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 \((\beta_0,\beta_1)=(0.75,0.50)\), the predicted wall shear is \(f''(0)=0.8997161394\), compared with the finite-domain reference \(0.8997168085\), giving an absolute error of \(6.691\times10^{-7}\). The residual MSE is \(6.357\times10^{-9}\), while the relative \(L_2\) errors in \(f'\) and \(f''\) are \(2.848\times10^{-6}\) and \(4.188\times10^{-5}\). 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.