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arxiv: 2606.27177 · v1 · pith:LHKCTHJTnew · submitted 2026-06-25 · 💻 cs.CE

A hybrid IFENN solver for generalizable modeling of phase-field fracture initiation and propagation

Panos Pantidis , Fouad Amin , Diab Abueidda , Mostafa E. Mobasher This is my paper

Pith reviewed 2026-06-26 01:47 UTC · model grok-4.3

classification 💻 cs.CE
keywords phase-field fracturehybrid solverfinite element neural networkcrack initiationcrack propagationphysics-informed traininggeneralizable geometries
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The pith

The IFENN hybrid couples a FEM solver to neural networks that approximate the phase-field equation, enabling fracture predictions on unseen geometries after one training run on a benchmark shape.

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

The paper establishes that a hybrid solver can capture the complete evolution of phase-field fracture, from the start of a crack through its growth, on shapes the model has never encountered before. It achieves this by training two networks once on a single benchmark geometry with a physics-informed loss that uses only the maximum strain energy and the phase-field variable at a small number of strategically chosen points. One network handles crack initiation while the other manages propagation, and artificial boundary conditions keep far-field values near zero during inference. A sympathetic reader would care because the method promises to cut the cost of repeated full simulations when the same trained networks are reused across different component shapes.

Core claim

The IFENN framework tightly couples a standard finite element solver for mechanical equilibrium with a pre-trained neural network that approximates the phase-field diffusion equation; a DeepOKAN network is used for the initiation stage and a CNN for the propagation stage, both trained physics-informed on one benchmark geometry with few increments and limited Gauss points sampled from the fracture process zone, then applied to both the training geometry and arbitrary unseen geometries.

What carries the argument

The Integrated Finite Element Neural Network (IFENN) hybrid scheme, which pairs a conventional FEM solver for equilibrium with a neural network surrogate for the phase-field equation.

If this is right

  • The method models both crack initiation and subsequent propagation within a single framework.
  • Training cost drops sharply because only a small number of increments and Gauss points from the fracture zone are required.
  • The same trained networks produce usable results on both the original benchmark and previously unseen geometries.
  • Artificial boundary conditions allow the networks to extrapolate near-zero phase-field values far from the crack tip.

Where Pith is reading between the lines

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

  • Engineering workflows that repeatedly analyze fracture in families of similar parts could replace many full simulations with a single training step plus fast inference.
  • The same hybrid coupling pattern could be tested on other coupled-field problems such as diffusion-reaction systems or thermo-mechanical problems.
  • Systematic checks on geometries whose topology differs markedly from the training shape would reveal the practical limits of the claimed generalizability.

Load-bearing premise

A neural network trained exclusively on one benchmark geometry with artificial boundary conditions will produce accurate phase-field predictions on arbitrary unseen geometries without retraining or accumulating large errors.

What would settle it

A side-by-side comparison on a new geometry with a different topology, measuring whether the hybrid solver's predicted phase-field values and crack paths deviate substantially from a full finite-element reference solution.

Figures

Figures reproduced from arXiv: 2606.27177 by Diab Abueidda, Fouad Amin, Mostafa E. Mobasher, Panos Pantidis.

Figure 1
Figure 1. Figure 1: Schematic of a cracked domain with a sample phase-field damage contour [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Architecture of the adopted DeepOKAN, with Radial Basis Functions (RBFs) as the activation functions for all the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: a. First sampling method (SM1), where colored Gauss points indicate sensor locations across zones A–D. b. Second sampling method (SM2), featuring uniformly and randomly selected Gauss points. c. Representative profiles of H and ϕ, illustrating the information contained within each zone. 8 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematic representation of the weight-function adopted for filtering the far-field [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: d maps the selected training sensors against their aligned testing counterparts. The spatial proximity between each matched pair is evident. As it will be shown in the numerical examples, this approach enables the DeepOKAN to readily predict the localized phase-field initiation on completely unseen geometries, without requiring re-training or architectural modifications [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of the physics-informed CNN setup and training procedure. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Schematic representations of a. train geometry (Model A) and b. test geometries (Model B). Once the FEM analysis is complete, we proceed with the one-time training of the networks. For the initiation stage we develop two variants of the DeepOKAN, one that is trained on TS1 and one on TS2. These are henceforth termed as DK-TS1 and DK-TS2. Both networks follow the general architecture of [PITH_FULL_IMAGE:fi… view at source ↗
Figure 8
Figure 8. Figure 8: Training performance of DK-TS1. a. True ϕ values. b. Predicted ϕ values. c. Absolute error ϕ values. d. Evolution of training loss. e. Metrics of ϕ, including its L2 norm and maximum value. The results of the DK-TS1 and DK-TS2 training are depicted in Figs. 8 and 9 respectively. In both figures, the top row displays the true ϕ profiles, the middle row presents the corresponding network predictions, and the… view at source ↗
Figure 9
Figure 9. Figure 9: Training performance of DK-TS2. a. True ϕ values. b. Predicted ϕ values. c. Absolute error ϕ values. d. Evolution of training loss. e. Metrics of ϕ, including its L2 norm and maximum value. Next, we develop the CNN model to capture the propagation stage of phase-field. The network consists of four consecutive convolution layers, with subsequent activation functions as presented in Section 3.2. The H profil… view at source ↗
Figure 10
Figure 10. Figure 10: Training performance of the developed CNN. The true and predicted [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Numerical results: Case 1. a. Reaction force-displacement curves. b. Evolution of phase-field second norm. c. 3D views of phase-field contours at various initiation increments from FEM (top) and IFENN (bottom). Next, we demonstrate the capability of the hybrid analysis to operate within a multi-pass staggered scheme, again on Model A. The numerical tolerance is set to tol = 10−3 , defining the threshold b… view at source ↗
Figure 12
Figure 12. Figure 12: Numerical results: Case 2. a. Reaction force-displacement curves. b. Evolution of phase-field second norm. c. 3D views of phase-field contours at various initiation increments from FEM (top) and IFENN (bottom). Having demonstrated the performance of IFENN under the weight-function approach, we now evaluate its implementation utilizing a DeepOKAN that is trained via the artificial boundary condition method… view at source ↗
Figure 13
Figure 13. Figure 13: Numerical results: Case 3. a. Reaction force-displacement curves. b. Evolution of phase-field second norm. c. 3D views of phase-field contours at various initiation increments from FEM (top) and IFENN (bottom). So far, all IFENN analyses have been conducted on Model A, which is the same geometry that was used for the training of both networks. In this final example, we extend the applicability of these pr… view at source ↗
Figure 14
Figure 14. Figure 14: Numerical results: Case 4. a. Reaction force-displacement curves. b. Evolution of phase-field second norm. c. 3D views of phase-field contours at various initiation increments from FEM (top) and IFENN (bottom). 6. Summary, conclusions and outlook In this work, we present for the first time a complete IFENN framework for capturing the entire evolution of phase-field fracture in quasi-brittle materials. Bui… view at source ↗
read the original abstract

In this paper we demonstrate how the Integrated Finite Element Neural Network (IFENN) framework can effectively model the entire evolution of phase-field fracture, including the initiation and propagation stage, across generalizable geometries. IFENN is a hybrid scheme for coupled computational mechanics problems, tightly coupling a standard FEM solver (mechanical equilibrium) with a pre-trained neural network (coupled field). In this work, the phase-field diffusion equation is approximated with: i) a DeepONet architecture with Kolmogorov-Arnold networks in the trunk and branch (DeepOKAN) for the initiation stage, and ii) a Convolution Neural Network (CNN) for the propagation stage. Both networks are trained only once, on a benchmark geometry, using a purely physics-informed approach based on the maximum strain energy and the phase-field variable. The training process utilizes an extremely small number of training increments and only a limited number of Gauss points that are strategically sampled from the fracture process zone. These features enable a substantial decrease of the offline training cost. To address the extrapolation of the DeepOKAN predictions in regions away from the crack tip during the inference stage, we implement a set of artificial boundary conditions to enforce the near-zero values in the far-field predictions. We showcase the flexibility and numerical accuracy of the proposed methodology across both the training and unseen geometries.

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

3 major / 2 minor

Summary. The manuscript introduces a hybrid IFENN framework that couples a standard FEM solver for mechanical equilibrium with pre-trained neural networks to solve the phase-field diffusion equation for fracture. DeepOKAN (Kolmogorov-Arnold networks in trunk and branch) handles the initiation stage and a CNN handles propagation; both are trained once on a single benchmark geometry via a physics-informed loss based on maximum strain energy evaluated at a small number of strategically sampled Gauss points inside the process zone. Artificial far-field boundary conditions are added during inference to enforce near-zero predictions away from the crack tip. The central claim is that this yields accurate modeling of the full fracture evolution (initiation and propagation) on both the training geometry and arbitrary unseen geometries.

Significance. If the generalizability claim is substantiated with quantitative validation, the work would demonstrate a practical route to low-cost, one-time training of hybrid physics-informed networks for phase-field fracture on arbitrary domains. The emphasis on extremely limited training increments and Gauss points, together with the explicit handling of extrapolation via artificial BCs, addresses a known cost bottleneck in data-driven fracture modeling and could be extended to other coupled mechanics problems.

major comments (3)
  1. [Abstract] Abstract and numerical-results section: the central claim of 'numerical accuracy' and 'flexibility' across unseen geometries is unsupported by any reported quantitative metrics (L2 errors on the phase field, crack-path deviation, global energy balance, or convergence under mesh refinement). Without these, the generalizability assertion cannot be evaluated.
  2. [Method] Method and inference description: the artificial boundary conditions introduced to correct DeepOKAN extrapolation away from the crack tip are presented as a fix, yet no analysis quantifies their effect on local accuracy near the process zone or on geometries whose far-field conditions differ from the artificial prescription.
  3. [Numerical examples] Training protocol: the networks are trained exclusively on one benchmark geometry with a fixed, small set of Gauss points; the manuscript provides no description of the number, geometric diversity, or loading conditions of the 'unseen' test cases used to demonstrate generalizability, which is load-bearing for the primary claim.
minor comments (2)
  1. [Introduction] Notation for the phase-field variable and the maximum-strain-energy loss should be defined explicitly at first use rather than assumed from prior literature.
  2. [Figures] Figure captions should state the number of Gauss points used and the precise locations of the artificial boundary conditions so that the limited-data regime is reproducible.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below with explanations and indicate the revisions we will make to strengthen the quantitative support for our claims and the clarity of the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract and numerical-results section: the central claim of 'numerical accuracy' and 'flexibility' across unseen geometries is unsupported by any reported quantitative metrics (L2 errors on the phase field, crack-path deviation, global energy balance, or convergence under mesh refinement). Without these, the generalizability assertion cannot be evaluated.

    Authors: We acknowledge that the current manuscript primarily demonstrates accuracy and flexibility through visual comparisons of phase-field contours and crack paths against reference FEM solutions. While these comparisons show close qualitative agreement on both the training geometry and the unseen cases, we agree that explicit quantitative metrics are needed to rigorously support the central claims. In the revised manuscript we will add L2 errors on the phase-field variable, crack-path deviation metrics, global energy balance checks, and a brief mesh-convergence study for the key examples. revision: yes

  2. Referee: [Method] Method and inference description: the artificial boundary conditions introduced to correct DeepOKAN extrapolation away from the crack tip are presented as a fix, yet no analysis quantifies their effect on local accuracy near the process zone or on geometries whose far-field conditions differ from the artificial prescription.

    Authors: The artificial far-field boundary conditions are introduced to enforce the expected near-zero phase-field values outside the process zone during inference. We recognize that the manuscript does not include a dedicated quantification of their influence. We will add a short analysis in the revised method section that compares DeepOKAN predictions with and without these boundary conditions on the benchmark geometry (reporting local errors inside the process zone) and will verify robustness on one additional unseen geometry whose far-field setup differs from the artificial prescription. revision: yes

  3. Referee: [Numerical examples] Training protocol: the networks are trained exclusively on one benchmark geometry with a fixed, small set of Gauss points; the manuscript provides no description of the number, geometric diversity, or loading conditions of the 'unseen' test cases used to demonstrate generalizability, which is load-bearing for the primary claim.

    Authors: The unseen geometries are shown in the numerical examples, yet we agree that a concise, systematic description of their number, geometric variations, and loading conditions is currently insufficient. In the revised manuscript we will expand Section 4 with a table listing each unseen case, its geometric parameters (e.g., notch location, specimen aspect ratio), and the applied boundary/loading conditions, thereby making the generalizability evaluation fully reproducible. revision: yes

Circularity Check

0 steps flagged

No circularity; physics-informed training derives from governing PDEs independent of model outputs

full rationale

The paper trains DeepOKAN and CNN components via physics-informed losses on the phase-field diffusion equation residuals using maximum strain energy, applied to one benchmark geometry with limited Gauss points. This setup follows standard PINN methodology and does not reduce any prediction to a fitted parameter or self-referential definition. Artificial boundary conditions are introduced explicitly as an engineering patch for extrapolation, not as a definitional closure. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work are described in the provided text. The generalization claim to unseen geometries is presented as an empirical outcome rather than a tautological reduction of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard phase-field fracture formulation and finite-element discretization; the neural-network approximations are the primary addition and are trained against the physics residual.

axioms (2)
  • domain assumption The phase-field model with maximum strain energy criterion accurately captures fracture initiation and propagation
    Invoked as the training target for both networks.
  • domain assumption Neural networks can approximate solutions to the phase-field diffusion equation to sufficient accuracy for engineering use
    Basis for replacing the FEM solve of the diffusion equation with DeepOKAN and CNN.

pith-pipeline@v0.9.1-grok · 5778 in / 1244 out tokens · 32954 ms · 2026-06-26T01:47:20.913162+00:00 · methodology

discussion (0)

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