Hard-Constrained Physics--Informed Neural Network with Adaptive Regional Residual Balancing for the Generalized Falkner--Skan Problem
Abstract
We present a physics--informed neural solver for the generalized Falkner--Skan boundary-value problem that combines an exact boundary--admissible trial representation with adaptive regional residual balancing (ARRB). The three prescribed boundary conditions are embedded analytically, eliminating boundary-condition penalty terms while allowing the finite--domain streamfunction value and wall shear to be determined by the governing equation. The residual is divided into wall, middle, and tail regions, and a physical global residual is reconstructed from regional mean-squared errors using region-length fractions. Safeguarded exponential--moving--average inverse--gradient coefficients adaptively balance the regional contributions during Adam optimization. A deterministic 800-point residual monitor is used for checkpoint selection, followed by two deterministic L--BFGS stages minimizing the physical global residual. The method is tested on the Blasius, favourable-pressure--gradient Falkner--Skan, and Pohlhausen cases. For (β0,β1)=(0.75,0.50), the predicted wall shear is f''(0)=0.8997161394, compared with the finite-domain reference 0.8997168085, giving an absolute error of 6.691×10−7. The residual MSE is 6.357×10−9, while the relative L2 errors in f' and f'' are 2.848×10−6 and 4.188×10−5. A matched single--seed ablation shows that ARRB reduces residual MSE by 52.81\% relative to global--residual training and by 50.24\% relative to equal-regional weighting. Wall--shear errors are reduced by 77.30\% and 72.37\%, respectively. These results support ARRB as an accuracy-oriented residual-conditioning strategy, while multi--seed experiments remain necessary to quantify optimization variability.
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BibTeX
@misc{endalew2026hardconstrained,
title = {{Hard-Constrained Physics--Informed Neural Network with Adaptive Regional Residual Balancing for the Generalized Falkner--Skan Problem}},
author = {Mehari Fentahun Endalew and Xiaoming John Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23493},
eprint = {2609.23493},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23493},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hard-Constrained Physics--Informed Neural Network with Adaptive Regional Residual Balancing for the Generalized Falkner--Skan Problem
AU - Mehari Fentahun Endalew
AU - Xiaoming John Zhang
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23493
UR - https://arxiv.org/abs/2609.23493
AB - We present a physics--informed neural solver for the generalized Falkner--Skan boundary-value problem that combines an exact boundary--admissible trial representation with adaptive regional residual balancing (ARRB). The three prescribed boundary conditions are embedded analytically, eliminating boundary-condition penalty terms while allowing the finite--domain streamfunction value and wall shear to be determined by the governing equation. The residual is divided into wall, middle, and tail regions, and a physical global residual is reconstructed from regional mean-squared errors using region-length fractions. Safeguarded exponential--moving--average inverse--gradient coefficients adaptively balance the regional contributions during Adam optimization. A deterministic 800-point residual monitor is used for checkpoint selection, followed by two deterministic L--BFGS stages minimizing the physical global residual. The method is tested on the Blasius, favourable-pressure--gradient Falkner--Skan, and Pohlhausen cases. For (β_0,β_1)=(0.75,0.50), the predicted wall shear is f''(0)=0.8997161394, compared with the finite-domain reference 0.8997168085, giving an absolute error of 6.691×10^−7. The residual MSE is 6.357×10^−9, while the relative L_2 errors in f' and f'' are 2.848×10^−6 and 4.188×10^−5. A matched single--seed ablation shows that ARRB reduces residual MSE by 52.81\% relative to global--residual training and by 50.24\% relative to equal-regional weighting. Wall--shear errors are reduced by 77.30\% and 72.37\%, respectively. These results support ARRB as an accuracy-oriented residual-conditioning strategy, while multi--seed experiments remain necessary to quantify optimization variability.
ER -