A proposed way of starting physics-informed neural networks, or PINNs, reported shorter total computing times in four neutron-diffusion benchmark problems while retaining comparable numerical accuracy overall. The abstract lists reported reductions of 25.4%, 38.2%, 49.4% and 28.9%, along with fewer anomalous results in the effective multiplication factor, the parameter solved for alongside neutron flux.
The tests covered 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 cubic case. The document is labeled an arXiv preprint.
Starting closer to a solution
JI-PINN, the paper’s joint-initialization approach, uses a low-resolution discrete K-eigenvalue solution to initialize both a neural network for groupwise flux and the effective multiplication factor. The two are then optimized together under physical constraints.
The evaluation examined the quality of the initialization information, the role of initializing both quantities together, sensitivity to random starting states, and computational performance under different problem conditions. Its comparisons included a randomly initialized PINN and R2-PINN.
For the setup used in later experiments, the researchers tested low-resolution grid sizes Nh of 17, 25, 33 and 49, and pretraining lengths Mpre of 1,000, 3,000, 5,000 and 8,000 steps. They selected Nh=25 and Mpre=5,000.
The clearest gain came with repeated starts
In a two-material comparison run with 15 random seeds for each method, JI-PINN’s reported effective-multiplication-factor values were concentrated near the reference, and its flux-error variability was lower. Average total time was reported as 25.4% lower than randomly initialized PINN and 38.8% lower than R2-PINN.
Across the 15 JI-PINN runs, the final effective multiplication factor averaged 1.066 ± 0.001. The relative error averaged (2.428 ± 1.511) × 10^-3, while total time averaged 2,467.7 ± 931.5 seconds. The figures are reported as mean ± standard deviation.
The reported summaries contain a percentage discrepancy. The abstract gives 38.2% among the overall reductions, while the random-seed results give 38.8% against R2-PINN in the two-material comparison.
Speed was not the whole story
An ablation on the IAEA benchmark separated the two parts of the initialization. The reported total time moved from 7,398.5 seconds to 7,106.3 seconds when only the effective multiplication factor was initialized, a 3.9% reduction. With flux pretraining, it was 5,194.9 seconds, a reported 29.8% reduction from the starting value. Adding the corresponding approximate effective multiplication factor brought the reported time to 4,573.0 seconds, a further 12.0% reduction.
In the two-dimensional, two-group, four-material problem, JI-PINN reported an effective multiplication factor of 0.793601 against a reference of 0.79572, a relative error of 2.66 × 10^-3, and total time of 1,394.9 seconds. Its time was 49.4% lower than randomly initialized PINN and 69.5% lower than R2-PINN, but its effective-multiplication-factor error was higher than both.
In the three-dimensional, single-group cubic case, JI-PINN’s total time was 496.1 seconds, approximately 28.9% lower than the randomly initialized PINN and 19.0% lower than R2-PINN. Its effective multiplication factor was 0.950660, with relative error 6.95 × 10^-4 and flux MSE 9.49 × 10^-7. The paper described solution accuracy as generally comparable in that case.
A benchmark result with a clear boundary
The authors interpret JI-PINN as a more efficient initialization strategy that maintains a favorable balance between efficiency and accuracy in heterogeneous and three-dimensional numerical settings.
The stated limitation is dependence on the quality of the low-resolution approximation. For complex material interfaces or pronounced local flux variations, that approximation may affect final accuracy. The paper reports only four numerical benchmark configurations, and the random-seed evidence is concentrated in one two-material case using 15 runs.
The document is an arXiv preprint. Its results point to a possible workflow for PINN neutron-diffusion calculations, while leaving the speed-accuracy balance tied to the quality of the starting approximation.
Paper data and sources
Original title: 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 et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text