{"id":"c503fc3b-7ecc-4d70-a92b-74ce7d9ca375","arxiv_id":"2506.01976","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A study of DeepONet and Fusion DeepONet as surrogates for particle-simulation crack propagation, with the claimed consistent advantage of Fusion not supported by the paper's own numbers.","lead":"This paper trains two neural network designs, vanilla DeepONet and Fusion DeepONet, on particle simulation data to predict how cracks grow in a brittle material with holes and notches. It reports that the Fusion variant is often more accurate, but one of its own tables shows it is worse in one of the three test cases.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's own Table 2 contradicts the 'consistently outperforms' claim (Fusion is worse in Case 3), and the comparison is confounded by differing architectures, activations, and training schedules, so the central empirical claim is unsupported.","rationale":"The reader's stated weakest assumption is that the problem is fully specified by two scalar parameters (h and r). That is a real limitation of the claimed generalization and of the 'operator learning' framing, but it is not where the central claim first fails. The central empirical claim—Fusion consistently outperforms vanilla—is already undermined by the paper's own Table 2 and by the heavily confounded comparison. A performance comparison with different network sizes, activations, optimizers, and iteration counts cannot isolate the architectural contribution of Fusion DeepONet. Additionally, the ambiguity between 'training error' and 'relative error' means the reader cannot determine whether the reported numbers even measure generalization to unseen geometries. This is a more direct and more load-bearing problem than the finite-dimensional parametrization of the branch network: even if the two-scalar parametrization were adequate for the family of geometries studied, the empirical superiority claim would still lack support. My proposed check is a matched-hyperparameter rerun with clear test-set metrics; if the rerun reverses or eliminates the gap, the strongest claim fails. Since the reader already recommended REJECT, my concern does not change that verdict, hence UNCHANGED.","tokens_in":14968,"tokens_out":6415,"duration_ms":69725,"concrete_test":"Re-run the comparison on all three cases with matched training protocols: identical number of layers and neurons, identical activation function, identical optimizer and learning-rate schedule, identical number of iterations, and identical train/test splits. Report per-time-step and time-averaged relative L2 errors on the held-out test samples, including standard errors over multiple random seeds. If Fusion is not below vanilla in Case 3 under matched conditions, the claim of consistent outperformance should be withdrawn.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Fusion DeepONet consistently outperforms vanilla DeepONet. This is the load-bearing result, and the paper's own reported numbers do not support it. In Table 2, the average relative L2 errors are: Case 1, 6.20e-05 (vanilla) vs 4.97e-05 (Fusion); Case 2, 9.86e-04 vs 5.13e-04; Case 3, 8.46e-04 vs 1.01e-03. Thus in Case 3, Fusion has the higher error, directly contradicting 'consistently outperforms.' The text tries to reconcile this by pointing to Figure 14(a), where Fusion is said to be better after tau=20, but no per-time-step test-error table or per-sample breakdown is provided, so the aggregate contradiction cannot be checked. The comparison is also confounded: Section 5 states that vanilla uses five hidden layers of width 100, tanh activation, a fixed learning rate of 1e-4, and 60,000 iterations, while Fusion uses three layers of width 64, the Rowdy activation, a decaying learning rate starting at 1e-3, and 50,000 iterations. Any observed difference could be due to capacity, activation, optimizer schedule, or training budget rather than the fusion mechanism. Finally, Table 2 is described in the text as 'training error' but titled 'relative error,' and Figures 10–13 show only visual overlays for one or two test samples; no clear held-out test statistics are reported. Without matched training protocols and unambiguous test-set metrics, the claim that Fusion consistently outperforms vanilla is not established.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15264,"tokens_out":8662,"duration_ms":74127,"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":[{"comment":"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.","section":"Table 2, Abstract, Section 5"},{"comment":"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.","section":"Section 5"},{"comment":"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.","section":"Section 5 vs. Section 6"},{"comment":"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.","section":"Table 2 and Figure 14"},{"comment":"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.","section":"Section 4.1"}],"minor_comments":[{"comment":"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.","section":"References"},{"comment":"In the paragraph after Figure 14(b), the phrase the solution begins to accountsfor material separation contains a typo; it should read account for.","section":"Section 5"},{"comment":"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.","section":"Equation (6)"},{"comment":"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.","section":"Section 3.1"},{"comment":"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.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The main issue is that the central claim of consistent superiority is contradicted by the paper's own Table 2. I believe this can be fixed by rerunning the comparison with matched training protocols and reporting test-set statistics with error bars; if the authors cannot do that, they should explicitly revise the abstract and Section 5 to reflect the actual rankings. Given the otherwise interesting application area, I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The genuinely useful part is the CPD data-generation pipeline and its validation (Westergaard, Kirsch, and an experimental crack path). The paper then trains vanilla and Fusion DeepONets on that data to predict particle displacements over time for three 2D geometry families. The new element is just this application: both architectures are prior work (Refs. 30, 44), so the novelty is an empirical benchmark, not a method.\n\nThe problem is the load-bearing claim. Abstract and Section 5 say Fusion consistently outperforms vanilla. Table 2 says otherwise: Case 3 average relative L2 error is 1.01e-03 for Fusion versus 8.46e-04 for vanilla. The text explains this by pointing to Figure 14(a) after tau=20, but no per-timestep or per-sample breakdown is given, so the aggregate contradiction stands. On top of that, the comparison is confounded: different depth/width (5x100 tanh vs 3x64 Rowdy), different learning-rate schedule, different iteration counts (60k vs 50k). Any gap could be architecture or training budget, not the fusion mechanism.\n\nThe operator-learning framing is also overstated. The branch network takes only two scalars (h and r), so the model is finite-dimensional parametric regression, not an operator mapping functions to functions. That is fine if stated honestly, but the paper leans on operator terminology throughout. The manuscript also contradicts itself on training-sample counts (35 vs 32 for Case 1 in Sections 5 and 6), and there are no error bars or released code/data.\n\nTo be fair, the paper is not a mess. The CPD simulations are carefully set up, the three cases are sensible stress tests, and the visual overlays suggest both models capture rough crack paths. With matched architectures, proper test-set metrics, and a softened claim, the empirical story could become credible. As it stands, the central claim is unsupported.\n\nThis is for people working on surrogate models for discrete particle fracture. I would send it to peer review, because the data and problem are real and the flaws are correctable, but I would not cite it in its current form. It is a serious work with a fixable evidence problem.","headline":"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.","tokens_in":15841,"tokens_out":2566,"would_cite":false,"duration_ms":23840,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","74R10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["crack propagation","operator learning","DeepONet","Fusion DeepONet","particle dynamics","fracture mechanics","surrogate modeling","discrete particle system"],"falsifier":"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.","tokens_in":14736,"feed_emoji":"💥","tokens_out":5157,"duration_ms":48196,"temperature":0.7,"pith_summary":"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.","feed_headline":"Fusion DeepONet predicts crack paths across unseen geometries","feed_subtitle":"Trained on particle simulations, it beats the vanilla DeepONet and tracks crack deflection toward holes.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the CPD formulation that generates the particle-displacement training data and enforces constitutive behavior through Delaunay-triangle energies.","marker":"[22]"},{"why":"Introduces the DeepONet branch-trunk architecture and universal operator approximation theorem that the vanilla baseline implements.","marker":"[38]"},{"why":"Introduces Fusion DeepONet and the multi-scale layer-wise conditioning mechanism that the paper adapts.","marker":"[44]"},{"why":"Demonstrates DeepONet-based fracture prediction for phase-field models, the prior application this work carries over to CPD data.","marker":"[30]"},{"why":"Provides the experimental crack-path reference used to validate CPD's fracture predictions before training the operators.","marker":"[55]"},{"why":"Supplies the crack-and-hole specimen geometry that defines the three case studies.","marker":"[56]"},{"why":"Introduces the Rowdy activation function used in the Fusion DeepONet training configuration.","marker":"[57]"}],"fun_headline_variants":["Fusion DeepONet outperforms vanilla on crack paths","Particle-trained operator learns crack paths","Trained on particles, DeepONet tracks cracks toward holes","Fusion DeepONet surrogates CPD crack simulations","Single operator replaces crack simulations for new geometries"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Fusion DeepONet outperforms vanilla on crack paths","Particle-trained operator learns crack paths","Trained on particles, DeepONet tracks cracks toward holes","Fusion DeepONet surrogates CPD crack simulations","Single operator replaces crack simulations for new geometries"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001751,"raw_usage":{"total_tokens":6912,"prompt_tokens":944,"completion_tokens":5968,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":560,"completion_tokens_details":{"reasoning_tokens":5892}},"tokens_in":560,"tokens_out":5968,"duration_ms":39683,"temperature":1.0,"reasoning_tokens":5892,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:04:19.246889+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Constitutively informed multi-body interactions for lattice particle models.Computer Methods in Applied Mechanics and Engineering, 366:113052, 2020","cited_arxiv_id":null,"evidence_quote":"Supplies the CPD formulation that generates the particle-displacement training data and enforces constitutive behavior through Delaunay-triangle energies."},{"cited_title":"Karniadakis","cited_arxiv_id":null,"evidence_quote":"Introduces the DeepONet branch-trunk architecture and universal operator approximation theorem that the vanilla baseline implements."},{"cited_title":"Predicting crack nucleation and propagation in brittle materials using deep operator networks with diverse trunk architectures.arXiv preprint arXiv:2501.00016, 2024","cited_arxiv_id":null,"evidence_quote":"Demonstrates DeepONet-based fracture prediction for phase-field models, the prior application this work carries over to CPD data."},{"cited_title":"An abaqus implementation of the ex- tended finite element method.Engineering fracture mechanics, 76(3):347–368, 2009","cited_arxiv_id":null,"evidence_quote":"Provides the experimental crack-path reference used to validate CPD's fracture predictions before training the operators."},{"cited_title":"Mechanics of the crack path formation.International Journal of Fracture, 47:291–305, 1991","cited_arxiv_id":null,"evidence_quote":"Supplies the crack-and-hole specimen geometry that defines the three case studies."},{"cited_title":"Deep Kro- necker neural networks: A general framework for neural networks with adaptive activation functions.Neurocomputing, 468:165–180, 2022","cited_arxiv_id":null,"evidence_quote":"Introduces the Rowdy activation function used in the Fusion DeepONet training configuration."}],"review_version":1}