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arxiv: 2605.02310 · v1 · submitted 2026-05-04 · 💻 cs.CE

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A Variational Kolosov--Muskhelishvili Network for Elasticity and Fracture

Christian H\"affner, Niklas Fehlemann, Sebastian M\"unstermann, Shuwei Zhou, Sophie Stebner, Zhichao Wei

Authors on Pith no claims yet

Pith reviewed 2026-05-08 02:20 UTC · model grok-4.3

classification 💻 cs.CE
keywords physics-informed neural networkslinear elasticityfracture mechanicsKolosov-Muskhelishvili potentialsvariational methodsstress intensity factorsmesh-free methodscomputational mechanics
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The pith

Representing elastic fields with two holomorphic potentials and training them solely on total potential energy solves 2D elasticity and fracture problems accurately with a single loss term.

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

The paper introduces a neural network that encodes the solution to two-dimensional linear elasticity using two holomorphic Kolosov-Muskhelishvili potentials rather than learning the displacement field directly. Training occurs through a single energy-based loss derived from the principle of minimum total potential energy, which enforces equilibrium and boundary conditions without separate residual terms. For cracked geometries the method adds a discontinuous representation of the stress potential that automatically satisfies traction-free crack faces and incorporates the crack-tip singularity into the network ansatz. Validation on benchmark problems shows accurate recovery of stress, displacement, and stress intensity factors. A reader would care because the approach simplifies loss construction, improves accuracy, and speeds convergence relative to networks that treat elasticity as a generic PDE residual problem.

Core claim

The central claim is that the variational Kolosov-Muskhelishvili informed neural network, whose solution is expressed by two holomorphic potentials and whose training minimizes a single total-potential-energy integral, accurately reproduces both smooth elastic fields and fields containing crack discontinuities, including reliable extraction of stress intensity factors, while requiring fewer loss terms than residual-based formulations.

What carries the argument

The representation of displacement and stress by two holomorphic Kolosov-Muskhelishvili potentials, optimized via a single energy integral from the minimum total potential energy principle; for cracks, a discontinuous stress-potential ansatz that embeds traction-free faces and tip singularities directly into the network.

If this is right

  • The network recovers stress and displacement fields to high accuracy on both cracked and uncracked benchmark problems.
  • Stress intensity factors are obtained directly as part of the solution without post-processing or special meshing.
  • Loss construction is reduced to a single energy term that automatically satisfies all boundary conditions.
  • Training converges faster than displacement-based neural networks on the tested cases.
  • The same framework applies uniformly to problems with and without cracks.

Where Pith is reading between the lines

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

  • The discontinuous-potential idea could be adapted to model other sharp interfaces such as material boundaries in composites.
  • If the energy-only loss remains stable under mesh-free sampling, the method may support real-time or parametric studies of crack paths without repeated remeshing.
  • Because the potentials are analytic, the framework might extend naturally to related plane problems in electrostatics or incompressible flow.
  • The approach suggests a route to hybrid solvers that combine neural networks with classical complex-variable methods for selected subdomains.

Load-bearing premise

The holomorphic Kolosov-Muskhelishvili potentials, even when made discontinuous across cracks, remain expressive enough to represent arbitrary two-dimensional elastic fields while the single energy integral alone enforces all boundary and interface conditions.

What would settle it

On the standard infinite plate with central crack under remote tension, if the network's predicted stress intensity factor deviates by more than a few percent from the known analytical value or if the computed displacement field shows large errors near the crack tip when compared with an independent finite-element solution.

Figures

Figures reproduced from arXiv: 2605.02310 by Christian H\"affner, Niklas Fehlemann, Sebastian M\"unstermann, Shuwei Zhou, Sophie Stebner, Zhichao Wei.

Figure 1
Figure 1. Figure 1: Overall architecture of the proposed vKMINN framework for two-dimensional linear elasticity and fracture problems. view at source ↗
Figure 2
Figure 2. Figure 2: Geometries, loading conditions, and boundary conditions of the benchmark problems without cracks, (a) circular tube under pressure view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of von Mises stress and displacement fields, and their absolute errors between vKMINN and PINN for the circular tube view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the von Mises stress and displacement fields, and their absolute errors between vKMINN and PINN for the square plate view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the stress and displacement components along the selected path for the square plate with a hole under tension: (a) selected view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of σyy and uy fields and the corresponding absolute errors for vKMINN, PIHNN, and PINN in the plate under non-uniform tension loading: (a) stress component σyy and its absolute error, and (b) displacement component uy and its absolute error. 17 view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the predicted stress and displacement components along the selected vertical path for the plate under non-uniform tension view at source ↗
Figure 8
Figure 8. Figure 8: Analysis of the SENT case: (a) geometry and loading conditions of the SENT specimen, (b) relative error of the crack opening displace view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of FEM, vKMINN, and KMINN results for the SENT case: (a) stress component view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the mode I stress intensity factor view at source ↗
Figure 12
Figure 12. Figure 12: The stress fields predicted by vKMINN and KMINN agree well with the FEM results, while the displacement view at source ↗
Figure 11
Figure 11. Figure 11: Geometry, loading conditions, domain decomposition, and training convergence for the OCCT case: (a) geometry and non-uniform view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the stress and displacement fields among FEM, vKMINN, and KMINN for the OCCT case: stress components view at source ↗
read the original abstract

Physics-informed neural networks provide a mesh-free framework for solving partial differential equation-governed problems in solid mechanics. However, most existing formulations in linear elasticity still learn the displacement field directly, which does not explicitly exploit the analytic structure of two-dimensional elasticity and becomes restrictive for fracture problems with crack face discontinuities and crack tip singularities. Moreover, existing Kolosov--Muskhelishvili informed neural network formulations still rely on residual-based loss functions with multiple boundary and interface terms, whereas a variational concept has not yet been established. To address these issues, a variational Kolosov--Muskhelishvili informed neural network framework for two-dimensional linear elastic problems with and without cracks is proposed in this work. The solution is represented by two holomorphic Kolosov--Muskhelishvili potentials and trained through an energy-based loss function derived from the principle of minimum total potential energy. For crack problems, a discontinuous stress potential representation is further introduced to embed the crack face condition and crack tip singularity directly into the solution ansatz. The proposed framework is validated on a series of benchmark problems with or without crack problems. The results show that variational Kolosov--Muskhelishvili informed neural network can accurately predict stress and displacement field as well as stress intensity factors. Compared with traditional neural network models, it achieves higher accuracy, simpler loss construction, and faster convergence in the considered cases. Overall, the proposed variational Kolosov--Muskhelishvili informed neural network provides an effective and physically consistent variational framework for two-dimensional linear elastic fracture analysis.

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 proposes a variational Kolosov-Muskhelishvili informed neural network (vKMNN) for two-dimensional linear elasticity and fracture problems. The solution is represented by two holomorphic potentials, trained by minimizing the total potential energy derived from the principle of minimum potential energy. For problems with cracks, a discontinuous stress potential representation is introduced to embed the traction-free crack face conditions and the 1/r^{1/2} singularity. The framework is validated on benchmark problems, with claims of accurate prediction of displacement and stress fields as well as stress intensity factors, and advantages over traditional neural networks in accuracy, loss simplicity, and convergence speed.

Significance. If the central claims hold, the work offers a mesh-free variational approach that exploits the analytic structure of 2D elasticity via holomorphic potentials and an energy-based loss, potentially simplifying PINN formulations for fracture by reducing explicit boundary residual terms. Embedding discontinuities and singularities in the ansatz while relying on the variational principle could yield more physically consistent predictions of fields and SIFs, with benefits for problems where mesh-based methods require heavy refinement.

major comments (2)
  1. In the section introducing the discontinuous stress potential representation for crack problems, the central claim is that the single total-potential-energy loss automatically enforces traction-free crack faces and far-field conditions once the ansatz is adopted. However, no explicit verification is provided (such as post-training residual evaluation on crack faces or a demonstration that the NN parameterization is dense enough in the admissible space) to show that deviations from the exact holomorphic functions are penalized by the energy integral alone; any mismatch in the branch-cut behavior would remain invisible to the loss and undermine the assertion that all interface conditions are satisfied without additional terms.
  2. In the validation and results section, the claims of higher accuracy, simpler loss construction, and faster convergence compared to traditional neural network models are load-bearing for the contribution. The manuscript must include quantitative error tables (e.g., L2 norms for displacements and stresses), convergence plots versus iterations or epochs, and direct baseline comparisons with specific metrics on the benchmark problems to substantiate these performance advantages; the absence of such data in the reported summary leaves the superiority assertions unverified.
minor comments (1)
  1. The abstract contains the redundant phrase 'benchmark problems with or without crack problems'; rephrase for conciseness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive review. The comments identify key areas where additional verification and quantitative evidence would strengthen the manuscript. We address each major comment below and will revise the manuscript accordingly to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: In the section introducing the discontinuous stress potential representation for crack problems, the central claim is that the single total-potential-energy loss automatically enforces traction-free crack faces and far-field conditions once the ansatz is adopted. However, no explicit verification is provided (such as post-training residual evaluation on crack faces or a demonstration that the NN parameterization is dense enough in the admissible space) to show that deviations from the exact holomorphic functions are penalized by the energy integral alone; any mismatch in the branch-cut behavior would remain invisible to the loss and undermine the assertion that all interface conditions are satisfied without additional terms.

    Authors: We appreciate the referee's emphasis on explicit verification. The variational loss is derived from the principle of minimum potential energy, which enforces traction-free conditions as natural boundary conditions when the ansatz is kinematically admissible. Nevertheless, to directly address the concern, we will add post-training residual evaluations of tractions on the crack faces and far-field boundaries in the revised manuscript. We will also include a brief discussion on the approximation properties of the neural network parameterization for holomorphic functions to confirm that deviations are penalized by the energy integral. These additions will substantiate that the embedded ansatz and variational loss together satisfy the interface conditions without supplementary residual terms. revision: yes

  2. Referee: In the validation and results section, the claims of higher accuracy, simpler loss construction, and faster convergence compared to traditional neural network models are load-bearing for the contribution. The manuscript must include quantitative error tables (e.g., L2 norms for displacements and stresses), convergence plots versus iterations or epochs, and direct baseline comparisons with specific metrics on the benchmark problems to substantiate these performance advantages; the absence of such data in the reported summary leaves the superiority assertions unverified.

    Authors: We agree that quantitative substantiation is essential for the performance claims. While the manuscript presents results on benchmark problems demonstrating accurate field predictions and SIFs, we acknowledge that more detailed tabular and graphical comparisons would better support the assertions of higher accuracy, simpler loss, and faster convergence. In the revision, we will add L2-norm error tables for displacements and stresses, convergence plots of loss and error versus epochs, and direct numerical comparisons against standard PINN baselines using specific metrics (e.g., relative L2 errors and iteration counts) for all considered cases. These enhancements will provide the required evidence without altering the core methodology. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard variational principle and classical elasticity theory

full rationale

The loss function is constructed directly from the principle of minimum total potential energy, a foundational and independent physical law in solid mechanics that does not depend on the neural network outputs or fitted parameters. The Kolosov-Muskhelishvili potentials are drawn from established analytic theory of 2D elasticity, and the discontinuous ansatz for cracks is introduced as a representation choice to embed known boundary behavior rather than being fitted or self-referential. Validation occurs via comparison to benchmark solutions on independent test cases, providing external falsifiability. No load-bearing step reduces by construction to a self-citation, a fitted input renamed as prediction, or an ansatz smuggled through prior work by the same authors. This is the common case of a self-contained numerical method.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review performed on abstract only; full text unavailable so ledger entries are inferred from stated claims.

axioms (1)
  • standard math Principle of minimum total potential energy
    Invoked to derive the single energy-based loss function that replaces multiple residual terms.
invented entities (1)
  • Discontinuous stress potential representation no independent evidence
    purpose: To embed crack-face traction-free condition and crack-tip singularity directly into the network ansatz
    Introduced specifically for fracture problems so that discontinuities do not have to be learned from data.

pith-pipeline@v0.9.0 · 5608 in / 1318 out tokens · 38519 ms · 2026-05-08T02:20:56.509453+00:00 · methodology

discussion (0)

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