Residual-Based Adaptive Refinement in Hybrid Variational-Collocation PINNs for Singular Elliptic PDEs with Mixed Dirichlet-Neumann Boundary Conditions

Somnath Giri; Muslim Malik; Rishikesh Yadav

math.NAOct 19, 2026

Residual-Based Adaptive Refinement in Hybrid Variational-Collocation PINNs for Singular Elliptic PDEs with Mixed Dirichlet-Neumann Boundary Conditions

Authors: Somnath Giri, Muslim Malik, Rishikesh Yadav

Organizations: School of Mathematical & Statistical Sciences, Indian Institute of Technology Mandi, Mandi, 175005, Himachal Pradesh, India

Abstract

Solutions of second-order elliptic PDE problems on nonconvex or cracked domains with mixed Dirichlet-Neumann conditions typically lie outside $H^2(Ω)$ . Near re-entrant corners, crack tips and points where the boundary condition changes type, the gradient grows like $r^{λ-1}$ with $λ<1$ . We study residual-based adaptive refinement in hybrid variational-collocation PINNs. A multiscale Fourier-feature network is trained on a composite loss. The loss combines a variational residual tested against Legendre bubble functions, a strong-form collocation term and penalties for the Dirichlet and Neumann data. The variational term needs only first derivatives of the network, whereas the collocation term requires second derivatives and provides a residual that can be evaluated at any point. After each training stage, candidate points selected by this residual are added to the collocation set. The method is tested on five two dimensional benchmarks, namely a Mode~III crack, a cross-shaped domain with four re-entrant corners with unequal amplitude, a reaction-diffusion problem with four localized peaks, a monotone semilinear problem with a boundary layer and the Motz problem. Compared with uniform point insertion under the same collocation and iteration budget, residual-based refinement reduced the relative $L^2$ error by up to a factor of $32$ , the relative $L^\infty$ error by up to a factor of $38$ and the relative $H^1$ semi-norm error by up to a factor of $15$ .

Figures & tables

Problem	Points of $\mathcal{S}$ that govern $\lambda$	Rays	$\omega$	$\lambda_{1}$
Crack (§ 5.1 )	crack tip	(N,N)	$2\pi$	$1/2$
Cross (§ 5.2 )	four re-entrant corners	(N,N)	$3\pi/2$	$2/3$
Reaction-diffusion (§ 5.3 )	vertices	(N,N)	$\pi/2$	$2$
Semilinear (§ 5.4 )	re-entrant corner	(D,D)	$3\pi/2$	$2/3$
Motz (§ 5.5 )	junction at $(0,0)$	(N,D)	$\pi$	$1/2$

Table 1: Leading singular exponents from ( 2.9 ). In the cross problem the corner singularity enters through the flux data

h

, the exponent is the same for (D,D) and (N,N).

Problem	$\Psi(\bm{x})$	$n_{\mathrm{in}}$
Crack (§ 5.1 )	$\bigl(x,\,y,\,r^{1/2}\cos\tfrac{\theta}{2},\,r^{1/2}\sin\tfrac{\theta}{2}\bigr)$	4
Cross (§ 5.2 )	$\bigl(x,\,y,\,r_{1}^{2/3},\,r_{2}^{2/3},\,r_{3}^{2/3},\,r_{4}^{2/3}\bigr)$	6
Reaction-diffusion (§ 5.3 )	$(x,\,y)$	2
Semilinear (§ 5.4 )	$\bigl(x,\,y,\,r^{2/3}\bigr)$	3
Motz (§ 5.5 )	$\bigl(x,\,y,\,r^{1/2}\cos\tfrac{\theta}{2}\bigr)$	3

Table 2: The enrichment map

\Psi

of ( 3.1 ). Here

r_{i}

is the distance to the

i

-th re-entrant corner. For the crack,

(r,\theta)

are polar coordinates centered at the tip, with

\theta\in(0,2\pi)

and the branch cut on the slit. For the Motz problem,

(r,\theta)

are centered at the Dirichlet-Neumann junction, with

\theta\in[0,\pi]

. In the semilinear row,

r

is the distance to the re-entrant corner at the origin. Every radius is evaluated through ( 3.2 ).

Panel of Figure 1 — Figure 1: Architecture of the proposed framework. A multiscale Fourier-feature embedding $\gamma$ , whose frequency matrices are drawn once and held fixed, acts on the enriched coordinate $\bm{z}=\Psi(\bm{x})$ and feeds a residual $\tanh$ network whose output, scaled by $c_{u}$ , is the trial function $\hat{u}(\bm{x};\bm{\theta})$ , $\Psi$ appends the leading singular mode where the corner behavior is known and is the identity otherwise. One application of automatic differentiation supplies $\nabla\hat{u}$ for the variational residual $\widetilde{\mathcal{R}}^{e}_{ij}$ , assembled on the fixed quadrature set $\mathcal{T}_{Q}$ against the Legendre bubble test functions $v^{e}_{ij}$ , a second, nested application supplies $\Delta\hat{u}$ for the strong-form residual $R(\hat{u})$ on the collocation set $\mathcal{T}_{f}$ . As every $v^{e}_{ij}$ vanishes on $\partial\Omega_{e}$ , the natural boundary functional $B^{e}_{ij}$ vanishes identically and the Neumann data reach the loss only through the penalty $\mathcal{L}_{N}$ . The composite loss is the weighted sum of the variational residual, the collocation term and the two boundary penalties. The RAR loop branches from the strong-form residual, the only loss component that can be evaluated at an arbitrary interior point. It scores a candidate pool $\mathcal{C}$ , discards candidates inside the guard radius $r_{\mathrm{g}}$ , inserts the $m$ highest ranked points into $\mathcal{T}_{f}$ and continues training from the current parameters. The element partition, the number $K$ of test functions per direction and the quadrature set $\mathcal{T}_{Q}$ are fixed throughout, so the scheme performs neither $h$ - nor $p$ -refinement, adaptivity acts on $\mathcal{T}_{f}$ alone.

Metric	VC-PINNs (uniform)	VC-PINNs-RAR
Relative $L^{2}$ error	$0.4138$	$\mathbf{0.3545}$
Relative $L^{\infty}$ error	$2.462$	$\mathbf{1.889}$
Relative $H^{1}$ semi-norm error	$2.374$	$\mathbf{1.442}$

Table 3: Crack problem on the slit square (Mode III). All errors are reported in percent (%) and bold marks the more accurate arm in each row. Both arms use the same collocation schedule (

|\mathcal{T}_{f}|=1500\to 3900

150{,}000

Adam iterations and thirteen L-BFGS stages, one training run per arm. Errors are evaluated on a uniform

301\times 301

grid on

[-1,1]^{2}

. The relative

L^{\infty}

error is

\max|u-\hat{u}|/\max|u|

over the grid, where

\max|u|=2^{1/4}\cos(\pi/8)\approx 1.099

is attained at the corners

(1,\pm 1)

, the

H^{1}

semi-norm excludes grid points with

r\leq 10^{-3}

Panel of Figure 2 — Figure 2: Crack problem on the slit square (Mode III). (A-C) Exact solution, VC-PINNs (uniform) prediction and pointwise absolute error. (D-F) Domain with boundary conditions and element partition, uniform collocation set and adaptive collocation set after twelve rounds, $|\mathcal{T}_{f}|=1500\to 3900$ . (G-I) Exact solution, VC-PINNs-RAR prediction and pointwise absolute error; panels C and I have separate color scales. (J-K) Adam loss components for the uniform and the RAR arm.

Metric	VC-PINNs (uniform)	VC-PINNs-RAR
Relative $L^{2}$ error	$5.078$	$\mathbf{0.1601}$
Relative $L^{\infty}$ error	$13.85$	$\mathbf{1.283}$
Relative $H^{1}$ semi-norm error	$32.36$	$\mathbf{2.248}$

Table 4: Cross problem. All errors are reported in percent (%) and bold marks the more accurate arm in each row. Both arms use the same collocation schedule (

|\mathcal{T}_{f}|=8000\to 10{,}400

150{,}000

Adam iterations and thirteen L-BFGS stages; one training run per arm. Errors are evaluated at the points of a uniform

301\times 301

grid on

[-1,1]^{2}

that lie in

\overline{\Omega}

. The relative

L^{\infty}

error is

\max|u-\hat{u}|/\max|u|

over these points, where

\max|u|\approx 3.577

; the

H^{1}

semi-norm uses the network gradient from automatic differentiation and excludes points within

10^{-3}

of a corner.

Panel of Figure 3 — Figure 3: Cross problem: Laplace equation on the cross-shaped domain. (A–C) Exact solution, VC-PINNs (uniform) prediction and pointwise absolute error. (D-F) Domain with boundary conditions and element partition, uniform collocation set and adaptive collocation set after twelve rounds, $|\mathcal{T}_{f}|=8000\to 10{,}400$ . (G-I) Exact solution, VC-PINNs-RAR prediction and pointwise absolute error; panels C and I have separate color scales. (J-K) Adam loss components for the uniform and the RAR arm.

Metric	VC-PINNs (uniform)	VC-PINNs-RAR
Relative $L^{2}$ error	$16.00$	$\mathbf{1.015}$
Relative $L^{\infty}$ error	$26.50$	$\mathbf{0.6986}$
Relative $H^{1}$ semi-norm error	$64.46$	$\mathbf{4.738}$

Table 5: Reaction-diffusion problem. All errors are reported in percent (%) and bold marks the more accurate arm in each row. The RAR arm grows its collocation set from

10{,}000

13{,}000

points over ten rounds of

300

insertions, with

140{,}000

Adam iterations and eleven L-BFGS stages. The uniform arm is trained on a fixed set of uniformly drawn collocation points without insertion rounds. One training run per arm. Errors are evaluated on a uniform

401\times 401

grid on

[-1,1]^{2}

and the

H^{1}

semi-norm is computed with central differences on the same grid. As

\max|u|=1

, the relative

L^{\infty}

error equals the absolute one.

Panel of Figure 4 — Figure 4: Reaction-diffusion problem with four localized peaks. (A-C) Exact solution, VC-PINNs (uniform) prediction and pointwise absolute error. (D-F) Domain with the four peak centers and element partition, uniform collocation set and adaptive collocation set after ten rounds, $|\mathcal{T}_{f}|=10{,}000\to 13{,}000$ . (G-I) Exact solution, VC-PINNs-RAR prediction and pointwise absolute error. (J-K) Adam loss components for the uniform and the RAR arm.

Metric	VC-PINNs (uniform)	VC-PINNs-RAR
Relative $L^{2}$ error	$2.403$	$\mathbf{0.1904}$
Relative $L^{\infty}$ error	$55.85$	$\mathbf{3.546}$
Relative $H^{1}$ semi-norm error	$39.00$	$\mathbf{3.366}$

Table 6: Semilinear problem. All errors are reported in percent (%) and bold marks the more accurate arm in each row. Both arms use the same collocation schedule (

|\mathcal{T}_{f}|=4000\to 7600

150{,}000

Adam iterations and thirteen L-BFGS stages, with no interior anchors; one training run per arm. Errors are measured against the finite element reference on its

601\times 601

grid and the

H^{1}

semi-norm excludes grid points within

10^{-3}

of the corner. As

0\leq u\leq 1

with

\max u=1

, the relative

L^{\infty}

error equals the absolute one.

Panel of Figure 5 — Figure 5: Semilinear problem on the L-shaped domain. (A-C) Finite element reference, VC-PINNs (uniform) prediction and pointwise absolute error against the reference. (D-F) Domain with boundary conditions and element partition, uniform collocation set and adaptive collocation set after twelve rounds, $|\mathcal{T}_{f}|=4000\to 7600$ . (G-I) Finite element reference, VC-PINNs-RAR prediction and pointwise absolute error. (J-K) Adam loss components for the uniform and the RAR arm.

Metric	VC-PINNs (uniform)	VC-PINNs-RAR
Relative $L^{2}$ error	$0.2407$	$\mathbf{0.2096}$
Relative $L^{\infty}$ error	$0.7872$	$\mathbf{0.4458}$
Relative $H^{1}$ semi-norm error	$1.666$	$\mathbf{0.9920}$
Leading coefficient $a_{0}$	$399.92050$	$\mathbf{400.15007}$

Table 7: Motz problem. The first three rows are relative errors in percent (%), the last row is the leading coefficient

a_{0}

. Bold marks the more accurate arm in each row. Both arms use the same collocation schedule (

|\mathcal{T}_{f}|=1500\to 3900

150{,}000

Adam iterations and thirteen L-BFGS stages, one training run per arm. Errors are measured against the quadratic (

\mathbb{P}_{2}

) finite element reference, whose own error floor is

0.000079\%

. As

0\leq u\leq 500

with

\max u=500

, the relative

L^{\infty}

error is the absolute error divided by

500

. The leading coefficient is extracted by ( 5.12 ) at

r_{0}=0.5

, its reference value is

a_{0}=401.16191

Panel of Figure 6 — Figure 6: The Motz problem. (A-C) Finite element reference solution, VC-PINNs (uniform) prediction and pointwise absolute error against the reference. (D-F) Domain with the mixed boundary conditions and element partition, uniform collocation set and adaptive collocation set after twelve rounds, $|\mathcal{T}_{f}|=1500\to 3900$ . (G-I) Finite element reference solution, VC-PINNs-RAR prediction and pointwise absolute error. (J-K) Adam loss components for the uniform and the RAR arm.

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Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers

Henry Kasumba, Ronald Katende

Department of Mathematics · Makerere University · 7062, Kampala, Uganda+2

PDE Solving Adaptive Mesh Refinement

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Classical finite-difference solvers remain reliable tools for partial differential equations, but their efficiency depends on where mesh resolution is placed. Uniform refinement can waste degrees of freedom when solution difficulty is localised near sharp gradients, fronts, oscillations, or constraint-sensitive regions. This paper studies a hybrid strategy in which a physics-informed neural network (PINN) is used not as the final solver, but as an off-grid residual probe for adaptive mesh refinement. The PINN residual is sampled over the domain, converted into cellwise indicators, and used to guide refinement before the final approximation is computed by a finite-difference solver. The method is evaluated on three benchmarks. The main full-solver validation uses the one-dimensional viscous Burgers equation with a nonuniform finite-difference solve on the adapted meshes. PINN-threshold refinement attains final relative

L^2

error

0.021067

with

60

degrees of freedom, compared with

0.022617

for uniform refinement with

192

degrees of freedom. At matched mesh size, PINN-threshold reduces the error by about

67.5\%

. PINN-Dörfler refinement gives similar performance, with error

0.021264

using

58

degrees of freedom. A gradient indicator remains slightly more accurate, so the result supports usefulness rather than universal superiority. Manufactured 2D and 3D proxy tests, based on a nonlinear Schrödinger equation and an incompressible Navier--Stokes system, show that PINN residuals can organise structured refinement and improve over random refinement, although they do not consistently outperform gradient or uniform baselines. The results support PINN-guided AMR as a residual-indicator strategy for transferring physics-informed diagnostic information into finite-difference mesh adaptation while preserving the classical solver as the final approximation engine.

Abstract

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