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REVIEW 3 major objections 6 minor 11 references

Physics-Informed Neural Networks for Irregular Domain Mapping and Partial Differential Equations solving

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that a PINN can learn an irregular-to-rectangular domain mapping, enabling finite-difference and convolutional neural network solvers on complex geometries.

desk verdict The grid-generation idea is promising, but the PDE-transformation formula in Eq. (13) is wrong, so the FDM/PICNN results solve a different equation than claimed. read the letter →

arxiv 2506.08622 v2 pith:ZRHOJGQY submitted 2025-06-10 physics.comp-ph

Cuizhi Zhou , Kaien Zhu This is my paper
classification physics.comp-ph
keywords physics-informedneuralnetworksirregulardomainsstructuredgridsdomainmappingfinitedifferencemethodconvolutionalJacobiandeterminantpartialdifferentialequations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes that a physics-informed neural network can learn the coordinate mapping between an irregular physical domain and a rectangular computational domain, replacing traditional elliptic grid generation. The network takes coordinates in the rectangle as input and outputs matching coordinates in the irregular domain, trained so boundaries align, the mapping is smooth, and cell Jacobians stay positive. Once the mapping is learned, PDEs stated on the irregular domain are rewritten in the rectangular coordinate system, so finite-difference and physics-informed convolutional neural network solvers can be used instead of unstructured-mesh solvers. The authors demonstrate the idea on heat conduction and steady Navier-Stokes examples, comparing against OpenFOAM finite-volume results. If the approach holds, one PINN training run yields both a reusable structured mesh and the transformation needed to solve a whole class of PDEs on complicated geometries.

What carries the argument

The load-bearing object is the learned coordinate map $G$, implemented as an MLP that takes computational-space coordinates $(\xi,\eta)$ and outputs physical-space coordinates $(x,y)$. Its quality is controlled by three loss terms: a Laplacian regularizer for smoothness, a boundary-matching term for conformity to the irregular boundary, and a Jacobian-determinant penalty enforcing $\det(J)>0$ to keep grid cells from folding. This map replaces the elliptic equations of classical body-fitted grid generation; requirements such as boundary-layer densification or local refinement are encoded directly as modifications to the loss rather than as new PDE systems.

What would settle it

Take the trained PINN mapping and evaluate $\det(J)=\frac{\partial x}{\partial \xi}\frac{\partial y}{\partial \eta}-\frac{\partial x}{\partial \eta}\frac{\partial y}{\partial \xi}$ on a dense grid much finer than the training samples; finding any interior point with $\det(J)\le 0$ would show the coordinate transformation folds and the mapped PDE (Eq. 13) is invalid there.

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Extended reading notes

Core claim

The central claim is that a single PINN, trained with a composite loss, produces a valid regular-to-irregular mapping $G:\Omega_c\to\Omega_p$ and its functional inverse, so that a structured grid on an irregular domain is obtained by pulling back the regular grid through $G$. The training loss combines a Laplacian smoothing term that keeps grid lines smooth, a boundary term that pins sampled boundary points to their true physical locations, and a Jacobian term that penalizes non-positive cell determinants, with the aim of ensuring $\det(J)>0$ everywhere. With the mapping in hand, the paper rewrites differential operators in computational coordinates—notably the Laplacian in the form $\nabla^2 T = \frac{1}{|J|}\left(\frac{\partial}{\partial \xi}\left(|J|\frac{\partial T}{\partial \xi}\right) + \frac{\partial}{\partial \eta}\left(|J|\frac{\partial T}{\partial \eta}\right)\right)$—and solves the transformed equation by FDM on a rectangle, mapping results back to the physical domain. The same PINN-generated mapping is also used as the grid for physics-informed convolutional neural networks on steady Navier-Stokes problems. The authors report that the resulting temperature fields and flow solutions match OpenFOAM reference results.

Load-bearing premise

The approach assumes that a soft penalty at finitely many sampled points is enough to guarantee the learned mapping is one-to-one and never folds over the entire domain, including regions between sample points.

Editorial extensions

If this is right

  • For any PDE on an irregular domain, the learned mapping converts the problem into a rectangular-domain problem, so FDM and spectral-type solvers become applicable without building an unstructured mesh.
  • Customized structured grids—denser boundary layers, refinement around specified points, and merged subgrids—can be produced by adding or modifying loss terms rather than by formulating new PDE-based grid generators.
  • The same PINN-generated mapping directly supplies the structured grid required by PICNN, so an AI-based PDE solver can run on complex geometries with no additional mesh-generation step.
  • Because the mapping is continuous, the resulting grids are reusable across different boundary conditions and source terms for the same geometry.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implication the paper leaves implicit is that the approach could extend to three dimensions by adding a third coordinate and a 3x3 Jacobian positivity penalty, since the loss formulation carries over.
  • The soft Jacobian penalty suggests a natural stress test: checking the learned map on a much finer grid than used in training would reveal whether non-folding holds between sample points.
  • The same loss-engineering idea could be pointed at time-dependent or moving-boundary problems, where the map would be re-learned or parameterized by time instead of fixed once.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes a physics-informed neural network (PINN) approach for generating structured grids in irregular physical domains by learning a mapping from a rectangular computational domain. The network is trained with a loss combining Laplace smoothing, boundary matching, and a soft Jacobian-determinant positivity penalty. The authors then use the mapping to transform PDEs to the computational domain, solving a heat conduction problem by finite differences and steady Navier-Stokes equations by physics-informed convolutional neural networks, with comparisons against OpenFOAM. The paper further demonstrates customized grids with denser boundary layers, denser/sparser regions around specified points, and merged subdomains.

Significance. The central idea---using a single PINN training to produce a structured grid and then applying classical or CNN-based solvers on a rectangle---is attractive and, if made correct, could be a useful alternative to elliptic grid generation. The loss-function formulation is flexible and the mesh examples in Figs. 3 and 7--10 are visually plausible. However, the paper as written does not establish the validity of the PDE-solving pipeline: the transformed Laplacian in Eq. (13) is incorrect for general non-affine maps, and the numerical comparisons are only qualitative. With a corrected transformation and quantitative validation, the method could be a worthwhile contribution.

major comments (3)
  1. [§2.2.1, Eq. (13)] Equation (13) is not the Laplacian in the computational coordinates for a general mapping. Starting from Eqs. (9)--(12), the sum ∂²T/∂x² + ∂²T/∂y² contains the metric coefficients g^{ξξ}=(x_η²+y_η²)/J², g^{ξη}=−(x_ξ x_η+y_ξ y_η)/J², and g^{ηη}=(x_ξ²+y_ξ²)/J²; the correct expression is ∇²T = (1/J)[∂_ξ(J g^{ξξ} T_ξ + J g^{ξη} T_η) + ∂_η(J g^{ξη} T_ξ + J g^{ηη} T_η)]. Equation (13) is recovered only if g^{ξξ}=g^{ηη}=1 and g^{ξη}=0, i.e., for an isometric mapping. For a polar map (x=ρ cosθ, y=ρ sinθ), Eq. (13) gives T_ρρ + T_ρ/ρ + T_θθ, whereas the correct Laplacian is T_ρρ + T_ρ/ρ + T_θθ/ρ². Consequently Eq. (18) and the FDM/PICNN results in Figs. 4--6 solve a different PDE from the intended heat and Navier--Stokes problems; this is an internal inconsistency with Eqs. (9)--(12) and invalidates the central claim.
  2. [§2.1, Eq. (6)] The Jacobian constraint is enforced only as a soft penalty at N_f sampled collocation points. This does not guarantee det J > 0 over the whole computational domain, and the transformation in Eq. (18) divides by J; a sign change or near-zero J between sample points would break both the grid validity and the PDE solve. The authors should report the minimum Jacobian over a dense validation grid or enforce the constraint in a way that provides a global guarantee.
  3. [§2.2.2, §2.3, Figs. 4--6] The comparisons with OpenFOAM are purely qualitative; no pointwise error, L2 norm, or convergence study is reported. Since the transformed equation is currently wrong, the apparent agreement cannot be taken as validation. After correcting Eq. (13), the authors should provide quantitative errors against OpenFOAM for the heat conduction and Navier--Stokes examples and a grid-convergence study for the FDM solver.
minor comments (6)
  1. [§2.1, Eq. (5)] The notation uθ(ξ,η) in the boundary loss is inconsistent with the mapping notation (x,y)=NN(ξ,η); please clarify the meaning of uθ.
  2. [§2.1, Eq. (6)] The summation in Eq. (6) is missing a lower limit, and the term inside the max should be written unambiguously as max(0, −(x_ξ y_η − x_η y_ξ)).
  3. [§2.1] The sentence 'The term Leq means solving a diffusion equation in computational space' is imprecise; Eq. (4) is a Laplace smoothing term, not a time-dependent diffusion equation.
  4. [§3.3, Eqs. (29)--(30)] The expressions for P and Q are singular at (ξ0,η0); please specify how the loss is evaluated at the densification point.
  5. [§2.1, §2.2] The manuscript does not report the MLP architecture, activation function, optimizer, learning rate, or training time; without these details the grid-generation and PDE-solving experiments are not reproducible.
  6. [Title and Abstract] The title and abstract contain grammatical errors ('PDEs solving', 'PDES'); please copyedit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the mapping is trained from boundary/smoothness constraints, and the PDE solvers are benchmarked against external OpenFOAM results.

full rationale

The claimed derivation chain does not reduce any prediction to fitted inputs. The PINN-generated map G is determined by boundary correspondence (Eqs. 1-2), the Laplacian smoothness losses (Eqs. 4, 7-8), and the Jacobian positivity penalty (Eq. 6); none of these terms involves the temperature or velocity fields solved later. The PDE results in Figs. 4-6 are compared with OpenFOAM FVM solutions, an external benchmark, and the coordinate-transformation framework is attributed to Anderson [11], not to the authors' own prior work. No uniqueness theorem is imported from a same-author citation, and no ansatz is smuggled in via self-citation. The reader's concern that the Jacobian constraint is only softly enforced, and the skeptic's point that Eq. (13) may not be the correct general-coordinate Laplacian for arbitrary non-affine maps, are correctness or validity issues rather than circularity: they do not make the predicted solution equivalent to the training data by construction. Therefore no circular step is identified.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The method introduces no new physical entities. It relies on standard elliptic grid generation theory and a soft Jacobian penalty, with several hand-tuned numerical parameters and unspecified training details.

free parameters (3)
  • Loss weights ω_eq, ω_bd, ω_jac
    Chosen by hand to balance boundary fit, smoothness, and Jacobian positivity; values are not reported, so the method cannot be exactly reproduced.
  • Densification parameters α, β, γ (Eqs. 25, 29, 30)
    Control grid density and distortion degree, selected by hand; the paper gives no automatic selection procedure.
  • Sampling counts N_f, N_b and network hyperparameters
    Number of collocation points, training iterations, learning rate, and MLP size are not specified, affecting grid quality and convergence.
assumptions (3)
  • standard math Coordinate transformation framework from [11]: the Laplacian transforms as in Eq. (13) for a differentiable, invertible mapping.
    Trusted from standard CFD texts; not re-derived in this paper.
  • domain assumption The elliptic (Laplace) equations for grid generation produce valid boundary-conforming grids when solved with boundary conditions.
    Standard in body-fitted grid generation; the paper relies on this instead of demonstrating convergence of the PINN solution to the true solution.
  • ad hoc to paper A soft penalty on negative Jacobian determinant at sampled points is sufficient to prevent grid folding globally.
    No mathematical proof or dense sampling analysis is provided; the paper assumes sample-level constraint extrapolates to the continuum.

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Cite this review

Pith. "Pith review of Physics-Informed Neural Networks for Irregular Domain Mapping and Partial Differential Equations solving." pith.science (2026). https://pith.science/paper/ZRHOJGQY

@misc{pith2026250608622,
  author       = {Pith},
  title        = {Pith review of: Physics-Informed Neural Networks for Irregular Domain Mapping and Partial Differential Equations solving},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZRHOJGQY}},
  note         = {Machine review of arXiv:2506.08622}
}
read the original abstract

The solution of partial differential equations (PDES) on irregular domains has long been a subject of significant research interest. In this work, we present an approach utilizing physics-informed neural networks (PINNs) to achieve irregular-to-regular domain mapping. Thus we can use finite difference method and physics-informed convolutional neural networks to solve PDEs on rectangular grids instead of the original irregular boundary. Structured grids on irregular domains are obtained by inverse mapping. We demonstrate PINN's versatile capability to produce customized structured grids tailored to diverse computational requirements, thereby significantly facilitating PDEs solving.

Figures

Figures reproduced from arXiv: 2506.08622 by the authors.

Figure 1
Figure 1. The PINN mapping approach. We follow the notation system in reference [10]. Grids are obtained by PINN method. The key point is to find the regular-to-irregular domain mapping G : Ωp → Ωc in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Schematic diagram of the PINN model. Ljac = 1 Nf X Nf i (max(0, −( ∂x ∂ξ ∂y ∂η − ∂x ∂η ∂y ∂ξ ))), (6) ∇2x = ∂ 2x ∂ξ2 + ∂ 2x ∂η2 , ∀(ξ, η) ∈ Ωc, (7) ∇2 y = ∂ 2 y ∂ξ2 + ∂ 2 y ∂η2 , ∀(ξ, η) ∈ Ωc. (8) The term Leq means solving a diffusion equation in computational space which is different from [10] where a diffusion equation is solved in physical space. Laplacian-based regularization term of the loss function enforces … view at source ↗
Figure 3
Figure 3. PINN-generated meshes on irregular domains. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Temperature distribution by OpenFOAM and FDM. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Comparison between our PINN-generated mesh and FDM-generated [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Navier-Stokes equations solved by: (left) PICNN on FDM-generated [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Mesh generation for a tokamak’s section. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Normal boundary layer (upper) vs. denser boundary layer (lower). [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Sparser and denser grids around specific points. [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: Grid merging 3.4 Mesh merging When solving physical problems, if different types of grids need to be gener￾ated for the interior and exterior of a device, PINN can be used to generate the structured grids separately, followed by grid merging process. In this way, the …

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