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arxiv: 2605.27308 · v1 · pith:PWTJWP5Inew · submitted 2026-05-26 · 💻 cs.GR · cs.NA· math.NA

PINNsur: Physics-Informed Neural Networks for PDEs on Curved Surfaces

Pranav Jain , Navami Kairanda , Peter Yichen Chen , Oded Stein This is my paper

Pith reviewed 2026-06-29 14:27 UTC · model grok-4.3

classification 💻 cs.GR cs.NAmath.NA
keywords physics-informed neural networkssurface PDEscurved surfacesneural fieldsdifferential operatorsconvergence analysisgeometry processing
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The pith

A neural field approximating surface normals lets PINNs solve PDEs on curved surfaces by projecting operators from R^3.

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

The paper presents PINNSur as a framework to solve partial differential equations on curved surfaces with physics-informed neural networks. It trains one neural field to represent the surface normals and then defines the needed surface operators by projecting the standard three-dimensional versions onto the surface. Because every orientable manifold has well-defined normals, the approach applies to surfaces of any curvature or topology without requiring a mesh. The authors also introduce a simple empirical test to check whether solutions converge to the true PDE despite the simultaneous approximation of both the unknown function and the surface geometry. If the method works, it supplies a continuous, mesh-free alternative to finite-element methods for surface PDEs in geometry processing.

Core claim

PINNSur trains a neural field to approximate the surface's normals and expresses surface differential operators using their projection from R^3 onto the surface. This construction works for all orientable manifolds regardless of curvature or topology and is accompanied by an empirical convergence test that examines whether the combined approximation of function and geometry still yields the correct PDE solution.

What carries the argument

Projection of differential operators from R^3 onto the surface using a neural approximation of the normals.

If this is right

  • Surface PDEs can be solved without generating or refining a mesh.
  • The same framework applies to surfaces with arbitrary topology.
  • Both the unknown function and the surface geometry are approximated together inside one training loop.
  • An empirical test now exists to monitor convergence for surface PINNs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The projection idea could be reused for other surface operators beyond the Laplacian or gradient.
  • If the normal field is learned jointly with the PDE solution, the method might handle cases where the surface itself is only approximately known.
  • Time-dependent surface PDEs would require only an additional time coordinate in the same projection setup.

Load-bearing premise

The neural field produces normals accurate enough that the projected operators still drive the PINN training to the true surface PDE solution.

What would settle it

On a surface PDE with a known analytic solution, the numerical error fails to decrease as network capacity or training time increases.

Figures

Figures reproduced from arXiv: 2605.27308 by Navami Kairanda, Oded Stein, Peter Yichen Chen, Pranav Jain.

Figure 1
Figure 1. Figure 1: PINNsur solves PDEs on curved surfaces embedded in 3D. We empirically show that the [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence plots for the Poisson and Helmholtz equations with different boundary [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview. To solve PDEs on a curved surface, we first train a normal net￾work to approximate the surface normals. The PDE network then leverages these normals to compute the surface Laplacian ∆Ωu, with boundary conditions added as soft constraints. For all experiments, we choose a starting learning rate of 1e −3 with the ReduceLROnPlateau scheduler. We uniformly sample points on the domain and train until … view at source ↗
Figure 4
Figure 4. Figure 4: Figure showing the solution and error as network width increases for Poisson and Helmholtz equations under various boundary conditions. We observe that error decreases with increasing width. the number of parameters of the Siren network that models the PDE solution uθ. Since Siren is a coordinate-based MLP, #W is determined by the depth and width of the MLP. We exclude the normal network nθˆ from this coun… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison with Williamson and Mitra [2025]. By directly training on surface normals, our method achieves a lower PDE solution error. Moreover, our approach applies to arbitrary topologies, whereas Williamson and Mitra [2025] is limited to surfaces homeomorphic to a sphere. high-frequency functions compared to using sigmoid or tanh activation functions. In [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) Result of the minimal sur￾face computed for the cylinder. (b) Re￾sult of harmonic interpolation in the in￾terior of the plane given Dirichlet bound￾ary values on the windows of the plane. Minimal Surfaces. A surface Ω ⊂ R 3 is minimal if its mean curvature is zero everywhere. The mean curvature re￾lates to the Laplace–Beltrami operator via ∆Ω = −2Hn, where H is the mean curvature and n the surface norm… view at source ↗
Figure 7
Figure 7. Figure 7: Poisson solve on a Euclidean domain. A comparison with FEM, Kairanda et al. [2026], Kharazmi et al. [2021], Yu et al. [2018] for similar degrees of freedom shows that the relative ℓ2 error and convergence for ours match more closely with FEM. Heat Equation. Laplacians can also be used to solve the heat equation. The implicit Euler finite difference scheme in time for the heat equation ∂u ∂t = ∆Ωu, results … view at source ↗
Figure 8
Figure 8. Figure 8: Our method can effectively solve PDEs even on mesh with intricate details. [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: While the FEM error is dependent on the mesh quality – error increases with mesh [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: An ablation study where we compare the Siren network, which uses sine activation function with Tanh and Sigmoid activation functions on a 2D domain. The figure shows that Tanh and Sigmoid fail to solve for a high-frequency function. we show that both choices result in the same error for equal number of tunable weights #W. In all our experiments we choose to increase the width of the network to increase #W… view at source ↗
Figure 11
Figure 11. Figure 11: Result of an ablation study where we compare the final error first by changing the width, [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Figure showing how the error varies when the tunable parameters of the normal network [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of our method with [Sugimoto et al., 2024] which is a Monte-Carlo based [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Figure showing the limitations of our work. Enforcing Neumann boundary is difficult at [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Figure showing the training loss for different [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
read the original abstract

Partial differential equations (PDEs) on surfaces are fundamental to scientific computing and geometry processing. A popular approach to solving PDEs on surfaces is the finite element method (FEM), where the surface is divided into discrete geometric elements (usually triangles). Recently, physics-informed neural networks (PINNs) have emerged as a continuous, mesh-free alternative that does not suffer from FEM's sensitivity to mesh quality or geometric discretization errors. We present PINNSur, a simple framework for using PINNs on curved surfaces: we train a neural field to approximate the surface's normals, and then we express surface differential operators using their projection from $\mathbb{R}^3$ onto the surface. Since every orientable manifold has well-defined normals, our method is suitable for all such surfaces, regardless of curvature or topology, enabling many geometry processing applications. Moreover, despite their empirical success in solving PDEs in flat Euclidean domains, PINNs lack convergence guarantees to the true solution of the underlying PDE, and there is limited systematic experimental evidence demonstrating such convergence. This gap restricts their adoption as reliable solvers compared to established methods like FEM, where convergence to the true solution is well understood and theoretically grounded. These surface PDEs are particularly challenging to solve convergently, as one must not only deal with the convergence of the function approximation, but also with the convergence of the geometric approximation of the surface itself. In this work, we empirically investigate the convergence behavior of PINNs for solving surface PDEs by introducing a simple empirical convergence test.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces PINNSur, a framework for solving PDEs on curved surfaces with PINNs: a neural field is trained to approximate surface normals, after which surface differential operators (gradient, divergence) are obtained by projecting Euclidean derivatives via P = I - n n^T. The approach is claimed to apply to any orientable manifold independent of curvature or topology. The paper additionally presents a simple empirical convergence test for PINNs on surface PDEs, motivated by the known absence of convergence guarantees for PINNs even in Euclidean domains and the extra difficulty of simultaneous geometric approximation.

Significance. If the projection-based operator construction and the empirical convergence test can be shown to produce reliable solutions, the method would supply a mesh-free alternative to FEM for surface PDEs that avoids meshing artifacts and extends immediately to arbitrary topologies. The explicit focus on convergence behavior is a positive step beyond typical PINN papers, but the absence of error propagation analysis between normal approximation and operator accuracy limits the strength of the claims.

major comments (2)
  1. [Abstract / method description] Abstract and method description: the central construction defines surface operators via the projected Euclidean derivatives of the solution network using the learned normal field N(x). No a-priori estimate or bound is supplied relating the normal approximation error ||N - n_true|| to the resulting operator error ||P_N abla u_N - P_true abla u_true||, nor is it shown that this composite error remains controlled relative to the PDE residual during joint (or sequential) optimization. This directly affects whether the empirical convergence test can establish convergence to the true surface PDE solution.
  2. [Empirical convergence test] Empirical convergence test section: the test is performed on the composite system (learned normals + learned solution) without isolating the contribution of normal approximation error. Consequently the reported convergence behavior does not separate geometry-induced operator error from function-approximation error, weakening the claim that the method addresses the additional convergence challenge of surface PDEs.
minor comments (1)
  1. [Abstract] The abstract states that the method is 'suitable for all such surfaces, regardless of curvature or topology' but does not discuss how the neural normal field is initialized or regularized on surfaces with high curvature or genus changes; a brief clarification would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. Below we address each major point.

read point-by-point responses

  1. Referee: [Abstract / method description] Abstract and method description: the central construction defines surface operators via the projected Euclidean derivatives of the solution network using the learned normal field N(x). No a-priori estimate or bound is supplied relating the normal approximation error ||N - n_true|| to the resulting operator error ||P_N abla u_N - P_true abla u_true||, nor is it shown that this composite error remains controlled relative to the PDE residual during joint (or sequential) optimization. This directly affects whether the empirical convergence test can establish convergence to the true surface PDE solution.

    Authors: We agree that the manuscript supplies no theoretical a-priori bound relating normal-field error to projected-operator error. The work is framed as an empirical investigation; we will revise the abstract and method description to state explicitly that no such guarantee is claimed and that the reported test evaluates the composite system in practice only. revision: yes

  2. Referee: [Empirical convergence test] Empirical convergence test section: the test is performed on the composite system (learned normals + learned solution) without isolating the contribution of normal approximation error. Consequently the reported convergence behavior does not separate geometry-induced operator error from function-approximation error, weakening the claim that the method addresses the additional convergence challenge of surface PDEs.

    Authors: The test evaluates the end-to-end pipeline that would be deployed. To address the isolation concern we will add an ablation that compares results obtained with the learned normal field against results obtained with ground-truth normals (where available) so that geometry-induced and function-approximation contributions can be distinguished. revision: yes

Circularity Check

0 steps flagged

No circularity; direct construction with empirical test only.

full rationale

The paper defines PINNSur by training a neural field on normals then projecting Euclidean operators; this is an explicit ansatz presented as the method itself, not derived from prior results or fits. The convergence investigation is introduced as a new empirical test without any equations, fitted parameters, or self-citations that reduce a claimed prediction back to the input. No load-bearing steps equate outputs to inputs by construction, and the text supplies no uniqueness theorems or renamings. The derivation chain is therefore self-contained as a proposed framework plus separate empirical check.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are described beyond the general claim that normals exist on orientable manifolds and that projection works.

pith-pipeline@v0.9.1-grok · 5819 in / 1090 out tokens · 33487 ms · 2026-06-29T14:27:45.381855+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 10 canonical work pages · 2 internal anchors

  1. [1]

    URLhttps://arxiv.org/abs/1804.10904. I. Baran and J. Popovi´c. Automatic rigging and animation of 3d characters.ACM Trans. Graph., 26 (3):72–es,

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    doi: 10.1145/3131280

    ISSN 0001-0782. doi: 10.1145/3131280. URL https://doi. org/10.1145/3131280. M. Desbrun, M. Meyer, P. Schröder, and A. H. Barr. Implicit fairing of irregular meshes using diffusion and curvature flow. InProceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, page 317–324,

    work page doi:10.1145/3131280

  3. [3]

    Edavamadathil Sivaram, T.-M

    10 V . Edavamadathil Sivaram, T.-M. Li, and R. Ramamoorthi. Neural geometry fields for meshes. In ACM SIGGRAPH 2024 Conference Papers,

    2024

  4. [4]

    , TITLE =

    ISBN 9781470411442. doi: 10.1090/gsm/019. URLhttp://dx.doi.org/10.1090/gsm/019. G. Fargion and O. Weber. Learning Metric Fields for Fast Low-Distortion Mesh Parameterizations. Computer Graphics Forum,

    work page doi:10.1090/gsm/019

  5. [5]

    Ferguson, P

    Z. Ferguson, P. Jain, D. Zorin, T. Schneider, and D. Panozzo. High-order incremental potential contact for elastodynamic simulation on curved meshes. InACM SIGGRAPH 2023 Conference Proceedings,

    2023

  6. [6]

    Asset li- censed under CC BY-NC

    https://sketchfab.com/3d- models/wooden-head-from-above-a-fireplace-2-aacc5088078743b39844b9331bb41eab. Asset li- censed under CC BY-NC. Z. Hao, S. Liu, Y . Zhang, C. Ying, Y . Feng, H. Su, and J. Zhu. Physics-informed machine learning: A survey on problems, methods and applications.arXiv preprint arXiv:2211.08064,

    work page arXiv

  7. [7]

    URL https://hal.science/ hal-01297648. P. Jain, Z. Qu, P. Y . Chen, and O. Stein. Neural monte carlo fluid simulation. InACM SIGGRAPH 2024 Conference Papers,

    2024

  8. [8]

    doi: 10.1145/2487228.2487237

    ISSN 1557-7368. doi: 10.1145/2487228.2487237. URL http://dx.doi.org/10.1145/2487228.2487237. E. Kharazmi, Z. Zhang, and G. E. Karniadakis. hp-vpinns: Variational physics-informed neural networks with domain decomposition.Computer Methods in Applied Mechanics and Engineering, 374:113547, Feb

    work page doi:10.1145/2487228.2487237

  9. [9]

    Karniadakis

    ISSN 0045-7825. doi: 10.1016/j.cma.2020.113547. URL http://dx. doi.org/10.1016/j.cma.2020.113547. J. Lai, S. Wang, C. Wang, and Y . Lu. Unveiling the scaling law of PINNs under non-euclidean geometry,

    work page doi:10.1016/j.cma.2020.113547 2020

  10. [10]

    URL http://dx.doi.org/10.21136/AM.2018.0095-18

    doi: 10.21136/am.2018.0095-18. URL http://dx.doi.org/10.21136/AM.2018.0095-18. 11 Z. Lu, H. Pu, F. Wang, Z. Hu, and L. Wang. The expressive power of neural networks: a view from the width. InProceedings of the 31st International Conference on Neural Information Processing Systems, NIPS’17, page 6232–6240,

    work page doi:10.21136/am.2018.0095-18 2018

  11. [11]

    Loper, N

    Downloaded modified version from odedstein- meshes github.com/odedstein/meshes/tree/master/objects/hand, adapted from M. Loper, N. Mahmood, J. Romero, G. Pons-Moll, M. Black "SMPL" (SIGGRAPH Asia 2015). Asset licensed under CC BY 4.0. I. Mehta, M. Chandraker, and R. Ramamoorthi. A level set theory for neural implicit evolution under explicit flows. InEuro...

    2015

  12. [12]

    Y . Shin, J. Darbon, and G. E. Karniadakis. On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type pdes.arXiv preprint arXiv:2004.01806,

    work page arXiv 2004

  13. [13]

    Sorkine, D

    O. Sorkine, D. Cohen-Or, Y . Lipman, M. Alexa, C. Rössl, and H.-P. Seidel. Laplacian surface editing. InProceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, page 175–184,

    2004

  14. [14]

    com/odedstein/meshes/tree/master/objects/hammer, originally from sketchfab.com/ 3d-models/hammer-rawscan-01e5b5acad9843818529d15663df5b9e

    Downloaded modified version from odedstein-meshes github. com/odedstein/meshes/tree/master/objects/hammer, originally from sketchfab.com/ 3d-models/hammer-rawscan-01e5b5acad9843818529d15663df5b9e . Asset licensed under CC BY 4.0. R. Sugimoto, N. King, T. Hachisuka, and C. Batty. Projected walk on spheres: A monte carlo closest point method for surface pde...

    2024

  15. [15]

    Tang and Z

    Z. Tang and Z. Fu. Physics-informed neural networks for elliptic partial differential equations on 3d manifolds.arXiv preprint arXiv:2103.02811,

    work page arXiv

  16. [16]

    Learning to Solve PDEs on Neural Shape Representations

    L. Welschinger, Y . Liu, Z. Wang, and N. Mitra. Learning to solve pdes on neural shape representations. arXiv preprint arXiv:2512.21311,

    work page internal anchor Pith review Pith/arXiv arXiv

  17. [17]

    good” mesh FEM error on “bad

    13 Physics-Informed Neural Networks for PDEs on Curved Surfaces —Appendices— A Further Experiments 14 A.1 Intricate Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 A.2 FEM Solution Dependence on Mesh Quality . . . . . . . . . . . . . . . . . . . . . 14 A.3 Further Ablations . . . . . . . . . . . . . . . . . . . . . . . . ...

    2024