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

Mehari Fentahun Endalew; Xiaoming John Zhang

cs.LGAug 28, 2026

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

Authors: Mehari Fentahun Endalew, Xiaoming John Zhang

Organizations: Beijing Institute of Mathematical Sciences and Applications (BIMSA), Beijing, China

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.

Figures & tables

Quantity	Value
Total weighted loss	$6.789\times 10^{-8}$
Raw ODE loss	$6.789\times 10^{-9}$
Raw wall loss	$5.574\times 10^{-12}$
Raw far-field loss	$1.435\times 10^{-14}$

Table 1: Final loss values after L-BFGS refinement.

Iteration	Weighted Loss	$\bm{f^{\prime\prime}(0)}$	$\bm{\lambda_{\mathrm{ODE}}}$	$\bm{\lambda_{\mathrm{Wall}}}$	$\bm{\lambda_{\mathrm{Far}}}$
0	$9.61\times 10^{-2}$	0.00130673	10.00	10.00	0.10
1000	$1.29\times 10^{-4}$	0.33104609	10.00	0.10	10.00
2000	$1.74\times 10^{-5}$	0.33163539	10.00	0.10	10.00
3000	$1.13\times 10^{-5}$	0.33170084	10.00	0.10	5.82
4000	$7.53\times 10^{-6}$	0.33247665	10.00	0.10	10.00
5000	$3.21\times 10^{-6}$	0.33231699	10.00	0.10	5.02

Table 2: Representative Adam training output.

Figure 1: Training-loss history during Adam and L-BFGS optimization. The red dotted vertical lines denote residual-based collocation updates, while the black dashed line marks the transition from Adam to L-BFGS. The horizontal coordinate contains the Adam iterations followed by successive outer L-BFGS calls.

Figure 2: Evolution of the adaptive ODE, wall, and far-field loss coefficients during Adam training. The coefficients are restricted to $0.1\leq\lambda_{k}\leq 10$ .

Outer Step	Weighted Loss	$\bm{f^{\prime\prime}(0)}$
0	$8.50\times 10^{-8}$	0.33207789
1	$8.44\times 10^{-8}$	0.33207733
2	$8.39\times 10^{-8}$	0.33207678
3	$8.35\times 10^{-8}$	0.33207634
4	$8.31\times 10^{-8}$	0.33207593
5	$8.28\times 10^{-8}$	0.33207566

Table 3: Representative L-BFGS outer-step results.

$\eta$	$f(\eta)$	$f^{\prime}(\eta)$	$f^{\prime\prime}(\eta)$
0	0.000002	0.000001	0.332076
1	0.165570	0.329764	0.322968
2	0.649993	0.629726	0.266733
3	1.396729	0.845988	0.161346
4	2.305609	0.955464	0.064253
5	3.283088	0.991496	0.015899

Table 4: Representative adaptive PINN solution for the Blasius equation.

Figure 3: Comparison of the Blasius stream function $f(\eta)$ predicted by the adaptive PINN and the independent numerical BVP solution.

Figure 4: Comparison of the nondimensional velocity profile

f^{\prime}(\eta)

predicted by the adaptive PINN and the numerical BVP solution.

Figure 5: Comparison of the shear profile

f^{\prime\prime}(\eta)

predicted by the adaptive PINN and the numerical BVP solution.

Figure 6: Pointwise absolute errors in $f(\eta)$ , $f^{\prime}(\eta)$ , and $f^{\prime\prime}(\eta)$ relative to the independent numerical BVP solution.

Figure 7: Comparison of the adaptive PINN nondimensional velocity profile with classical Blasius reference values [ 2 ] .

Figure 8: Pointwise absolute Blasius-equation residual over the computational domain. The maximum residual over the $500$ evaluation points is approximately $3.24\times 10^{-4}$ .

Model	Architecture	$\bm{f^{\prime\prime}(0)}$	Absolute Error	Raw ODE Loss	Weighted Loss
$f_{1}$	$[1,100,100,1]$	0.3320557068	$1.629\times 10^{-6}$	$4.072\times 10^{-8}$	$3.089\times 10^{-7}$
$f_{2}$	$[1,100,100,100,1]$	0.3321201301	$6.279\times 10^{-5}$	$1.209\times 10^{-8}$	$1.209\times 10^{-7}$
$f_{3}$	$[1,90,90,90,90,1]$	0.3320457565	$1.158\times 10^{-5}$	$2.017\times 10^{-8}$	$2.017\times 10^{-7}$
$f_{4}$	$[1,90,90,90,90,90,1]$	0.3318284466	$2.289\times 10^{-4}$	$2.424\times 10^{-7}$	$2.424\times 10^{-8}$

Table 5: Full-training results for the four adaptive PINN architectures.

Feature	Krishna et al. [ 13 ]	Present representative PINN
Computational domain	$[0,8]$	$[0,8]$
Representative network	2 hidden layers, 100 neurons each	$[1,100,100,1]$
Base collocation	100 equidistant points	499 unique nonuniform points
Near-wall concentration	Uniform distribution	60% of nominal base points in $[0,3]$
Residual-based refinement	Not reported	Yes
Loss weighting	Non-adaptive combined objective	Gradient-norm adaptive weighting

Table 6: Comparison with the Blasius PINN formulation of Krishna et al. [ 13 ] .

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