Boundary-Layer-Induced Failure of Standard Physics-Informed Neural Networks: A Legendre Wavelet Collocation Benchmark for Singularly Perturbed Transport Problems

Abstract: Boundary layers provide a demanding test for numerical solvers because the solution may remain almost constant over most of the domain while changing rapidly in a narrow region near the boundary. This paper studies a singularly perturbed one-dimensional transport boundary-value problem with increasing Peclet number $(\mathrm{Pe})$. A local Legendre wavelet collocation method (LWM) is compared with a standard soft-boundary physics-informed neural network (PINN) for this benchmark. The wavelet approximation uses locally supported Legendre polynomial basis functions and converts the problem into a square algebraic collocation system with residual, boundary, and interface-continuity equations. Numerical experiments are performed for $\mathrm{Pe}=1,10,100,$ and $1000$. The LWM captures all four cases, with the largest error remaining below $5\times 10<sup>{-3}$. The standard soft-boundary PINN performs well for the mild cases but fails to resolve the sharp boundary layer for the larger Peclet numbers. The results show that local wavelet collocation is more reliable than the standard soft-boundary PINN for this benchmark, while dense near-boundary evaluation helps reveal errors that may be missed on coarse grids.

• The paper demonstrates that Legendre wavelet collocation resolves singular boundary layers across Pe=1–1000, maintaining a maximum error below 5×10−3 at Pe=1000.
• The standard soft-boundary PINN performs well for Pe≤10 but degenerates to an approximately constant 0.5 solution for Pe≥100, producing errors near 5×10−1.
• The paper shows that local basis support and layer-aware evaluation are crucial, while warning that wavelet-system conditioning rises to 5.9×10^9 and that adaptive or hard-boundary PINNs remain untested.

This paper presents a controlled numerical comparison between a direct two-sum Legendre wavelet collocation method (LWM) and a standard soft-boundary physics-informed neural network (PINN) on a singularly perturbed one-dimensional advection–diffusion boundary-value problem. The central finding is stark: the wavelet collocation scheme resolves the boundary layer for all tested Peclet numbers up to $\mathrm{Pe}=1000$ with maximum error below $5\times10^{-3}$, while the baseline PINN collapses to an essentially constant approximation with errors of order $5\times10^{-1}$ for $\mathrm{Pe}\ge 100$ (2606.09676).

Benchmark problem

The test problem is the steady transport equation

$$u''(x)-\mathrm{Pe}\,u'(x)=0,\qquad u(0)=1,\quad u(1)=0,$$

with exact solution $$u(x)=\bigl(1-e^{\mathrm{Pe}(x-1)}\bigr)/(1-e^{-\mathrm{Pe}})$$. Four cases are considered, $\mathrm{Pe}=1,10,100,1000$ (equivalently $\nu=1/(\mathrm{Pe})$ from $10^{0}$ to $10^{-3}$), spanning smooth solutions through strongly layer-dominated regimes in which the solution stays near unity over most of $5\times10^{-3}$0 and drops sharply within an $5\times10^{-3}$1 region adjacent to $5\times10^{-3}$2. Both methods are evaluated against this exact solution on a dense grid augmented with points clustered near $5\times10^{-3}$3, a deliberate design choice: because the dominant error is confined to a vanishingly narrow layer, coarse uniform sampling can mask failure entirely.

Legendre wavelet collocation formulation

The LWM partitions $5\times10^{-3}$4 into $5\times10^{-3}$5 equal cells of width $5\times10^{-3}$6, each mapped affinely to $5\times10^{-3}$7, and expands the solution as

$5\times10^{-3}$8

where $5\times10^{-3}$9 is the normalized mapped Legendre polynomial of degree $5\times10^{-1}$0, supported only on cell $5\times10^{-1}$1. With $5\times10^{-1}$2 modes per cell, the residual is collocated at $5\times10^{-1}$3 Gauss–Legendre points per cell; these $5\times10^{-1}$4 equations are supplemented by the two boundary conditions and value- and derivative-continuity conditions at the $5\times10^{-1}$5 internal interfaces, yielding exactly $5\times10^{-1}$6 equations for the $5\times10^{-1}$7 unknowns — a square linear system $5\times10^{-1}$8 solved directly, with no optimization involved.

The cell count is increased with Peclet number ($5\times10^{-1}$9 for $\mathrm{Pe}\ge 100$0). The summary results are:

Pe	Unknowns	cond($\mathrm{Pe}\ge 100$1)	Alg. residual	$\mathrm{Pe}\ge 100$2	$\mathrm{Pe}\ge 100$3
1	24	2.949E+04	4.441E−16	1.056E−09	4.341E−10
10	48	3.243E+05	2.842E−14	6.327E−06	1.312E−06
100	240	4.458E+07	1.819E−12	2.213E−04	2.021E−05
1000	1200	5.879E+09	1.310E−10	4.735E−03	2.006E−04

The algebraic residuals confirm exact solves, but the condition number grows roughly linearly with $\mathrm{Pe}\ge 100$4, reaching $\mathrm{Pe}\ge 100$5 at $\mathrm{Pe}\ge 100$6. This progressive ill-conditioning is the principal caveat on the LWM result: at still larger Peclet numbers, roundoff amplification could erode the accuracy reported here, and the paper does not address preconditioning or scaling. Pointwise data show that even at $\mathrm{Pe}\ge 100$7 the error peaks at about $\mathrm{Pe}\ge 100$8 inside the layer while remaining at machine precision over the flat portion of the domain — precisely the localized-error structure that motivates the dense near-boundary evaluation protocol.

Standard soft-boundary PINN formulation

The baseline network is a fully connected feed-forward architecture (one input, four hidden layers of 64 neurons, tanh activation, scalar output) trained on the loss

$\mathrm{Pe}\ge 100$9

with $$u''(x)-\mathrm{Pe}\,u'(x)=0,\qquad u(0)=1,\quad u(1)=0,$$0 uniform collocation points, $$u''(x)-\mathrm{Pe}\,u'(x)=0,\qquad u(0)=1,\quad u(1)=0,$$1, derivatives via automatic differentiation, Adam (15,000 epochs, learning rate $$u''(x)-\mathrm{Pe}\,u'(x)=0,\qquad u(0)=1,\quad u(1)=0,$$2) followed by L-BFGS refinement. No hard-constraint construction, adaptive weighting, domain decomposition, or layer-adapted sampling is used, so the comparison characterizes the vanilla Raissi-style formulation rather than physics-informed methods generally — a limitation the authors state explicitly.

The outcome bifurcates cleanly:

Pe	Total loss	Residual loss	BC loss	$$u''(x)-\mathrm{Pe}\,u'(x)=0,\qquad u(0)=1,\quad u(1)=0,$$3
1	8.763E−08	8.541E−08	2.225E−11	1.391E−05
10	1.145E−06	1.145E−06	3.469E−14	4.592E−06
100	4.975E+01	2.477E−01	4.950E−01	5.019E−01
1000	5.000E+01	2.603E−03	4.999E−01	5.000E−01

For $$u''(x)-\mathrm{Pe}\,u'(x)=0,\qquad u(0)=1,\quad u(1)=0,$$4 and $$u''(x)-\mathrm{Pe}\,u'(x)=0,\qquad u(0)=1,\quad u(1)=0,$$5, the PINN matches the exact profile with errors comparable to or slightly better than the LWM's $$u''(x)-\mathrm{Pe}\,u'(x)=0,\qquad u(0)=1,\quad u(1)=0,$$6 at $$u''(x)-\mathrm{Pe}\,u'(x)=0,\qquad u(0)=1,\quad u(1)=0,$$7. For $$u''(x)-\mathrm{Pe}\,u'(x)=0,\qquad u(0)=1,\quad u(1)=0,$$8 and $$u''(x)-\mathrm{Pe}\,u'(x)=0,\qquad u(0)=1,\quad u(1)=0,$$9, training fails qualitatively: the network converges to a nearly constant value near $$u(x)=\bigl(1-e^{\mathrm{Pe}(x-1)}\bigr)/(1-e^{-\mathrm{Pe}})$$0, violating both boundary conditions by approximately $$u(x)=\bigl(1-e^{\mathrm{Pe}(x-1)}\bigr)/(1-e^{-\mathrm{Pe}})$$1 each. Notably, at $$u(x)=\bigl(1-e^{\mathrm{Pe}(x-1)}\bigr)/(1-e^{-\mathrm{Pe}})$$2 the interior residual loss is tiny ($$u(x)=\bigl(1-e^{\mathrm{Pe}(x-1)}\bigr)/(1-e^{-\mathrm{Pe}})$$3) — a constant function satisfies $$u(x)=\bigl(1-e^{\mathrm{Pe}(x-1)}\bigr)/(1-e^{-\mathrm{Pe}})$$4 pointwise almost everywhere except in the layer, which occupies a measure-$$u(x)=\bigl(1-e^{\mathrm{Pe}(x-1)}\bigr)/(1-e^{-\mathrm{Pe}})$$5 fraction of the uniform collocation set. The optimizer therefore finds a low-residual, high-boundary-violation stationary point, illustrating the competing-objective pathology of soft penalty enforcement when the solution's variation is concentrated below the resolution of the collocation distribution. This is consistent with previously documented PINN failure modes under sharp gradients [De Florio et al., J. Comput. Appl. Math. 436 (2024)].

Direct comparison and interpretation

Pe	LWM $$u(x)=\bigl(1-e^{\mathrm{Pe}(x-1)}\bigr)/(1-e^{-\mathrm{Pe}})$$6	PINN $$u(x)=\bigl(1-e^{\mathrm{Pe}(x-1)}\bigr)/(1-e^{-\mathrm{Pe}})$$7	LWM $$u(x)=\bigl(1-e^{\mathrm{Pe}(x-1)}\bigr)/(1-e^{-\mathrm{Pe}})$$8	PINN $$u(x)=\bigl(1-e^{\mathrm{Pe}(x-1)}\bigr)/(1-e^{-\mathrm{Pe}})$$9
1	1.056E−09	1.391E−05	4.341E−10	7.792E−06
10	6.327E−06	4.592E−06	1.312E−06	2.313E−06
100	2.213E−04	5.019E−01	2.021E−05	4.949E−01
1000	4.735E−03	5.000E−01	2.006E−04	4.995E−01

At $\mathrm{Pe}=1,10,100,1000$0 the gap in $\mathrm{Pe}=1,10,100,1000$1 spans more than two orders of magnitude of relative degradation between the methods. The authors attribute the divergence to representation structure rather than optimization per se: the wavelet basis admits local refinement simply by increasing $\mathrm{Pe}=1,10,100,1000$2 in the layer region, whereas the globally supported tanh network has no analogous mechanism, and the uniform collocation distribution provides no gradient information where it matters. A secondary methodological contribution is the evaluation protocol itself — the paper demonstrates that without dense near-boundary sampling, the PINN's failure would be invisible on a coarse uniform grid, since the erroneous constant solution agrees with the exact solution over most of the interval.

Limitations and open questions

Several caveats bound the strength of the conclusions. First, the comparison targets the standard soft-boundary PINN specifically; hard-constraint methods (constrained expressions, X-TFC), adaptive-collocation schemes, and domain-decomposed variants were not tested, so the result does not establish that physics-informed neural approaches cannot handle this problem class. Second, the LWM requires manual tuning of $\mathrm{Pe}=1,10,100,1000$3 per Peclet number, and its conditioning degrades steadily with $\mathrm{Pe}=1,10,100,1000$4; whether the approach remains viable beyond $\mathrm{Pe}=1,10,100,1000$5 without preconditioning is unresolved. Third, the benchmark is a linear one-dimensional ODE with a known exact solution; extension to nonlinear or multidimensional singularly perturbed problems, where both the collocation system assembly and the PINN loss landscape behave differently, is left open. Finally, no systematic study of initialization sensitivity, network depth, or collocation density for the failing PINN cases is provided, so the precise threshold in $\mathrm{Pe}=1,10,100,1000$6 at which the baseline formulation breaks down is not characterized.

Conclusion

On a shared exact-solution benchmark with dense layer-aware error evaluation, local Legendre wavelet collocation solves the singularly perturbed transport problem reliably across four decades of Peclet number, while the standard soft-boundary PINN resolves only the mild cases and degenerates to a constant for $\mathrm{Pe}=1,10,100,1000$7. The study isolates locality of basis support and boundary enforcement mechanism as the operative differences, and its evaluation protocol underscores that error assessment for layer-dominated problems must itself be layer-resolved. The open question the work poses directly is whether hard-boundary, adaptive-collocation, or domain-decomposed PINN variants can close the gap on this same benchmark.