Mehari Fentahun Endalew

arXiv2026

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 $(β_0,β_1)=(0.75,0.50)$, the predicted wall shear is $f''(0)=0.8997161394$, compared with the finite-domain reference $0.8997168085$, giving an absolute error of $6.691\times10^{-7}$. 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.

physics.flu-dyn↗

arXiv2026

Adaptive Physics-Informed Neural Networks for the Blasius Boundary-Layer Problem

Physics-informed neural networks (PINNs) provide a mesh-free approach for solving differential equations, but their performance can depend strongly on loss weighting, collocation placement, and optimization strategy. This study develops an adaptive PINN framework for the Blasius boundary-layer equation using gradient-norm-based adaptive loss weighting, nonuniform and residual-based collocation, and sequential Adam--L-BFGS optimization. In the representative run using the architecture $[1,100,100,1]$, the model predicts $f''(0)=0.3320762918$, compared with the high-accuracy benchmark $0.332057336215$, giving an absolute error of $1.896\times10^{-5}$. The final weighted loss is $6.789\times10^{-8}$, and the predicted stream-function, velocity, and shear profiles agree closely with an independent numerical boundary-value solution. A separate full-training architecture study shows that the two-hidden-layer model achieves the smallest wall-shear error among the four tested architectures, $1.629\times10^{-6}$, whereas the deepest network attains the smallest weighted objective but a substantially larger wall-shear error. Compared with the previously reported PINN value $f''(0)=0.33165$, the representative run reduces the wall-shear error by approximately a factor of $21.5$. The results show that the combined adaptive training framework can achieve high accuracy for the Blasius problem and that weighted loss alone is insufficient for identifying the most physically accurate PINN. Because the adaptive components are applied jointly, their individual contributions cannot be isolated from the present results and would require a controlled ablation study for separate assessment.

cs.LG↗