{"id":"f42d71f3-4732-4fdd-85a4-d8e2105fc080","arxiv_id":"2506.11761","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A new DeepONet variant, FExD, outputs all 26 bridge-deck response time histories in a single forward pass and reports lower error and faster training than vanilla and Extended DeepONets.","lead":"This paper proposes a new neural network design, the full-field Extended DeepONet, that predicts structural responses at many sensor locations at once from an earthquake ground motion input. It claims the model is more accurate and much faster to train than two standard DeepONet variants on a cable-stayed bridge simulation.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation 13d defines R² with the prediction variance in the denominator; as written it is not the coefficient of determination, so the headline R²=98.29% must be recomputed before the accuracy claim is accepted.","rationale":"The paper's contribution is best read as a fixed-sensor multi-output surrogate: FExD maps a ground-motion history and time t to a 26-dimensional acceleration vector. Within that scope, the 2-s and 10-s experiments, the convergence-with-data-size curve, and the FFT/FDD spectral checks provide real supporting evidence that the model learns the dominant dynamics. I did not find a flaw in the RRMSE or MAE computations as defined. The load-bearing concern is the R² statistic: Eq. 13d defines a quantity that is not the coefficient of determination, and the headline claim explicitly relies on R²=98.29%. This is an internal correctness issue rather than a disagreement with field consensus. It is also distinct from the reader's weakest assumption about fixed sensor locations: that assumption limits the claimed 'full-field' generality but does not make the reported numbers wrong; the R² formula can. The lack of code and the 'data available on request' statement add verification risk, but the concrete remedy is simple: recompute with the standard target-variance denominator. Because this is a fixable evaluation error and the other metrics still support a strong but more modest claim, the conditional verdict remains appropriate.","tokens_in":29023,"tokens_out":8155,"duration_ms":80230,"concrete_test":"Recompute every R² entry in Tables 3, 5, B2-B4 using the standard formula R² = 1 - Σ(ŷ_i - y_i)² / Σ(y_i - mean(y_target))² on the same held-out predictions. If FExD's aggregate R² stays near 98% and the ordering FExD > ExD > VD is unchanged, the concern is resolved; if the aggregate changes by more than a percentage point or any ordering flips, the headline comparison must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation 13d defines R² = 1 - Σ(ŷ_i - y_i)² / Σ(ŷ_i - \\bar{y})², with \\bar{y} stated to be the time-average of the response prediction. The standard coefficient of determination uses the target variance, Σ(y_i - \\bar{y}_target)², in the denominator. Because the printed denominator is the variance of the predictions, the reported values are not comparable across models with different prediction variances, and they are particularly distorted for low-amplitude sensors: FExD has R²=85.05% at position 7 and 77.65% at position 19 despite RRMSE of 37.48% and 45.78%. The paper's strongest claim quotes R²=98.29%, and Tables 3, 5, B2-B4 all rely on Eq. 13d, so this is not a cosmetic typo. It directly affects the conclusion that FExD improves and stabilizes R². The fixed-26-sensor scope is a limitation, but the mislabeled R² is a correctness issue in the reported evidence; no code or data are released, so the computation cannot be checked from the paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a full-field Extended DeepONet (FExD) that, unlike vanilla DeepONet and Ex-DeepONet, processes only time coordinates in the trunk and outputs a vector of responses at all 26 selected sensor locations in a single forward pass. The model is trained on 30,000 simulated responses of a cable-stayed bridge FE model to stochastic Kanai-Tajimi ground motions. The empirical study compares FExD against the two baselines on 2-s and 10-s test data, reporting error metrics, worst-case behavior, and frequency-domain agreement. The central claim is that FExD is more accurate and much faster to train than either baseline.","tokens_in":29350,"tokens_out":6018,"duration_ms":55308,"significance":"If the results hold, the FExD design is a practically useful way to build multi-output spatio-temporal surrogates for structural dynamics, and the reported training-time reduction (0.92 h vs. 13.45 h and 20.85 h) is striking. The paper is thorough in its evaluation: location-wise tables, worst-case analysis, a 10 s generalization study, and FFT/FDD checks. However, the evidence is currently weakened by a non-standard definition of R² and by a capacity-mismatched comparison, and the 'full-field' terminology overstates the fixed-sensor output. No code or data are provided, which limits reproducibility. With corrected metrics and matched-capacity ablations, the contribution would be meaningful.","major_comments":[{"comment":"Equation 13d defines R² with the variance of the predictions in the denominator rather than the variance of the targets; as written it is not the coefficient of determination. Because the denominator depends on the model's own output variance, the reported R² values are not comparable across models and do not measure goodness-of-fit in the standard sense. This affects the headline statistic in Table 3 (FExD R²=98.29%), the location-wise Tables B2–B4, and Table 5 for the 10 s case; for example, Table B4 reports FExD R²=85.05% at position 7 and 77.65% at position 19 despite RRMSE values of 37.48% and 45.78%, a combination that is inconsistent with the usual interpretation of R². Please recompute all R² values with the conventional denominator Σ(y_i − ȳ_target)² and confirm whether the qualitative conclusions—for instance that FExD improves and stabilizes R²—survive.","section":"§5.1, Eq. (13d)"},{"comment":"The three architectures are not matched in capacity: VD has branch and trunk output dimensions of 10, while ExD and FExD have branch output dimensions of 1200 (with trunk outputs 1 and 26, respectively). Since the branch output dimension controls the width of the final linear layer and the number of interaction coefficients, ExD and FExD contain far more parameters than VD in the branch and trunk components. The superior accuracy of ExD/FExD relative to VD may therefore reflect model capacity rather than the proposed architectural modifications. The claim in Section 4.1.1 of a 'fair comparison' should be supported by reporting parameter counts and by including a matched-capacity VD baseline (e.g., p = 1200) or an ExD with a reduced branch output dimension.","section":"§4.1.1, Table 2"},{"comment":"The term 'full-field' overstates what FExD predicts. Because the trunk consumes only temporal coordinates and the output dimension is fixed to the 26 pre-selected sensor locations (p = N_x), the model cannot evaluate the response at an arbitrary spatial point without retraining. The operator learned is therefore defined on a fixed sensor array, not on the continuum or fine FE mesh implied by Eqns. (1)–(2) and by the 'full-field' language used in the abstract, Section 2.4, and the conclusions. Please either qualify the claims throughout or demonstrate that the model can interpolate to unsampled spatial locations.","section":"§2.4, §5 and §6"}],"minor_comments":[{"comment":"In Eq. (6), the basis function in the sum is written as ψ_i(χ_j) but the summation index is k; this should be ψ_k(χ_j).","section":"§2.2, Eq. (6)"},{"comment":"There are several typographical errors that should be corrected, including 'exctied' and 'sujecting' in Section 3.3, 'interrograted' in Section 3, 'estimes' in Section 5.2.1, 'covnentional' in Section 2.4, and 'disply' in Section 5.3.","section":"§3.3, §4, §5"},{"comment":"The paper does not provide code, data, random seeds, or explicit hyperparameter search results, and the data availability statement says only 'Data will be made available on request.' For reproducibility, please deposit the data generation scripts, trained model configurations, and the exact seed(s) used in the parametric study.","section":"§4 and Data availability"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of stat.ML and the reference list appears appropriate. The main concerns are technical: the mislabeled R² metric and the capacity-mismatched comparison directly affect the paper's central accuracy claim. The 'full-field' terminology may overstate the fixed-sensor output and should be revised; otherwise there is a risk of misleading readers about the operator's spatial generality."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know about arXiv:2506.11761. First, the proposed FExD is a small but genuinely useful variant: it keeps the Ex-DeepONet nonlinear branch–trunk interaction, sets the trunk input to time only, and outputs the response at all 26 sensor locations in one forward pass. That removes a lot of redundant per-location evaluation and explains the dramatic training-time reduction. Second, the paper has a real correctness problem in its headline metric: Eq. 13d defines R² with the prediction variance in the denominator, not the target variance. As written, the reported R² values are not a coefficient of determination, and they are not comparable across models or across sensor locations with different response amplitudes. This affects Table 3, Table 5, and the appendix tables, so it is not a typo.\n\nWhat the paper does well: the evaluation is unusually thorough for this subfield. Per-location error metrics, worst-case realizations, training-data-size scaling, and frequency-domain comparisons (FFT and FDD) are all reported. The FExD numbers at the 26 fixed sensors are clearly better than the two baselines on MAE/RMSE/RRMSE, and the stability across locations is plausible. The authors also correctly identify the small-amplitude pier locations as hard cases, which is honest.\n\nThe soft spots beyond the R² issue: the baseline comparison is confounded. The vanilla DeepONet has a 10-dimensional branch output while ExD and FExD use 1200; part of the performance gap may simply be capacity. Also, the training-time comparison mixes an architectural difference (vectorized output vs. per-location evaluation) with the fact that ExD processes 26x more sample pairs per epoch. The fixed-26-sensor scope also makes the \"full-field\" claim narrower than it sounds; arbitrary spatial queries would require retraining. No code or data are released, only \"on request,\" which limits reproducibility but is common in this area.\n\nWho should read this: structural engineers and applied ML people working on operator surrogates for stochastic dynamics. The R² error is fixable, the architecture is sensible, and the empirical design is mostly careful. I would not cite it as is, but I would send it to a serious referee. If I were the editor, I would ask the authors to recompute R² with the standard definition, rerun the vanilla DeepONet with a comparable-capacity branch, and release code or data. Conditional accept after those fixes.","headline":"Useful architectural tweak with thorough structural-dynamics evaluation, but the R² metric in Eq. 13d is misdefined and the baselines are not cleanly controlled.","tokens_in":29770,"tokens_out":2257,"would_cite":false,"duration_ms":24254,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a single network, the full-field Extended DeepONet, can learn the full spatio-temporal response operator of a cable-stayed bridge under stochastic ground motion and evaluate all 26 deck sensors in one forward pass…","keywords":["deep operator network","DeepONet","full-field Extended DeepONet","surrogate modeling","spatio-temporal prediction","structural dynamics","cable-stayed bridge","stochastic ground motion"],"falsifier":"Train the FExD on the 26 deck sensors, then query it at a deck coordinate that is not among the 26, for instance midway between sensors 5 and 6. Because the trunk receives only time as input and the output layer is hard-wired to the 26 sensors, the network has no representation of that query point; if accurate responses can be produced only by retraining or by adding interpolation, the 'full-field' operator is a fixed-array evaluator rather than a continuous spatial operator. A second check is to feed ground motions with Kanai–Tajimi parameters far outside the training ranges (for example $\\omega_g \\neq 10\\pi$) and compare RRMSE, which would show whether the operator generalizes physically or only interpolates within the training distribution.","tokens_in":28867,"feed_emoji":"🌉","tokens_out":6995,"duration_ms":68616,"temperature":0.7,"pith_summary":"The paper proposes a DeepONet variant called the full-field Extended DeepONet (FExD) to serve as a spatio-temporal surrogate for dynamical systems under uncertainty. The central claim is that one trained network can map a stochastic ground-motion time history directly to vertical acceleration responses at 26 locations along a cable-stayed bridge deck in a single forward pass, replacing separate evaluations at each sensor. On 2-second test data, the FExD reports an average RRMSE of 7.77% and an $R^2$ of 98.29%, while training in 0.92 hours versus 13.45 hours for a vanilla DeepONet and 20.85 hours for the Extended DeepONet. The paper argues this removes a bottleneck in surrogate modeling, where full spatial fields previously required either one network per degree of freedom or repeated evaluations across spatial coordinates.","feed_headline":"One forward pass predicts bridge response at 26 sensors","feed_subtitle":"A DeepONet variant maps stochastic ground motion to all 26 vertical accelerations at once, reaching 98.3% accuracy.","key_machinery":"The full-field Extended DeepONet architecture is the load-bearing mechanism: a branch network encodes the input ground-motion history and outputs per-neuron multiplicative coefficients for every trunk hidden layer, while a trunk network takes only time coordinates and has an output layer of $p = 26$ units, one per sensor. Each trunk layer is scaled elementwise by the branch-derived coefficients before activation, so the basis functions themselves depend on the input, and the final trunk layer emits the vector of all sensor responses in a single pass. This combination is what lets the same operator represent inter-degree-of-freedom correlations without per-coordinate evaluations.","core_discovery":"The central discovery is architectural: by taking the Extended DeepONet's branch network, which modulates every hidden layer of the trunk, and setting the trunk output dimension equal to the number of spatial sensors ($p = N_x = 26$) while feeding the trunk only time coordinates, the entire spatial field can be emitted in one evaluation. This is claimed to encode spatial correlations directly into the shared output layer of the trunk, eliminating the need to replicate spatio-temporal coordinates for every sensor and evaluate the network separately at each location. On the benchmark cable-stayed bridge, the FExD achieves lower mean and median errors than both the vanilla and Extended DeepONets across all four reported metrics (MAE, RMSE, RRMSE, and $R^2$), with the largest relative gains in RRMSE and in error consistency across sensor locations. The paper also reports that the FExD preserves dominant modal frequencies and produces a worst-case test error that stays below the average of the comparators.","pith_inferences":["This suggests that the multiplicative branch-to-trunk interaction is the main source of both the accuracy and the speedup, so the same full-field trick could transfer to other operator architectures that produce one output vector per time step, such as Fourier neural operators or transformer-based operators.","Because the trunk never receives a spatial coordinate, the trained operator is tied to the predefined 26-sensor array; a natural extension would combine FExD with an interpolation layer or implicit neural representation to query arbitrary deck locations without retraining, but that goes beyond what the paper demonstrates.","The paper tests only ground motions drawn from the same Kanai–Tajimi parameter ranges and zero initial conditions, so whether the architecture generalizes to out-of-distribution excitations or nonzero initial states remains open; a clean test of that would separate architectural benefit from training-distribution benefit."],"forward_implications":["A single FExD replaces $N_x$ separate DeepONet evaluations for multi-degree-of-freedom systems, so training and inference cost scale with time steps rather than with time multiplied by sensor count.","Spatial correlations among sensors are captured implicitly through the shared trunk output layer, which the paper shows improves accuracy at low-amplitude locations near the piers compared with per-location baselines.","Training time for structural surrogate models becomes short enough (under an hour on a single GPU for the 2-second case) to make operator learning practical inside optimization and uncertainty-quantification loops.","The same architecture extends to longer 10-second simulations with only moderate error growth, suggesting the method scales in time as well as in sensor count.","The FExD approximates the vector-valued solution operator $\\ddot{u}_g(t) \\mapsto y(t)$ for the full field rather than for individual time histories, which is the form needed for full spatial-temporal surrogate modeling."],"supporting_citations":[{"why":"Introduces the DeepONet branch-trunk operator architecture that the paper adapts and extends.","marker":"[18]"},{"why":"Defines the Extended DeepONet whose branch-to-trunk multiplicative interactions the FExD builds upon.","marker":"[31]"},{"why":"States the universal approximation theorem for operators that underpins DeepONet-based operator learning.","marker":"[33]"},{"why":"Supplies the benchmark finite-element cable-stayed bridge model used as the testbed dynamical system.","marker":"[37]"},{"why":"Identifies the sensing locations along the bridge deck from which the 26 used sensors are drawn.","marker":"[41]"},{"why":"Provides the Kanai–Tajimi spectrum and envelope function used to generate the stochastic ground-motion inputs.","marker":"[42]"}],"fun_headline_variants":["DeepONet variant emits full spatial field in one pass","One forward pass yields 26-sensor bridge response","Full-field DeepONet surrogate outshines standard DeepONets","Spatio-temporal surrogate maps uncertainty to all sensors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the sensor locations can be fixed beforehand: the FExD's trunk never receives a spatial coordinate, so the 'full field' is really the 26 predefined points, and the model also assumes quiescent initial conditions and test ground motions drawn from the same stochastic family used for training.","fun_headline_variants_meta":{"raw":{"variants":["DeepONet variant emits full spatial field in one pass","One forward pass yields 26-sensor bridge response","Full-field DeepONet surrogate outshines standard DeepONets","Spatio-temporal surrogate maps uncertainty to all sensors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000187,"raw_usage":{"total_tokens":1333,"prompt_tokens":953,"completion_tokens":380,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":313}},"tokens_in":569,"tokens_out":380,"duration_ms":4122,"temperature":1.0,"reasoning_tokens":313,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:03:34.536521+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the FExD on the 26 deck sensors, then query it at a deck coordinate that is not among the 26, for instance midway between sensors 5 and 6. Because the trunk receives only time as input and the output layer is hard-wired to the 26 sensors, the network has no representation of that query point; if accurate responses can be produced only by retraining or by adding interpolation, the 'full-field' operator is a fixed-array evaluator rather than a continuous spatial operator. A second check is to feed ground motions with Kanai–Tajimi parameters far outside the training ranges (for example $\\omega_g \\neq 10\\pi$) and compare RRMSE, which would show whether the operator generalizes physically or only interpolates within the training distribution.","supporting_citations":[{"cited_title":"Learning nonlinear op- erators via deeponet based on the universal approximation theorem of operators.Nature Machine Intelligence, 3(3):218–229, March 2021","cited_arxiv_id":null,"evidence_quote":"Introduces the DeepONet branch-trunk operator architecture that the paper adapts and extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Extended DeepONet whose branch-to-trunk multiplicative interactions the FExD builds upon."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the universal approximation theorem for operators that underpins DeepONet-based operator learning."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the benchmark finite-element cable-stayed bridge model used as the testbed dynamical system."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the sensing locations along the bridge deck from which the 26 used sensors are drawn."},{"cited_title":"System identification through nonstationary data using time–frequency blind source separation.Journal of Sound and Vibration, 371:110–131, June 2016","cited_arxiv_id":null,"evidence_quote":"Provides the Kanai–Tajimi spectrum and envelope function used to generate the stochastic ground-motion inputs."}],"review_version":1}