Hard-Constrained Physics--Informed Neural Network with Adaptive Regional Residual Balancing for the Generalized Falkner--Skan Problem

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 $f^{\mathrm{\prime }\mathrm{\prime }}(0)=0.8997161394$, compared with the finite-domain reference $0.8997168085$, giving an absolute error of $6.691\times {10}^{-7}$. The residual MSE is $6.357\times {10}^{-9}$, while the relative $L_2$ errors in $f^{\mathrm{\prime }}$ and $f^{\mathrm{\prime }\mathrm{\prime }}$ are $2.848\times {10}^{-6}$ and $4.188\times {10}^{-5}$. 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.