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

Abstract: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.

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