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REVIEW 3 major objections 3 minor 47 references

Using Deep Operators to Create Spatio-temporal Surrogates for Dynamical Systems under Uncertainty

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict Useful architectural tweak with thorough structural-dynamics evaluation, but the R² metric in Eq. 13d is misdefined and the baselines are not cleanly controlled. read the letter →

arxiv 2506.11761 v1 pith:QHHCPD3D submitted 2025-06-13 stat.ML cs.LG

Jichuan Tang , Patrick T. Brewick , Ryan G. McClarren , Christopher Sweet This is my paper · ORCID
classification stat.MLcs.LG
keywords deepoperatornetworkONetfull-fieldExtendedsurrogatemodelingspatio-temporalpredictionstructuraldynamicscable-stayedbridgestochasticgroundmotion
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 3 minor

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.

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 (3)
  1. [§5.1, Eq. (13d)] 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.
  2. [§4.1.1, Table 2] 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.
  3. [§2.4, §5 and §6] 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.
minor comments (3)
  1. [§2.2, Eq. (6)] 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).
  2. [§3.3, §4, §5] 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.
  3. [§4 and Data availability] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the FExD surrogate is trained and evaluated on external FE data with held-out test realizations; the few self-citations are contextual and not load-bearing.

full rationale

The paper is an empirical operator-learning study. The target responses come from an external finite-element benchmark bridge model (Dyke et al., Refs. [37, 41]), and the ground-motion inputs are generated from an independent stochastic model (Kanai-Tajimi spectrum plus envelope function, Refs. [42]). The FExD architecture is defined in Section 2.4 from the previously cited Ex-DeepONet construction of Li et al. [31], which is not a self-citation. Training minimizes an L2 loss (Eq. 6) against FE-simulated responses, and all headline metrics (Table 3, 5, B2-B4) are computed on test realizations not used in training or validation. No fitted parameter is renamed as a prediction, and no uniqueness theorem or ansatz is imported from the authors' prior work. The self-citations to De and Brewick [23] and McClarren [44] are background citations in the introduction and data-format discussion, respectively, and neither supports a load-bearing premise. The trunk's restriction to a fixed set of 26 sensor locations is a scope limitation, not a circular derivation: the model is trained and tested on those sensors, and the claim of 'full-field' prediction is defined relative to those output dimensions (p=Nx). The R2 definition in Eq. 13d uses prediction variance in the denominator rather than target variance, which is a correctness and comparability issue in the reported evidence, not a circularity: the metric is still measured against held-out targets. Overall, the central accuracy and efficiency claims are self-contained external evaluations, so the circularity burden is very low.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physics or entities. Its free parameters are standard neural network hyperparameters and baseline choices. The main domain assumptions are linearity, zero initial conditions, and distributional stationarity.

free parameters (3)
  • Hidden layer width of branch and trunk networks = 300 neurons per layer
    Section 4.1.1: The architecture was chosen via a parametric study on the validation set; this is a form of fitting to data.
  • Hidden layer depth of branch and trunk networks = 4 layers
    Section 4.1.1: Depth was selected through parametric experiments to minimize validation loss.
  • Number of basis functions p for vanilla DeepONet = 10
    Table 2: The baseline VD uses p=10 while ExD and FExD use branch output 1200, creating a capacity mismatch.
assumptions (3)
  • domain assumption The structural response depends only on the ground acceleration history via the convolution integral y(t) = integral of u_g(tau) h(t-tau) d tau, with zero initial conditions.
    Section 3.1, Eq. 9: The surrogate is trained on responses from zero initial conditions, so it cannot represent free vibrations or nonzero initial states.
  • ad hoc to paper A single forward pass through the FExD with temporal input t can produce the full spatial response vector because spatial dependencies are encodable in the output layer without explicit spatial coordinates.
    Section 2.4: This architectural principle is the core of FExD and is not proven to generalize to arbitrary sensor layouts.
  • domain assumption The training distribution, defined by the Kanai-Tajimi spectrum with given parameter ranges and envelope function, is representative of all test cases of interest.
    Section 3.3: All test realizations are sampled from the same stochastic process, so out-of-distribution performance is untested.

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Pith. "Pith review of Using Deep Operators to Create Spatio-temporal Surrogates for Dynamical Systems under Uncertainty." pith.science (2026). https://pith.science/paper/QHHCPD3D

@misc{pith2026250611761,
  author       = {Pith},
  title        = {Pith review of: Using Deep Operators to Create Spatio-temporal Surrogates for Dynamical Systems under Uncertainty},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QHHCPD3D}},
  note         = {Machine review of arXiv:2506.11761}
}
read the original abstract

Spatio-temporal data, which consists of responses or measurements gathered at different times and positions, is ubiquitous across diverse applications of civil infrastructure. While SciML methods have made significant progress in tackling the issue of response prediction for individual time histories, creating a full spatial-temporal surrogate remains a challenge. This study proposes a novel variant of deep operator networks (DeepONets), namely the full-field Extended DeepONet (FExD), to serve as a spatial-temporal surrogate that provides multi-output response predictions for dynamical systems. The proposed FExD surrogate model effectively learns the full solution operator across multiple degrees of freedom by enhancing the expressiveness of the branch network and expanding the predictive capabilities of the trunk network. The proposed FExD surrogate is deployed to simultaneously capture the dynamics at several sensing locations along a testbed model of a cable-stayed bridge subjected to stochastic ground motions. The ensuing response predictions from the FExD are comprehensively compared against both a vanilla DeepONet and a modified spatio-temporal Extended DeepONet. The results demonstrate the proposed FExD can achieve both superior accuracy and computational efficiency, representing a significant advancement in operator learning for structural dynamics applications.

Figures

Figures reproduced from arXiv: 2506.11761 by the authors.

Figure 1
Figure 1. Schematic illustration of the vanilla DeepONet model considering (a) individual DoF responses and (b) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Schematic illustration of an Ex-DeepONet architecture considering (a) individual DoF responses and (b) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Data structuring for MDoF of vanilla vs. Ex-DeepONet vs. Full-field Ex-DeepONet. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: Full-field Ex-DeepONet. 3 Application to Structural Dynamics The capabilities of the proposed full-field Ex-DeepONet, as well as those of the vanilla DeepONet and conventional Ex-DeepONet, are interrograted through a structural dynamics application. A cable-stayed brid…
Figure 5
Figure 5. Figure 5: Finite element model of the bridge structure showing (a) the full model and (b) top view of bridge deck with [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Sample ground motion realization. 4 Implementation Details for MDoF responses We generate 3×104 realizations of ground accelerations by sampling from the distributions of the uncertain parameters that parameterize the K-T spectrum and evelope function. The ground motio…
Figure 7
Figure 7. Figure 7: Training and validation mean squared error (MSE) losses for each of the considered DeepONet models. [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: Comparison of testing MAE error for the chosen spatial-temporal DeepONets. [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Comparison of testing RMSE error for the chosen spatial-temporal DeepONets. [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: Comparison of testing RRMSE error for the chosen spatial-temporal DeepONets. [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Bar plot for mean RRMSE errors across locations for different models with whiskers for one standard [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: Prediction comparison of Full-field Ex-DeepONet with numerical responses at selected sensor positions. [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: Prediction comparison of Full-field Ex-DeepONet with numerical responses with 2s. [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]
Figure 14
Figure 14. Figure 14: Comparison of numerical responses with predicted responses from the FExD across all 26 locations. [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]
Figure 15
Figure 15. Figure 15: Comparison of response predictions at sensor positions (a) 5 and (b) 7 for the “worst case” from testing for [PITH_FULL_IMAGE:figures/full_fig_p016_15.png]
Figure 16
Figure 16. Figure 16: Performance vs. number of realizations for FExD. [PITH_FULL_IMAGE:figures/full_fig_p017_16.png]
Figure 17
Figure 17. Figure 17: Comparison of testing error for the chosen spatial-temporal DeepONets with 10s. [PITH_FULL_IMAGE:figures/full_fig_p018_17.png]
Figure 18
Figure 18. Figure 18: Comparison of FExD predictions at sensor positions (a) 5, (b) 12, (c) 7, and (d) 19 when trained on 10 s of [PITH_FULL_IMAGE:figures/full_fig_p019_18.png]
Figure 19
Figure 19. Figure 19: Fast Fourier Transform analysis comparing numerical and predicted responses. [PITH_FULL_IMAGE:figures/full_fig_p019_19.png]
Figure 20
Figure 20. Figure 20: Singular value decomposition of the cross-power spectral density matrices derived from both numerical [PITH_FULL_IMAGE:figures/full_fig_p020_20.png]
Figure 21
Figure 21. Figure 21: Prediction comparison of Full-field Ex-DeepONet against numerical responses with 10s at worst cases. [PITH_FULL_IMAGE:figures/full_fig_p020_21.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.