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REVIEW 5 major objections 5 minor 57 references

Crack Path Prediction with Operator Learning using Discrete Particle System data Generation

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that a Fusion DeepONet trained on particle-simulation data can predict crack paths for unseen notch heights and hole radii, and that it consistently beats the vanilla DeepONet, with the largest advantage when no fracture…

desk verdict Useful CPD-data surrogate study whose main claim—Fusion consistently beats vanilla—is contradicted by the paper's own Table 2; worth a revision, not a desk reject. read the letter →

arxiv 2506.01976 v2 pith:UMQV4UOS submitted 2025-05-15 cs.LG cond-mat.mtrl-scics.AI

Elham Kiyani , Venkatesh Ananchaperumal , Ahmad Peyvan , Mahendaran Uchimali , Gang Li , George Em Karniadakis This is my paper · ORCID
classification cs.LGcond-mat.mtrl-scics.AI MSC 68T0774R10
keywords crackpropagationoperatorlearningDeepONetFusionparticledynamicsfracturemechanicssurrogatemodelingdiscretesystem
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 particular neural operator, the Fusion DeepONet, can learn the mapping from a specimen's geometry—a pre-crack notch height h and a hole radius r—to the full time-resolved displacement field of a fracturing particle system. The training data come from Constitutively Informed Particle Dynamics (CPD), a discrete-particle method that grows cracks by zeroing interaction forces on failed Delaunay triangles. If the claim holds, engineers could replace expensive per-geometry fracture simulations with one trained network that answers new geometries without retraining. Across three case studies, the paper reports that the Fusion architecture predicts particle displacements and crack paths more accurately than the vanilla DeepONet, especially when no fracture occurs, while fracture events still produce error spikes.

What carries the argument

The load-bearing object is the layer-wise fusion mechanism in Fusion DeepONet: $a_T^{(l)} = S_B^{(l)} \odot \sigma(W_T^{(l)} a_T^{(l-1)} + b^{(l)})$, where the cumulative branch features $S_B^{(l)}$ element-wise modulate each trunk hidden layer, so geometric information conditions the spatial-temporal basis at every scale. In the vanilla DeepONet the branch and trunk interact only in a final inner product; in Fusion they interact throughout, which the paper argues lets coarse geometry guide low- and high-frequency components of the displacement field. The operator target is $G: (h,r) \mapsto u(x,t)$, with $u$ defined as $y_i(t)-x_i$ relative to the CPD reference configuration.

What would settle it

Train on Cases 1-3 exactly as reported, then hold out all specimens near the crack-hole interaction threshold, such as r = 1 cm with h around 0.4 cm where CPD predicts crack deflection into the hole, and compare the predicted crack path at tau = 99 against a fresh CPD simulation. If the Fusion network produces a straight, undeflected crack or does not reproduce coalescence, the two-parameter encoding is insufficient.

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Extended reading notes

Core claim

The central claim is that both vanilla and Fusion DeepONets can serve as surrogates for CPD crack-path simulations, and that the Fusion variant does so more faithfully. The networks take geometry parameters (h, r) in the branch and coordinate (x,t) in the trunk, and output particle displacements over 100 deformation steps. The paper reports lower relative L2 errors for Fusion after the onset of fracture, with Case 1 (elastic, no fracture) reaching training MSE near $10^{-5}$ versus $10^{-3}$ for vanilla, and with Cases 2 and 3 showing sharper error growth exactly when elements begin to fail. The intended upshot is that a single trained operator, not a new simulation, can predict crack deflection toward a hole across varying notch heights and hole radii.

Load-bearing premise

The prediction problem is assumed to be fully specified by two scalar geometry parameters, the notch height h and the hole radius r, plus the query point (x,t); if the true displacement field depends on particle configuration details or geometry features the branch never sees, the claimed generalization to new specimens fails.

Editorial extensions

If this is right

  • A single trained Fusion DeepONet can predict 100-step displacement and crack evolution for notch heights and hole radii not used in training, within the studied ranges.
  • The non-fracturing case is nearly solved: elastic displacement prediction stabilizes at low error, so geometry-to-displacement operators are practical for elastic specimens.
  • Fracture onset remains the accuracy bottleneck; errors jump at nucleation and re-nucleation events, so operator surrogates are currently safest before crack initiation or after equilibrium.
  • For fracture cases, the Fusion architecture's advantage over vanilla appears mainly after time step 20, meaning the fusion conditioning specifically helps during the fracture regime, not just elastic pre-loading.
  • The trained operators inherit CPD's ability to model crack deflection, arrest, and hole coalescence without continuum assumptions, because the training data carry those mechanisms implicitly.

Reading between the lines

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

  • Beyond the paper, the same two-scalar parameterization could be extended to crack length, loading rate, or material constants; since the branch currently sees only h and r, adding parameters would test whether the fusion mechanism scales to higher-dimensional geometry spaces.
  • A natural experiment is to feed a damage indicator, such as the cumulative failed-triangle fraction up to time t, as an additional trunk input; this may suppress the error spikes the paper observes at nucleation.
  • Inverse use is an untested corollary: because the operator maps (h,r) to crack path, a trained model could be searched over (h,r) to design specimens that arrest or deflect cracks, turning the surrogate into a design tool.
  • The reported Case 3 average error leaves room to check whether fusion's late-time advantage is consistent across all hold-out radii; a per-sample breakdown would clarify where the architecture still struggles.
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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

5 major / 5 minor

Summary. This paper trains two DeepONet variants (vanilla and Fusion) on displacement fields from Constitutively Informed Particle Dynamics (CPD) simulations of a pre-cracked specimen containing a circular hole, across three geometry families: varying notch height without fracture (Case 1), varying notch height with fracture (Case 2), and varying hole radius with fracture (Case 3). The branch network receives scalar geometric parameters (h and/or r) and the trunk receives (x,t); the output is the particle displacement field u(x,t). The authors claim that Fusion DeepONet consistently outperforms vanilla DeepONet, with lower relative L2 errors and lower training cost, and that the models generalize across the parameterized geometries. The paper presents visual overlays of predicted and true particle positions, training-loss curves, and a table of errors and training times.

Significance. If the central claim were solid, this would be a useful contribution: operator-learning surrogates for discrete-particle fracture simulations are relatively rare, and the paper addresses the challenging regime of crack paths interacting with holes. The work also demonstrates the use of CPD as a data generator for operator learning. However, the reported evidence does not support the headline claim: Table 2 shows that Fusion is worse in Case 3, the two models are compared under different architectures and training schedules, test-set statistics are not reported with error bars, and the low-dimensional branch input raises open questions about the generality of the learned operator. The paper's strengths are the three geometry families, the direct architecture comparison, and the grounding in a physically consistent particle method; these are undermined by the quantitative reporting issues.

major comments (5)
  1. [Table 2, Abstract, Section 5] Table 2 reports average relative L2 errors for Case 3 as 8.46e-04 for vanilla DeepONet and 1.01e-03 for Fusion DeepONet, meaning the Fusion model is less accurate in that case. The abstract and Section 5 nevertheless claim that Fusion consistently outperforms vanilla, and the explanation in Section 5 refers to Figure 14(a) to argue that Fusion is better after tau=20. No per-time-step test-error table or per-sample breakdown is provided, so the reader cannot verify this explanation against the aggregate numbers. Please provide per-time-step test errors with error bars over all test samples and reconcile the aggregate contradiction, or revise the claim to reflect the actual ranking.
  2. [Section 5] Section 5 specifies that vanilla DeepONet uses five hidden layers of 100 neurons with tanh, a fixed learning rate of 1e-4, and 60,000 iterations, while Fusion DeepONet uses three hidden layers of 64 neurons with Rowdy activation, a decaying learning rate starting at 1e-3, and 50,000 iterations. With so many differences, the observed error gap cannot be attributed to the fusion mechanism per se. Please retrain both models under matched conditions (same depth, width, activation, learning-rate schedule, and iteration count), or include ablations that isolate the fusion layers, before drawing conclusions about architectural superiority.
  3. [Section 5 vs. Section 6] Section 5 states that Case 1 uses 35 training and 5 test samples, Case 2 uses 45 and 5, and Case 3 uses 45 and 6, whereas Section 6 states that the training sample counts are 32, 45, and 45 for Cases 1-3. Please correct the inconsistency and specify the exact train/test split used in every reported result.
  4. [Table 2 and Figure 14] The text introducing Table 2 calls the reported quantity training error, while the table caption says relative error; it is therefore unclear whether these numbers are from the training set or a held-out test set. Since the abstract claims generalization, test-set errors are required. Additionally, no error bars, standard deviations, or multiple-seed results are given, and Figure 14(a) appears to show a single error curve per case without stating over which test samples it is averaged. Please report means and standard deviations across multiple seeds and across all test samples, and clarify the train/test status of the error numbers.
  5. [Section 4.1] Section 4.1 and Figure 8 show that the branch network receives only scalar geometric parameters (h and/or r), while the trunk receives (x,t); the model never sees the actual particle configuration or the full geometry. The claim that the method generalizes across complex, geometry-varying crack propagation therefore depends on the unstated assumption that the displacement field is fully determined by (h,r) and (x,t). This assumption is not validated, for example by testing on geometries outside the training ranges or by comparing against a branch that consumes the full geometry. Please state this limitation explicitly and provide evidence that the low-dimensional parametrization is sufficient for the studied family of geometries.
minor comments (5)
  1. [References] In the Introduction, reference [35] is cited for an RNN model of powder-mixing particle trajectories, but the bibliographic entry is about residual stress characterization in composite structures; please correct the citation.
  2. [Section 5] In the paragraph after Figure 14(b), the phrase the solution begins to accountsfor material separation contains a typo; it should read account for.
  3. [Equation (6)] Equation (6) presents the failure criterion as sigma1 <= sigma_U_t ; sigma2 >= -sigma_U_c, which is confusing because the preceding sentence states sigma1 > sigma2; please clarify the notation for ultimate tensile and compressive strengths.
  4. [Section 3.1] The threshold (r+h)/r <= 1.4 introduced in Section 3.1 is stated without derivation or statistical support; please either provide a derivation or label it as an empirical observation.
  5. [Section 5] The paper does not quantify the computational speed-up of the surrogate models relative to the CPD simulations, which is part of the stated motivation; a runtime comparison would strengthen the contribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's comparisons are empirical benchmarks against externally generated CPD simulation data, with no fitted parameter renamed as a prediction and no load-bearing self-citation chain.

full rationale

The paper's central claim is an empirical comparison between two trained neural operators, vanilla DeepONet and Fusion DeepONet, evaluated against CPD simulation outputs. The models are trained on displacement fields produced by the CPD particle simulations, and the reported errors in Table 2 and Figure 14 are measured losses or relative L2 errors on those data; they are not derived quantities. The branch inputs (notch height h and hole radius r) and trunk inputs (spatial coordinate x and time t) are genuine model inputs, not outputs of a fit that is then relabeled as a prediction. The architectures are adopted from prior work — CPD from [22] and Fusion DeepONet from [44] — but these citations provide construction and training details, not a uniqueness theorem or an ansatz whose validity is the paper's conclusion. No equation in the paper defines its target in terms of the fitted parameters, and no claim reduces to a self-citation. The text's assertion that Fusion DeepONet 'consistently outperforms' is weakened by its own Table 2, where vanilla has lower average error in Case 3 (8.46e-04 vs 1.01e-03), and the comparison is confounded by different architectures, activations, learning-rate schedules, and iteration counts; these are correctness-evidence problems, not circularity. Similarly, the paper's inconsistent training-sample counts (40/50/51 in Table 1 versus 32/45/45 in Section 6) and its use of 'training error' to caption Table 2's 'relative error' are reporting inconsistencies rather than circular reductions. Because the reported results are empirical evaluations against external simulation data, there is no self-definitional step, no fitted-input-called-prediction, and no load-bearing self-citation chain. The appropriate circularity score is therefore 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the fidelity of CPD simulations, the scalar-parameter input encoding, and the chosen network configurations. No invented physical entities are introduced. The main free parameters are the hand-tuned network architectures and training budgets, plus an empirically observed crack-hole interaction threshold.

free parameters (2)
  • Network configuration and training budget = Vanilla 5x100, tanh, 60k iters; Fusion 3x64, Rowdy, 50k iters
    Chosen empirically based on loss plateau (Section 5); because the two models differ in all of these choices, the comparison is not controlled.
  • Crack-hole interaction threshold (r+h)/r = 1.4
    Section 3.1 states crack deflection occurs when (r+h)/r <= 1.4; this threshold is an empirical observation from the simulated data without uncertainty or independent validation.
assumptions (5)
  • domain assumption CPD simulations faithfully reproduce brittle crack propagation.
    All training and test data are generated by CPD (Section 5); the only experimental check is one qualitative crack-path comparison in Section 3, so the network quality is bounded by CPD fidelity.
  • domain assumption Maximum principal stress failure criterion with ultimate strengths sigma_t^U and sigma_c^U governs fracture.
    Section 2.1, Eq. (6): a triangle fails when its principal stress exceeds these thresholds; crack paths in Cases 2 and 3 are defined by this rule.
  • ad hoc to paper Scalar inputs h and r, plus (x,t), determine the entire displacement field.
    Section 4.1: the branch network receives only h_i and r_i, so the model assumes no other geometric or configurational variability matters for the output.
  • domain assumption Fixed Delaunay triangulation in the reference configuration defines particle neighbors for all time.
    Section 2: the interaction network is computed once from the reference configuration, and it is not updated after failure, which affects how cracks unfold.
  • standard math DeepONet architectures can approximate the target displacement map.
    The paper relies on the universal approximation theorem for operators [38] and on the Fusion DeepONet design [44]; this is background theory, not proven in this paper.

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Cite this review

Pith. "Pith review of Crack Path Prediction with Operator Learning using Discrete Particle System data Generation." pith.science (2026). https://pith.science/paper/UMQV4UOS

@misc{pith2026250601976,
  author       = {Pith},
  title        = {Pith review of: Crack Path Prediction with Operator Learning using Discrete Particle System data Generation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UMQV4UOS}},
  note         = {Machine review of arXiv:2506.01976}
}
read the original abstract

Accurately modeling crack propagation is critical for predicting failure in engineering materials and structures, where small cracks can rapidly evolve and cause catastrophic damage. The interaction of cracks with discontinuities, such as holes, significantly affects crack deflection and arrest. Recent developments in discrete particle systems with multibody interactions based on constitutive behavior have demonstrated the ability to capture crack nucleation and evolution without relying on continuum assumptions. In this work, we use data from Constitutively Informed Particle Dynamics (CPD) simulations to train operator learning models, specifically Deep Operator Networks (DeepONets), which learn mappings between function spaces instead of finite-dimensional vectors. We explore two DeepONet variants: vanilla and Fusion DeepONet, for predicting time-evolving crack propagation in specimens with varying geometries. Three representative cases are studied: (i) varying notch height without active fracture; and (ii) and (iii) combinations of notch height and hole radius where dynamic fracture occurs on irregular discrete meshes. The models are trained using geometric inputs in the branch network and spatial-temporal coordinates in the trunk network. Results show that Fusion DeepONet consistently outperforms the vanilla variant, with more accurate predictions especially in non-fracturing cases. Fracture-driven scenarios involving displacement and crack evolution remain more challenging. These findings highlight the potential of Fusion DeepONet to generalize across complex, geometry-varying, and time-dependent crack propagation phenomena.

Figures

Figures reproduced from arXiv: 2506.01976 by the authors.

Figure 1
Figure 1. Discretization and mapping of deformation of the domain through particle displacement. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The particle i and its neighbors interacting through Delauney triangles Ti ; i = 1, 2 . . . 5. The strain energy density of T1 at a particular instance shown satisfies the failure criteria σ T1 1 = σ t U . 3 Modeling geometric discontinuity and brittle crack To demonstrate the model’s ability to capture stress concentrations around discontinuities, three benchmark cases were considered: a pre-existing sharp crack (F… view at source ↗
Figure 3
Figure 3. Stress field concentrations around geometric discontinuities in the domain. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Comparison of predicted crack path with experimental results and domain setup. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: (a) Geometry of domain with hole and pre-existing crack along with the loading condi [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Case 2: Crack paths simulated by varying the height (h) of pre-existing crack from the hole of constant radius of r = 1 cm. dotted box in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Case 3: Crack paths simulated by varying the radius (r) of the hole positioned at h = 1.5 cm below the pre-existing crack. where both the input u(ξ) and the output v(ξ) are functions defined on a continuous domain [30, 38, 44]. This framework is particularly well-suite…
Figure 8
Figure 8. Figure 8: Schematic of the DeepONet architecture for displacement field prediction. The branch [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Schematic of the Fusion DeepONet architecture for displacement field prediction. The [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: Comparison of predicted and true particle positions at two time steps, [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Case 1: Comparison of training loss between the Fusion DeepONet and the vanilla DeepONet over 50000 iterations. also of size 64 for trunk subnetwork and 2×64 for branch subnetwork. It utilizes the Rowdy activa￾tion function [57], a modified form of tanh designed to en…
Figure 12
Figure 12. Figure 12: Predicted and true crack paths at two time steps, [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: Predicted and true crack paths at two time steps, [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]
Figure 14
Figure 14. Figure 14: Comparison of (a) the relative L2 error over time for three case studies reported in [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.