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

cs.LGAug 26, 2026

Joint Initialization of Flux Networks and Effective Multiplication Factor for Physics-Informed Neural Networks Solving Neutron Diffusion Problems

Authors: Qin Hang, Yangdi Yi, Jiayi Li, Xu Wang, Heng Zhang

Organizations: School of Computer Science and Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China · Center for Scientific Intelligence Innovation, University of Science and Technology of China, Hefei 230026, China · Institute of Advanced Technology, University of Science and Technology of China, Hefei 230088, China

Abstract

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.

Figures & tables

Fig. 2: Geometry and material distribution of the two-dimensional two-group two-material case.

Material	$g$	$D_{g}\,(\mathrm{cm})$	$\Sigma_{\mathrm{a}}\,(\mathrm{cm}^{-1})$	$\nu\Sigma_{\mathrm{f}}\,(\mathrm{cm}^{-1})$	$\Sigma_{1\rightarrow 2}\,(\mathrm{cm}^{-1})$
Material 1	1	1.268	0.007181	0.004609	0.02767
Material 1	2	0.1902	0.07047	0.08675	$-$
Material 2	1	1.255	0.008252	0.004602	0.02533
Material 2	2	0.211	0.1003	0.1091	$-$

TABLE I: Material parameters of the two-dimensional two-group two-material case

Experimental Variable	Parameter Value	$L_{0}$	$T_{\mathrm{total}}\,(\mathrm{s})$	$\varepsilon_{\mathrm{k}}$
$N_{h}$	17	$4.929\times 10^{-3}$	797.9	$6.693\times 10^{-3}$
	25	$3.974\times 10^{-3}$	1575.3	$2.532\times 10^{-3}$
	33	$5.946\times 10^{-3}$	1631.0	$2.730\times 10^{-3}$
	49	$7.665\times 10^{-3}$	1688.4	$2.825\times 10^{-3}$
$M_{\mathrm{pre}}$	1000	$4.699\times 10^{-1}$	8045.4	$0.133\times 10^{-3}$
	3000	$7.413\times 10^{-3}$	1986.1	$2.93\times 10^{-3}$

TABLE II: Results of the Hyperparameter Analysis for the Two-Dimensional Two-Group Two-Material Case

Method	Final $k_{eff}$	$\varepsilon_{\mathrm{k}}$	Fast Group MSE	Thermal Group MSE	$T_{\mathrm{total}}\,(\mathrm{s})$
Randomly Initialized PINN	$1.037\pm 0.039$	$(3.141\pm 3.512)\times 10^{-2}$	$(1.088\pm 1.299)\times 10^{-1}$	$(1.634\pm 1.835)\times 10^{-2}$	$3309.6\pm 366.2$
$R^{2}$ -PINN	$1.056\pm 0.027$	$(1.252\pm 2.532)\times 10^{-2}$	$(2.276\pm 5.791)\times 10^{-2}$	$(3.389\pm 8.454)\times 10^{-3}$	$4033.4\pm 806.2$
JI-PINN	$1.066\pm 0.001$	$(2.428\pm 1.511)\times 10^{-3}$	$(5.615\pm 3.246)\times 10^{-3}$	$(8.330\pm 4.892)\times 10^{-4}$	$2467.7\pm 931.5$

TABLE III: Statistical Results of the Random Seed Experiments for the Two-Dimensional Two-Group Two-Material Case

Fig. 3: Solution results and computational time for different random seeds: (a) final

k_{eff}

; (b) total computational time; (c) MSE of the neutron flux for the fast and thermal groups.

Fig. 4: Geometric layout of the IAEA 2D benchmark.

Region	$g$	$D_{g}\,(\mathrm{cm})$	$\Sigma_{\mathrm{a}}\,(\mathrm{cm}^{-1})$	$\nu\Sigma_{\mathrm{f}}\,(\mathrm{cm}^{-1})$	$\Sigma_{1\rightarrow 2}\,(\mathrm{cm}^{-1})$
$\Omega_{1}$	1	1.5	0.010	0.000	0.02
$\Omega_{1}$	2	0.4	0.080	0.135	$-$
$\Omega_{2}$	1	1.5	0.010	0.000	0.02
$\Omega_{2}$	2	0.4	0.085	0.135	$-$
$\Omega_{3}$	1	1.5	0.010	0.000	0.02
$\Omega_{3}$	2	0.4	0.130	0.135	$-$

TABLE IV: Material parameters of the IAEA 2D Benchmark

Initialization Method	$k_{eff}^{(0)}$	$k_{eff}$	$L_{0}$	$\varepsilon_{\mathrm{k}}$	$T_{\mathrm{total}}\,(\mathrm{s})$
Random Initialization	1	1.029586	$6.420\times 10^{5}$	$3.48\times 10^{-4}$	7398.5
Random Initialization	1.029240	1.030006	$6.420\times 10^{5}$	$6.02\times 10^{-5}$	7106.3
Flux Initialization	1	1.030104	$8.858\times 10^{3}$	$1.55\times 10^{-4}$	5194.9
Flux Initialization	1.029240	1.029367	$8.842\times 10^{3}$	$5.61\times 10^{-4}$	4573.0

TABLE V: Computational Results for the IAEA 2D Benchmark under Different Initialization Configurations

Fig. 5: Convergence histories of $k_{eff}$ under different initialization configurations.

Fig. 6: Geometry and material distribution of the two-dimensional two-group four-material case.

Material	$g$	$D_{g}\,(\mathrm{cm})$	$\Sigma_{\mathrm{a}}\,(\mathrm{cm}^{-1})$	$\nu\Sigma_{\mathrm{f}}\,(\mathrm{cm}^{-1})$	$\Sigma_{1\rightarrow 2}\,(\mathrm{cm}^{-1})$
Material 1	1	1.255	0.008252	0.004602	0.02533
Material 1	2	0.2110	0.100300	0.10910	$-$
Material 2	1	1.268	0.007181	0.004609	0.02767
Material 2	2	0.1902	0.070470	0.08675	$-$
Material 3	1	1.259	0.008002	0.004663	0.02617
Material 3	2	0.2091	0.083440	0.10210	$-$

TABLE VI: Material parameters of the two-dimensional two-group four-material case

Method	$k_{eff}$	$\varepsilon_{\mathrm{k}}$	$T_{\mathrm{total}}\,(\mathrm{s})$
Randomly Initialized PINN	0.796108	$4.88\times 10^{-4}$	2757.6
$R^{2}$ -PINN	0.795609	$1.39\times 10^{-4}$	4578.0
JI-PINN	0.793601	$2.66\times 10^{-3}$	1394.9

TABLE VII: Computational Results for the Two-Dimensional Two-Group Four-Material Case

Fig. 7: Convergence history of

k_{eff}

for the two-dimensional two-group four-material case.

Method	$k_{eff}$	$\varepsilon_{\mathrm{k}}$	Flux MSE	$T_{\mathrm{total}}\,(\mathrm{s})$
Randomly Initialized PINN	0.9505335	$5.62\times 10^{-4}$	$6.15\times 10^{-7}$	697.4
$R^{2}$ -PINN	0.950510	$5.37\times 10^{-4}$	$8.17\times 10^{-7}$	612.6
JI-PINN	0.950660	$6.95\times 10^{-4}$	$9.49\times 10^{-7}$	496.1

TABLE VIII: Computational Results for the Three-Dimensional Single-Group Cubic Case

Fig. 8: Geometry of the three-dimensional single-group cubic case.

Fig. 9: Convergence history of $k_{eff}$ for the three-dimensional single-group cubic case.

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PE-EK-PINN: Physics Embedding with Evolving Kernel for Scalable Physics-Informed Neural Networks

Huiwen Zhang, Feng Ye, Chu Ma

Department of Electrical & Computer Engineering Madison, WI, 53706, USA

Neural PDE Solvers Physics-Informed ML

Open paper AskPDF arXiv

Physics-Informed Neural Networks (PINNs) embed governing equations into deep learning, but enforce them only through loss residuals, leaving highly oscillatory wave behavior to be discovered by optimization. As a result, methods that achieve relative

L_2

errors below

10^{-3}

on standard manufactured Helmholtz benchmarks can fail on practical radiation problems involving singular excitations, absorbing boundaries, and wave fields spanning tens of wavelengths. Architectural physics embedding addresses this limitation by factorizing the field into analytically derived oscillatory kernels and learnable envelopes. However, the kernel dictionary must be manually constructed and scales with the number of elementary units, growing exponentially with the depth of hierarchically structured systems such as antenna arrays and metasurfaces. We propose PE-EK-PINN (Physics Embedded with Evolving Kernels), which treats physics kernels as reusable learned representations rather than fixed analytical inputs. A converged subsystem field is frozen and promoted to an evolved kernel, whose transformed copies are reused to represent higher-level configurations without deriving new governing equations. The resulting hierarchy makes the peak number of active kernels independent of system size and reduces cumulative training cost from

O(N)

O(\log N)

. Experiments on dipole arrays, composite line-source geometries, and cross arrays demonstrate the dramatic training cost reduction, while achieving a reduced or comparable relative

L_2

error. One notable example is PE-EK-PINN solves a

256

-dipole array more than 30 times faster than direct PE-PINN.

Abstract

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