff, then jointly optimizes them under physical constraints.
Results
JI-PINN reduced total computational time by 25.4%–49.4% versus randomly initialized PINNs across four cases while maintaining comparable solution accuracy.
Takeaways & Limitations
JI-PINN provides a more efficient and robust initialization strategy for steady-state neutron-diffusion K-eigenvalue problems, including heterogeneous and three-dimensional cases.
Takeaways & Limitations
JI-PINN performance depends on the low-resolution approximate solution, especially with complex interfaces and pronounced local flux variations that can affect final accuracy.
Abstract
from arXiv · showhide
Efficient determination of the effective multiplication factor (keff) is an important computational task in reactor core neutronics analysis. Physics-informed neural networks (PINNs) incorporate neutron diffusion equations and boundary conditions into network training to efficiently determine the neutron flux distribution and keff. To further improve the efficiency of keff calculations using PINNs, a Joint Initialization Physics-Informed Neural Network (JI-PINN) is proposed in this work. In this method, a low-resolution approximate solution to the K-eigenvalue problem is used to construct a joint initial state for the flux network parameters and keff, and both are then jointly optimized under physical constraints. The proposed method was validated on a two-dimensional two-group two-material case, the IAEA 2D benchmark, a two-dimensional two-group four-material case, and a three-dimensional single-group case. For these test cases, the total computational time was reduced by 25.4%, 38.2%, 49.4%, and 28.9%, respectively, while comparable solution accuracy was maintained. The occurrence of anomalous results associated with marked deviations of keff from the reference value was also reduced. The proposed method provides a more efficient and robust initialization strategy for solving neutron diffusion K-eigenvalue problem with PINNs.
I. INTRODUCTION
Efficiently determining keff and neutron flux is important for reactor criticality and power-distribution analysis, but conventional methods and PINNs face computational or convergence challenges. The paper proposes JI-PINN, which jointly initializes flux-network parameters and keff using a low-resolution approximate solution before physics-constrained training.
- Motivation: keff and neutron flux distribution support reactor criticality analysis and core power distribution calculations.For fixed geometry, materials, and boundary conditions, keff indicates whether a reactor is subcritical, critical, or supercritical.
- Existing methods: Spatial discretization methods combined with power iteration provide reliable accuracy but can require costly repeated updates for high-dimensional problems.Finite difference, finite element, and nodal methods are identified as established approaches.
- Related PINN research: PINNs have been applied to neutron diffusion and K-eigenvalue problems through architectural changes, surrogate modeling, trainable keff, prior data, and interface treatments.Related approaches include GHO Hybrid PINN, DEPINN, inverse-power integration, and FC-PINNs.
- Research gap: Random flux initialization and empirical keff initialization can produce divergent optimization behavior, convergence difficulties, higher training cost, or large keff errors.A large discrepancy between the initial state and target solution is identified as a source of these difficulties.
- Proposed approach: JI-PINN uses a low-resolution approximate K-eigenvalue solution to initialize flux-network parameters and keff jointly before physics-constrained optimization.The method then jointly updates both quantities during training.
II. METHODS
JI-PINN combines a low-resolution discrete K-eigenvalue approximation with a PINN that jointly optimizes neutron flux network parameters and keff under physical constraints.
- Neutron diffusion formulation: JI-PINN treats neutron fluxes in all energy groups as neural-network outputs and keff as an independent trainable parameter.The flux network approximates the groupwise neutron flux, while keff is solved simultaneously.
- Low-resolution initialization: A low-resolution spatial grid provides approximate neutron flux and keff values at relatively low computational cost.The discrete problem uses discretized non-fission and fission operators to obtain the approximate K-eigenvalue solution.
- Low-resolution initialization: The discrete flux is interpolated into a continuous approximation and fitted to construct the initial flux-network parameters.This fitting supplies initialization information before physics-constrained training begins.
- Joint initialization: The fitted flux-network parameters and approximate keff form a joint initial state for subsequent optimization.The initial keff is set to the low-resolution approximate value ekeff,h.
- Physics-constrained optimization: Flux-network parameters and keff are jointly optimized by minimizing losses for the governing equation, boundary conditions, and flux scaling.The resulting optimization produces the neutron flux distribution and keff.
III. EXPERIMENTS
The experiments assess JI-PINN across initialization quality, joint-initialization effects, random-seed sensitivity, and computational performance under varied neutron-diffusion conditions.
- Evaluation design: JI-PINN experiments evaluate initialization quality, joint initialization, sensitivity to random initialization, and computational performance.These perspectives are examined separately to assess initialization quality against its additional computational cost.
- Evaluation design: The two-dimensional two-group two-material case varies approximate-solution resolution and flux-network pretraining while testing stability across multiple random seeds.The study also examines how these choices affect initialization effectiveness and computational cost.
A. Two-Dimensional Two-Group Two-Material Case
The two-dimensional two-group two-material case examines how low-resolution flux information and pretraining shape JI-PINN initialization, efficiency, accuracy, and robustness across random seeds.
- Initialization analysis: The case varies low-resolution grid resolution Nh and pretraining steps Mpre to balance initialization quality against computational cost.Nh controls retained spatial information, while Mpre controls how closely the network fits the approximate flux.
- Initialization analysis: Increasing Nh from 17 to 25 markedly reduced initial loss L0 and final relative keff error εk by preserving the approximate flux structure more adequately.Further increases to Nh=33 and 49 did not reduce L0 or εk but continued increasing total computational time.
- Random-seed robustness: Across 15 random seeds, JI-PINN concentrated keff values near the reference and reduced variability in fast- and thermal-group flux errors.Randomly initialized PINN showed substantial variability, while R2-PINN had a narrower distribution but retained some large deviations.
- Computational performance: 25.4% was the reduction in average total computational time for JI-PINN relative to randomly initialized PINN, while flux accuracy showed no noticeable loss.The comparison used total time including approximate-solution calculation, pretraining, and physics-constrained training.
- Initialization analysis: Appropriate spatial resolution and pretraining preserved the main flux features while controlling additional overhead, balancing initialization quality and computational efficiency.The selected settings were Nh=25 and Mpre=5000 for subsequent experiments.
B. IAEA 2D Benchmark
The IAEA 2D benchmark evaluates how flux-network pretraining and keff initialization affect convergence and computational cost. Jointly initializing both from the same approximate K-eigenvalue solution produced the most favorable training behavior.
- The quarter-core benchmark spans a 170 cm × 170 cm domain with four material regions and mixed Neumann and vacuum boundaries.
- All four initialization configurations reached final keff values close to the reference value, but differed in starting points and total computational times.
- 3.9% reduction: changing only the initial keff from 1.0 to the approximate value reduced Ttotal from 7398.5 s to 7106.3 s under random flux initialization.Random flux initialization still required gradual adjustment of the spatial flux distribution.
- 29.8% reduction: approximate flux pretraining reduced Ttotal from 7398.5 s to 5194.9 s when k(0)eff=1.The pretrained network already represented the main spatial distribution of the approximate flux.
- 12.0% further reduction: initializing keff with the value obtained alongside the approximate flux reduced Ttotal from 5194.9 s to 4573.0 s after flux pretraining.The initial loss remained nearly unchanged, while the paired initialization further shortened training.
- Joint initialization formed a more favorable starting state because the initial flux representation and keff came from the same approximate solution.Approximate flux pretraining improved the flux function representation before formal physics-constrained optimization.
C. Two-Dimensional Two-Group Four-Material Case
The heterogeneous four-material case tests JI-PINN in a domain with multiple material interfaces. JI-PINN converged faster, although its efficiency advantage was accompanied by some loss of keff accuracy.
- The test case is a heterogeneous two-dimensional, two-group domain with four materials and zero-flux outer boundaries.
- 49.4% reduction: JI-PINN reduced Ttotal relative to the randomly initialized PINN in the four-material case.The comparison used the computational results reported in Table VII.
- 69.5% reduction: JI-PINN reduced Ttotal relative to R2-PINN.
- JI-PINN began closer to the reference keff, fluctuated less initially, and entered a stable range more rapidly than the comparison methods.The improved starting state reduced early adjustment of both the flux representation and keff.
- The relative keff error for JI-PINN was higher than for the two comparison methods in this case.Shorter subsequent training did not fully correct local deviations retained in the approximate initial state.
- Increasing material interfaces made local flux variations harder for the low-resolution approximate solution to represent, leaving possible local deviations after joint initialization.How much later physics-constrained training corrected these deviations affected final solution accuracy.
D. Three-Dimensional Single-Group Cubic Case
The three-dimensional single-group cubic case examines whether joint initialization remains effective beyond the two-dimensional benchmarks. JI-PINN reduced computational time while maintaining generally comparable solution accuracy.
- The three-dimensional test uses a 1 m cube with zero-flux boundaries on all six surfaces and a center flux-scaling constraint of ϕ(0, 0, 0) = 0.5.
- Comparable solution accuracy: the three methods produced keff relative errors and neutron-flux MSE values of the same order of magnitude.
- 28.9% reduction: JI-PINN lowered Ttotal relative to the randomly initialized PINN.
- 19.0% reduction: JI-PINN lowered Ttotal relative to R2-PINN.
- JI-PINN reduced early keff correction and reached the stopping criterion with fewer training steps, although brief early fluctuations remained.The low-resolution approximate state still required further correction under physical constraints.
IV. CONCLUSION
JI-PINN jointly initializes the flux network and keff using a low-resolution approximate solution, improving computational efficiency across increasingly complex neutron diffusion cases. Its effectiveness depends on approximation quality, especially near complex interfaces and strong local flux variations.
- Conclusion: JI-PINN jointly initializes flux-network parameters and keff before physics-constrained optimization.The joint initial state is constructed from a low-resolution approximate solution to the K-eigenvalue problem.
- Conclusion: 25.4%–49.4% lower total computational time was achieved across four benchmark cases than with randomly initialized PINNs.The cases included two-dimensional two-group two-material, IAEA 2D, two-dimensional two-group four-material, and three-dimensional single-group problems.
- Conclusion: Appropriate approximate-solution resolution and pretraining balanced retained flux information against added computational cost.These choices also reduced differences between networks with different random initializations when formal training began.
- Conclusion: JI-PINN maintained a favorable balance between computational efficiency and solution accuracy in heterogeneous and three-dimensional settings.The conclusion states that this advantage persisted as problems included multiple material regions and three-dimensional domains.
- Conclusion: Performance remains sensitive to low-resolution solution quality, particularly for complex interfaces and pronounced local flux variations.Poor representation of local spatial information can affect physics-constrained optimization and final solution accuracy.