Critical heat flux is a major thermal limit in water-cooled nuclear reactors because heat-transfer deterioration can produce a rapid rise in fuel-cladding temperature and potentially damage the cladding. Under non-uniform axial heating, the local margin is not determined solely by pressure, mass flux, geometry, or equilibrium quality. It also depends on where heat has been deposited earlier along the channel, which alters the evolving two-phase flow and thermodynamic state. Engineering methods commonly begin with a critical heat-flux value for equivalent uniform heating and then apply a correction representing the axial power history. The study keeps that structure, using a frozen neural network trained on a large uniform-heating database as the baseline and training a compact transformer only to estimate the additional non-uniform-heating correction.

The experimental basis was a benchmark-ready database derived from non-uniform critical-heat-flux measurements compiled in a Korea Atomic Energy Research Institute report. Its 888 experiments were divided among inlet-peaked, middle-peaked, outlet-peaked, and spike-like axial power profiles. Before modeling, the researchers excluded 39 cases with two-phase inlet conditions because the frozen uniform-heating backbone produced unreliable predictions for them. They also excluded 123 cases that required the backbone to extrapolate beyond its diameter, mass-flux, or equilibrium-quality ranges. The resulting 726 experiments were randomly divided at the experiment level into training, validation, and test sets at 70%, 15%, and 15%. These exclusions define the model’s applicability domain and are not intrinsic limitations of the transformer architecture. A separate leave-one-profile-family-out evaluation tested what happened when an entire heating-profile family was withheld from training and validation.

For each axial location, the transformer received the normalized axial position, local equilibrium quality, and normalized axial heat-flux shape as a sequence token. Average heat flux and mass flux were also included, while mass flux was treated as a global operating-condition variable because it was conserved along the steady, constant-area test channels. The model used one encoder layer, a 32-dimensional embedding, two attention heads, a 64-unit feed-forward layer, GELU activation, and 0.10 dropout. No index-based positional encoding was added because physical axial position was already represented explicitly. A causal attention mask prevented downstream locations from influencing the correction at an upstream location, allowing the correction to depend on the current and accessible upstream history without using information from the future. After attention, a two-layer feed-forward head predicted an unconstrained value that was mapped to a positive correction factor through a bounded exponential transformation.

Training targeted the experiment-level minimum departure from nucleate boiling ratio, the minimum local thermal margin that determines the limiting critical-heat-flux condition. The objective penalized squared deviation of that minimum from a value of one while regularizing corrections away from unity along the sequence. This differs from pointwise training on the local critical heat flux, because the experimental data provide an experiment-level critical condition rather than separate local targets at every axial position. The researchers stated that alternative objectives, including logarithmic percentage-error losses, were not evaluated. Predicted correction factors ranged from 0.354 to 2.948 across all locations in the primary test set and from 0.814 to 2.493 at the transformer-selected limiting locations, so the numerical bounds were not approached closely. The authors also emphasized that the broad bounds were safeguards against extreme training values, not mechanistic limits on physical critical heat flux.

On the primary random split, the frozen uniform-heating neural network without any non-uniform correction substantially overpredicted the limiting critical heat flux. Its prediction-to-measurement mean was 1.247, its standard deviation was 0.388, and its mean absolute percentage error was 0.313. Applying the classical correction reduced the standard deviation to 0.143 and the mean absolute percentage error to 0.122. The Tong F-factor correction also improved on the uncorrected neural network but was less accurate on this test set, with a standard deviation of 0.246 and mean absolute percentage error of 0.178. Look-up-table predictions had still greater scatter, although they were evaluated on 108 of the 109 test cases because one middle-peaked case selected a limiting location outside the usable quality range. On the common 108-case subset, the transformer retained lower standard deviation and error than the classical correction, so the conclusion did not depend on the excluded look-up-table case. Paired bootstrap comparisons using 200,000 experiment-level resamples produced 95% confidence intervals below zero for differences in error and scatter, although no multiple-comparison adjustment was applied.

Results differed by axial profile. The transformer had lower scatter and mean absolute percentage error for the inlet-peaked and middle-peaked cases, which made up most of the test set. For 15 outlet-peaked cases, its standard deviation was close to that of the classical correction, while its error and mean were slightly better. Only one spike-like case appeared in the primary test set, so its result could not support a profile-level conclusion. The leave-one-family-out analysis revealed the model’s most important limitation. It retained lower scatter and error when the middle-peaked family was hidden, but performed worse than the classical correction when either the inlet-peaked or outlet-peaked family was held out. For the unseen inlet-peaked family, transformer scatter and mean absolute percentage error were 0.332 and 0.349, versus 0.235 and 0.200 for the classical correction. The corresponding held-out outlet-peaked values were 0.250 and 0.224 for the transformer and 0.182 and 0.152 for the baseline. Generalization to an entirely unseen profile family was therefore uneven.

Input ablations showed that the transformer relied on more than a single thermal variable. Removing global mass flux increased prediction scatter from 0.115 to 0.150, indicating that channel-level mass flux supplies important scaling information. Removing the normalized heat-flux profile raised scatter to 0.144, making the model comparable to the classical correction in that measure. Removing equilibrium-quality history produced the largest mean absolute percentage error among the ablations, 0.115, and shifted the prediction-to-measurement mean below unity. The complete model therefore provided the best balance of scatter, bias, and absolute error in the reported test set. The model also transferred between axial resolutions without retraining: a model trained at 200 locations was evaluated at 40, 100, 200, and 400 locations, and its prediction-to-measurement standard deviation remained between 0.114 and 0.115. The study did not investigate alternative global-conditioning strategies, and the results did not establish that the particular conditioning strategy used was optimal.

Robustness testing also exposed variation associated with model initialization. Across four training seeds, the transformer produced a lower mean absolute percentage error than the classical correction in every run. It reduced prediction scatter in three runs, but one seed yielded a standard deviation of 0.164, above the classical correction’s 0.143, although that seed still had a lower mean absolute percentage error of 0.115. The researchers said that engineering use would require further characterization or mitigation of this stochastic variability. Ensembling was suggested as a possible strategy but was not tested. Attention diagnostics at selected limiting locations showed that the model generally distributed attention over multiple accessible upstream positions rather than depending on one point. Weighting changed with the power-profile shape, including concentrations nearer earlier high-power regions for some profiles and closer to the downstream high-power region in an outlet-peaked case. These visualizations were treated as qualitative representations of learned information flow, not direct proof of physical causality or a complete account of any prediction.

Overall, the study shows that a small causal transformer can learn useful sequence-dependent corrections for critical heat flux under non-uniform axial heating while preserving a familiar engineering formulation. Its strongest reported advantage was narrower prediction error on the primary benchmark, with additional support from ablations, resolution-transfer tests, seed checks, and paired bootstrap analysis. Its weaknesses were equally concrete: the non-uniform database was small and imbalanced, unseen profile families produced inconsistent results, and the baseline was specific to one frozen uniform-heating neural network. The study did not compare alternative baseline predictors or establish an improved allowable design margin, because the statistical treatment of prediction errors and other uncertainties needed for such a limit was not developed. The authors characterized the result as a data-driven correction-factor model rather than a mechanistic model of critical heat flux.

Subject of Research: Machine-learning correction of critical heat-flux predictions under non-uniform axial reactor heating

Article Title: A transformer-based correction factor for critical heat flux prediction under non-uniform axial heating

Article References: Bourisaw, R., & Iskhakov, A. S. (2026). A transformer-based correction factor for critical heat flux prediction under non-uniform axial heating. Frontiers in Mechanical Engineering, 12, Article 1980739. https://doi.org/10.3389/fmech.2026.1980739

Image Credits: AI Generated

DOI: 10.3389/fmech.2026.1980739

Keywords: critical heat flux, non-uniform axial heating, causal transformer, neural networks, nuclear reactors, two-phase flow, minimum DNBR, model robustness, correction factor, transformer-based, correction, factor