Pith. sign in

REVIEW 2 major objections 5 minor 89 references

Single vs. Multiple Branches in DeepONet and S-DeepONet: Network Architecture Follows Coupling in Multiphysics Systems

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Shared branch tops split branches on coupled multiphysics

desk verdict A well-controlled empirical study whose central architectural recommendation is confounded by the Hadamard fusion rule, and whose 18,000x speedup claim is a factor-of-~1,000 arithmetic error. read the letter →

arxiv 2507.03660 v2 pith:SYUWUMOT submitted 2025-07-04 cs.LG

Jaewan Park , Kazuma Kobayashi , Qibang Liu , Seid Koric , Diab Abueidda , Syed Bahauddin Alam This is my paper · ORCID
classification cs.LG
keywords deepoperatornetworksneuralarchitecturemultiphysicscouplingsingle-branchvsmulti-branchS-DeepONetgatedrecurrentunitssurrogatemodelinglearning
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 asks whether the internal topology of a neural operator should track the physical coupling of the system it emulates. Across a reaction–diffusion problem, a coupled electro-thermal system, and a viscoplastic thermo-mechanical steel-solidification model, the authors compare architectures that encode multiple input functions through one shared branch against designs that give each input its own branch and fuse them by Hadamard product. The consistent result: in tightly coupled regimes the single-branch design is more accurate, while in uncoupled or single-physics settings the multi-branch design wins by a modest margin. The authors argue this mirrors the classical distinction between monolithic and staggered solvers in finite-element analysis, and report full-field inference up to roughly 18,000 times faster than high-fidelity FEA once trained.

What carries the argument

The central object is the branch–trunk decomposition of a DeepONet: a branch network encodes the input functions, a trunk network encodes the query coordinates, and their dot product yields the field prediction. The comparison hinges on where multiple inputs meet in the branch. In the single-branch design the inputs are concatenated and processed by one set of shared parameters, so cross-terms between physical fields can be represented early in the latent space. In the multi-branch (MIONet-style) design each input is encoded independently and fused only at the end by the element-wise Hadamard product $B_d \odot B_m$ before the trunk dot product. For sequential path-dependent inputs the paper uses GRU-based encoder–decoder branches (S-DeepONet), and the same shared-versus-factorized comparison carries through. That fusion point — early shared latent versus late factorized product — is the mechanism the ablation studies isolate.

What would settle it

Train a two-branch S-DeepONet on the coupled electro-thermal dataset with per-branch pipelines enlarged so total parameter count matches the single-branch model, then compare test errors; if the enlarged two-branch model matches or beats the single-branch one, the claimed shared-latent advantage is not specific to branch topology.

Watch

Extended reading notes

Core claim

Within the DeepONet and sequential S-DeepONet families, shared-parameter single-branch encoders outperform factorized multi-branch encoders for strongly coupled multiphysics problems, while the reverse holds when the physics decouples. In the coupled electro-thermal benchmark, the single-branch S-DeepONet gives relative L2 errors near 0.5–0.8% versus 3.8–4.2% for the two-branch variant; in the coupled thermo-mechanical benchmark it gives roughly half the stress MAE. When the same systems are run uncoupled, the two-branch model is slightly more accurate. The paper presents this as an empirical architectural principle, not a formal guarantee, and explicitly scopes it to DeepONet and S-DeepONet topologies.

Load-bearing premise

The conclusion assumes that holding every training setting identical while the multi-branch model carries roughly twice the parameters isolates branch topology; if the larger model is under-tuned or under-regularized for its capacity, the measured advantage of shared branches in coupled regimes could be a training artifact rather than a property of shared representations.

Editorial extensions

If this is right

  • In DeepONet-style surrogates for tightly coupled multiphysics, architecture selection should default to a single shared branch; the paper finds this for both feedforward and sequential GRU formulations.
  • For decoupled or single-physics tasks, separate per-input branches give a small but consistent accuracy advantage, so modular encoding remains useful when inputs are independent.
  • Trained S-DeepONet surrogates can replace expensive FEA solves for new input histories, with roughly 18,000x speedup and per-sample inference around 20 ms, enabling real-time digital twins and uncertainty propagation.
  • The coupling-aware pattern persists under input noise in the tested range: the single-branch model degrades more gracefully under 3–15% measurement noise in coupled settings, and in uncoupled settings it overtakes the two-branch model above roughly 12% noise.
  • The conclusions are scoped to DeepONet and S-DeepONet; whether other operator families show the same coupling-dependent topology effect remains an open question.

Reading between the lines

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

  • Beyond the paper: the shared-latent advantage should transfer to other operator families, such as Fourier or attention-based operators, that admit a factorized multi-input design, since the mechanism is generic.
  • Beyond the paper: the crossover point between single- and multi-branch preference could be read as a quantitative measure of coupling strength, giving practitioners a threshold based on the performance gap rather than physical intuition alone.
  • Beyond the paper: the noise-robustness crossover suggests shared representations act as an implicit regularizer, which motivates testing explicit regularization or ensemble strategies for multi-branch models in noisy sensing settings.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper asks whether the branch architecture of DeepONet and its sequential variant S-DeepONet should be chosen according to the strength of coupling in a multiphysics system. In three benchmark problems (a single-physics reaction–diffusion system, a coupled electro-thermal system, and a coupled thermo-mechanical steel solidification model, each with coupled and uncoupled data variants), the authors compare single-branch networks, in which all input functions are concatenated and encoded jointly, with multi-branch MIONet-style networks, in which each input is encoded independently and the branch outputs are merged by a Hadamard product. They report that single-branch models are more accurate for coupled multiphysics tasks, especially for stress and electric potential, while multi-branch models are slightly better for uncoupled or single-physics tasks. The paper also reports large inference speedups over finite-element solvers and includes supplementary ablations over hidden dimension, depth, sequence cell type, dataset size, and input noise.

Significance. If the central conclusion is correct, the paper provides practical guidance: for strongly coupled multiphysics problems, DeepONet/S-DeepONet users should prefer a single shared branch, while multi-branch designs are appropriate when inputs are independent. The study has notable strengths: public code and data, five-seed means and standard deviations, held-out test splits, and a broader set of ablations (GRU/LSTM, depth, hidden dimension, training-set size, and noise robustness) than is typical. The two coupled multiphysics benchmarks are nontrivial and the thermo-mechanical steel solidification problem is a realistic, challenging case. However, the headline causal claim is not fully isolated from the choice of fusion operation, and one reported quantitative claim (the speedup) contains a clear arithmetic error, both of which are load-bearing for the abstract's promise.

major comments (2)
  1. [Model architectures and Eq. (11); also Discussion] The claim that single-branch consistency depends on shared latent representations is an interpretation; the experiments as designed cannot rule out the alternative that the Hadamard fusion is simply too weak. As such, the current evidence supports an empirical observation about two specific architectures, not the stated mechanism.
  2. [Training Performance and Inference] While the corrected speedup is still substantial (over an order of magnitude), it does not justify the four-orders-of-magnitude statement in the present form.
minor comments (5)
  1. [Methods, Training and Evaluation protocols] The loss is described as a 'variance–normalized squared–error (COP) loss' but the acronym COP is not defined anywhere in the manuscript or supplement; please define it or provide a reference.
  2. [Model architectures and Supplementary Note 1] The trunk for the electro-thermal benchmark is described as taking spatial coordinates (x, y), but the problem is one-dimensional in space and the main text elsewhere states that the trunk receives (x, t) coordinate pairs. Please clarify whether the trunk input is (x, t), (x, y), or both, and unify the notation across the main text, figure captions, and supplementary specifications.
  3. [Table 3 vs. Methods] The iteration counts for the reaction-diffusion two-branch model are inconsistent: the Methods state 100,000 iterations for the two-branch variant, while Table 3 lists 110,000. Harmonize these numbers so that the 'same optimization settings' claim is verifiable.
  4. [Results, Single Physics Reaction-Diffusion Model] The reaction-diffusion results are reported as single mean L2 errors (3.40% vs. 2.55%) without standard deviations or number of seeds, whereas the multiphysics results report mean ± std over five runs; please state whether these numbers are from a single run or averaged over seeds, and add the corresponding measure of variability.
  5. [Discussion and Conclusions] The conclusions state that the results provide 'modest accuracy improvements' for multi-branch in uncoupled settings; this is accurate, but the abstract's phrase 'multi-branch designs remain favorable' might overstate the size of the effect for the uncoupled temperature field, where the difference between 0.081% and 0.070% is within the reported standard deviation. A more cautious formulation would reflect the statistical overlap in these cases.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: empirical branch-topology comparison on held-out FEM data; self-citations are contextual, not load-bearing.

full rationale

The paper is an empirical ablation study, not a derivation. Single-branch and multi-branch DeepONet/S-DeepONet variants are trained on FEM-generated data, and accuracy is reported on held-out test splits using relative L2 error and MAE. No fitted parameter is renamed as a prediction, and the uncoupled datasets function as controls rather than as evidence for the coupled conclusion. The central claim (single-branch outperforms multi-branch under coupled physics, while multi-branch is favored when inputs are independent) is an observed empirical pattern, not a consequence of how the architectures are defined. The self-citations, including S-DeepONet (ref 36) and the solidification model (ref 83), supply the baseline architecture and data-generation framework, but the branch comparison is self-contained within the present experiments and does not reduce to those citations. The mechanistic interpretation (shared latent representations, monolithic-vs-staggered analogy) is post hoc and not used to produce the reported numbers. A legitimate experimental concern is that the multi-branch variant changes both branch count and fusion operation (Hadamard product) simultaneously, so the causal attribution to branch separation alone is not fully isolated; however, this is an internal-validity confound, not circular reasoning. No uniqueness theorem, ansatz smuggled by citation, or self-definitional reduction appears in the paper.

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

The paper's central claim is an empirical architecture comparison, so the ledger contains no invented entities or fitted scientific constants. The listed free parameters are hand-chosen hyperparameters and dataset generators that the comparison depends on. The main load-bearing assumption is that fixed hyperparameters across branches of different capacity constitute a fair test of topology.

free parameters (5)
  • Hidden dimension h = 100
    Chosen by hand for all branch and trunk networks; the central comparison holds h fixed across single- and multi-branch designs, so the conclusion depends on this capacity choice.
  • Trunk MLP widths = (101,101,101,101,101,h*c)
    Fixed by hand for all benchmarks; c=1 for reaction-diffusion and c=2 for multiphysics cases.
  • Optimizer and learning rate = Adam, 1e-3 with inverse-time decay
    Chosen by hand and held identical across architectures; the fairness of the topology comparison relies on this.
  • Iteration budgets = 110k/100k (reaction-diffusion), 310k/310k (electro-thermal), 310k/110k (thermo-mechanical)
    Budgets differ by benchmark and coupling regime but are equal across branch designs; training length could affect relative performance.
  • Heat-flux profile sampling ranges = A in [3,8], B in [0.3,0.7], C in [-0.5,0.5]
    Used to generate thermo-mechanical boundary condition histories; the dataset, and therefore the comparison, depends on these hand-chosen ranges.
assumptions (5)
  • domain assumption FEniCSx and Abaqus finite element solutions are treated as exact ground truth for the PDE systems.
    All error metrics compare predictions to FEM/FEA outputs; if the reference solutions are inaccurate, the reported errors and rankings inherit that error.
  • domain assumption Equal hyperparameters and training schedules across single- and multi-branch architectures isolate the effect of branch topology.
    This is the core methodological premise of the ablation; it is stated in Training and Evaluation protocols but not proven, and multi-branch models have roughly twice the parameters.
  • standard math The universal approximation theorem for operators justifies the expressivity of the DeepONet family used here.
    Invoked in the Introduction via refs 32 and 33; the paper does not prove it and relies on it as background.
  • domain assumption The three benchmarks represent the coupling-strength spectrum relevant to the architectural claim.
    The paper generalizes from reaction-diffusion, electro-thermal, and steel solidification; representativeness is asserted, not established.
  • domain assumption Kozlowski and Zhu constitutive laws and the material constants in Eq. 9 are valid for the steel solidification regime.
    Taken from prior literature (refs 79, 80) and embedded in the UMAT; the thermo-mechanical dataset inherits these modeling choices.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Single vs. Multiple Branches in DeepONet and S-DeepONet: Network Architecture Follows Coupling in Multiphysics Systems." pith.science (2026). https://pith.science/paper/SYUWUMOT

@misc{pith2026250703660,
  author       = {Pith},
  title        = {Pith review of: Single vs. Multiple Branches in DeepONet and S-DeepONet: Network Architecture Follows Coupling in Multiphysics Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SYUWUMOT}},
  note         = {Machine review of arXiv:2507.03660}
}
abstract

`Real-time prediction of complex physical systems requires surrogate models that learn from data while representing strong multiphysics coupling. Deep Operator Networks have shown success in single-physics problems, yet their effectiveness in capturing nonlinear interactions in coupled systems (such as thermo-mechanical or electro-thermal coupling) remains underexplored. Here we pose a practical question: should the architecture of a neural operator reflect the strength of physical coupling it aims to model? We compare single-branch and multi-branch designs, in both feedforward and sequential recurrent forms, across three representative systems: a reaction--diffusion problem with heterogeneous sources, a nonlinear thermo-electrical problem with temperature-dependent conductivity and Joule heating, and a viscoplastic thermo-mechanical model of steel solidification. Single-branch networks consistently outperform multi-branch variants in tightly coupled regimes by encouraging shared latent representations, whereas multi-branch designs remain favorable for decoupled or single-physics tasks. Once trained, these surrogates deliver full-field predictions up to $1.8 \times 10^4$ times faster than physics-based solvers.

Figures

Figures reproduced from arXiv: 2507.03660 by the authors.

Figure 1
Figure 1. Two sequential deep operator network (S-DeepONet) architectural variants compared in this study. (a), Single-branch (coupled) design. A gated recurrent unit (GRU)-based encoder–decoder processes the sequential inputs jointly, producing branch output Bh that is combined with trunk encoding Tnhc, where m is the input time series length and HD is the hidden dimension. (b),Two-branch (multiple-input operator network, MI… view at source ↗
Figure 2
Figure 2. Reaction–diffusion benchmark comparing single- and two-branch deep operator network (DeepONet) architectures. Target finite-element solutions, model predictions, and absolute error contours are shown for representa￾tive test samples at the 55th (top), 85th (middle), and 99th (bottom) error percentiles. The multi-branch (multiple-input operator network, MIONet-style) design yields consistently smaller error fields th… view at source ↗
Figure 3
Figure 3. Predictions of electrical potential and temperature in the coupled electro-thermal benchmark. (a)–(c), Electrical potential ϕ for the best, 55th percentile, and 98th percentile test cases. (d)–(f), Temperature T for the same percentile cases. Finite-element method (FEM) reference solutions (Target) are compared with predictions from the single-branch and two-branch sequential deep operator network (S-DeepONet) model… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Test-error distributions for the electro-thermal benchmark under coupled and uncoupled regimes. Coupled setting (a), (b): histograms of L2 relative error for electrical potential (ϕ) and temperature (T). The single￾branch model concentrates errors near zero, whereas th…
Figure 5
Figure 5. Figure 5: Thermo–mechanical solidification context and boundary value problems for data generation. (a), Modeling slice within the continuous-casting context. A representative section of the slab moves through the mold in the casting direction; the highlighted slice denotes the …
Figure 6
Figure 6. Figure 6: Temperature and stress profiles along the slice for the coupled thermo–mechanical benchmark. (a)–(c), Temperature along the slice at the 25th (a), 45th (b), and 85th (c) test percentiles. Both single-branch and two-branch sequential deep operator network (S-DeepONet) p…
Figure 7
Figure 7. Figure 7: Mean absolute error (MAE) distributions for the thermo–mechanical benchmark under coupled and uncoupled regimes. Coupled setting (a), (b): histograms of MAE for stress (a) and temperature (b). Single-branch predictions (red) concentrate near zero, whereas two-branch pr…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

89 extracted references · 47 canonical work pages

  1. [1]

    Ai-driven non-intrusive uncertainty quantification of advanced nuclear fuels for digital twin-enabling technology.Progress in Nuclear Energy, 172:105177, 2024

    Kazuma Kobayashi et al. Ai-driven non-intrusive uncertainty quantification of advanced nuclear fuels for digital twin-enabling technology.Progress in Nuclear Energy, 172:105177, 2024

    2024

  2. [2]

    Multi-criteria decision making under uncertainties in composite materials selection and design.Composite Structures, 279:114680, 2022

    Dinesh Kumar et al. Multi-criteria decision making under uncertainties in composite materials selection and design.Composite Structures, 279:114680, 2022

    2022

  3. [3]

    Stefano Pagani and Andrea Manzoni. Enabling forward uncertainty quantification and sensitivity analysis in cardiac electrophysiology by reduced order modeling and machine learning.International Journal for Numerical Methods in Biomedical Engineering, 37(6):e3450, 2021

    2021

  4. [4]

    Shantanu Shahane, Erman Guleryuz, Diab W Abueidda, Allen Lee, Joe Liu, Xin Yu, Raymond Chiu, Seid Koric, Narayana R Aluru, and Placid M Ferreira. Surrogate neural network model for sensitivity analysis and uncertainty quantification of the mechanical behavior in the optical lens-barrel assembly.Computers & Structures, 270:106843, 2022

    2022

  5. [5]

    Adaptive data-driven deep-learning surrogate model for frontal polymerization in dicyclopentadiene.The Journal of Physical Chemistry B, 128(5):1220–1230, 2024

    Qibang Liu, Diab Abueidda, Sagar Vyas, Yuan Gao, Seid Koric, and Philippe H Geubelle. Adaptive data-driven deep-learning surrogate model for frontal polymerization in dicyclopentadiene.The Journal of Physical Chemistry B, 128(5):1220–1230, 2024

    2024

  6. [6]

    Advanced deep operator networks to predict multiphysics solution fields in materials processing and additive manufacturing

    Shashank Kushwaha, Jaewan Park, Seid Koric, Junyan He, Iwona Jasiuk, and Diab Abueidda. Advanced deep operator networks to predict multiphysics solution fields in materials processing and additive manufacturing. Additive Manufacturing, 88:104266, 2024

    2024

  7. [7]

    Chemnet: A deep neural network for advanced composites manufacturing.The Journal of Physical Chemistry B, 124(42):9428–9437, 2020

    Elyas Goli, Sagar Vyas, Seid Koric, Nahil Sobh, and Philippe H Geubelle. Chemnet: A deep neural network for advanced composites manufacturing.The Journal of Physical Chemistry B, 124(42):9428–9437, 2020

    2020

  8. [8]

    Deep learning for topology optimization of 2d metamaterials.Materials & Design, 196:109098, 2020

    Hunter T Kollmann, Diab W Abueidda, Seid Koric, Erman Guleryuz, and Nahil A Sobh. Deep learning for topology optimization of 2d metamaterials.Materials & Design, 196:109098, 2020

    2020

Show all 89 references
  1. [9]

    Inverse design of nonlinear mechanical metamaterials via video denoising diffusion models.Nature Machine Intelligence, 5(12):1466–1475, 2023

    Jan-Hendrik Bastek and Dennis M Kochmann. Inverse design of nonlinear mechanical metamaterials via video denoising diffusion models.Nature Machine Intelligence, 5(12):1466–1475, 2023. 17 Multi-Input DeepONets in Physics and Multiphysics

    2023

  2. [10]

    Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational physics, 378:686–707, 2019

    2019

  3. [11]

    Surrogate modeling for fluid flows based on physics- constrained deep learning without simulation data.Computer Methods in Applied Mechanics and Engineering, 361:112732, 2020

    Luning Sun, Han Gao, Shaowu Pan, and Jian-Xun Wang. Surrogate modeling for fluid flows based on physics- constrained deep learning without simulation data.Computer Methods in Applied Mechanics and Engineering, 361:112732, 2020

    2020

  4. [12]

    A deep learning energy-based method for classical elastoplasticity.International Journal of Plasticity, 162:103531, 2023

    Junyan He, Diab Abueidda, Rashid Abu Al-Rub, Seid Koric, and Iwona Jasiuk. A deep learning energy-based method for classical elastoplasticity.International Journal of Plasticity, 162:103531, 2023

    2023

  5. [13]

    The mixed deep energy method for resolving concentration features in finite strain hyperelasticity.Journal of Computational Physics, 451:110839, 2022

    Jan N Fuhg and Nikolaos Bouklas. The mixed deep energy method for resolving concentration features in finite strain hyperelasticity.Journal of Computational Physics, 451:110839, 2022

    2022

  6. [14]

    A deep energy method for finite deformation hyperelasticity.European Journal of Mechanics-A/Solids, 80:103874, 2020

    Vien Minh Nguyen-Thanh, Xiaoying Zhuang, and Timon Rabczuk. A deep energy method for finite deformation hyperelasticity.European Journal of Mechanics-A/Solids, 80:103874, 2020

    2020

  7. [15]

    Characterizing possible failure modes in physics-informed neural networks.Advances in neural information processing systems, 34:26548–26560, 2021

    Aditi Krishnapriyan, Amir Gholami, Shandian Zhe, Robert Kirby, and Michael W Mahoney. Characterizing possible failure modes in physics-informed neural networks.Advances in neural information processing systems, 34:26548–26560, 2021

    2021

  8. [16]

    When and why pinns fail to train: A neural tangent kernel perspective.Journal of Computational Physics, 449:110768, 2022

    Sifan Wang, Xinling Yu, and Paris Perdikaris. When and why pinns fail to train: A neural tangent kernel perspective.Journal of Computational Physics, 449:110768, 2022

    2022

  9. [17]

    Neural operator: Learning maps between function spaces with applications to pdes.Journal of Machine Learning Research, 24(89):1–97, 2023

    Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: Learning maps between function spaces with applications to pdes.Journal of Machine Learning Research, 24(89):1–97, 2023

    2023

  10. [18]

    Neural operators for accelerating scientific simulations and design.Nature Reviews Physics, 6(5):320–328, 2024

    Kamyar Azizzadenesheli, Nikola Kovachki, Zongyi Li, Miguel Liu-Schiaffini, Jean Kossaifi, and Anima Anand- kumar. Neural operators for accelerating scientific simulations and design.Nature Reviews Physics, 6(5):320–328, 2024

    2024

  11. [19]

    Fourier neural operator for parametric partial differential equations

    Zongyi Li, Nikola Borislavov Kovachki, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar, et al. Fourier neural operator for parametric partial differential equations. InInternational Conference on Learning Representations, 2021

    2021

  12. [20]

    Enhancing fourier neural operators with local spatial features.arXiv preprint arXiv:2503.17797, 2025

    Chaoyu Liu, Davide Murari, Chris Budd, Lihao Liu, and Carola-Bibiane Schönlieb. Enhancing fourier neural operators with local spatial features.arXiv preprint arXiv:2503.17797, 2025

    2025 arXiv

  13. [21]

    Amortized fourier neural operators.Advances in Neural Information Processing Systems, 37:115001–115020, 2024

    Zipeng Xiao, Siqi Kou, Hao Zhongkai, Bokai Lin, and Zhijie Deng. Amortized fourier neural operators.Advances in Neural Information Processing Systems, 37:115001–115020, 2024

    2024

  14. [22]

    Tapas Tripura and Souvik Chakraborty. Wavelet neural operator for solving parametric partial differential equations in computational mechanics problems.Computer Methods in Applied Mechanics and Engineering, 404:115783, 2023

    2023

  15. [23]

    Mitigating spectral bias for the multiscale operator learning

    Xinliang Liu, Bo Xu, Shuhao Cao, and Lei Zhang. Mitigating spectral bias for the multiscale operator learning. Journal of Computational Physics, 506:112944, 2024

    2024

  16. [24]

    Toward a better understanding of fourier neural operators: Analysis and improvement from a spectral perspective.arXiv e-prints, pages arXiv–2404, 2024

    Shaoxiang Qin, Fuyuan Lyu, Wenhui Peng, Dingyang Geng, Ju Wang, Naiping Gao, Xue Liu, and Liangzhu Leon Wang. Toward a better understanding of fourier neural operators: Analysis and improvement from a spectral perspective.arXiv e-prints, pages arXiv–2404, 2024

    2024

  17. [25]

    U-fno—an enhanced fourier neural operator-based deep-learning model for multiphase flow.Advances in Water Resources, 163:104180, 2022

    Gege Wen, Zongyi Li, Kamyar Azizzadenesheli, Anima Anandkumar, and Sally M Benson. U-fno—an enhanced fourier neural operator-based deep-learning model for multiphase flow.Advances in Water Resources, 163:104180, 2022

    2022

  18. [26]

    Factorized fourier neural operators

    Alasdair Tran, Alexander Mathews, Lexing Xie, and Cheng Soon Ong. Factorized fourier neural operators. In The Eleventh International Conference on Learning Representations, 2023

    2023

  19. [27]

    Enhancing solutions for complex pdes: Introducing com- plementary convolution and equivariant attention in fourier neural operators.arXiv preprint arXiv:2311.12902, 2023

    Xuanle Zhao, Yue Sun, Tielin Zhang, and Bo Xu. Enhancing solutions for complex pdes: Introducing com- plementary convolution and equivariant attention in fourier neural operators.arXiv preprint arXiv:2311.12902, 2023

    2023 arXiv

  20. [28]

    Ht-net: Hierarchical transformer based operator learning model for multiscale pdes

    Xinliang Liu, Bo Xu, and Lei Zhang. Ht-net: Hierarchical transformer based operator learning model for multiscale pdes. 2022

    2022

  21. [29]

    D2no: Efficient handling of heterogeneous input function spaces with distributed deep neural operators.Computer Methods in Applied Mechanics and Engineering, 428:117084, 2024

    Zecheng Zhang, Christian Moya, Lu Lu, Guang Lin, and Hayden Schaeffer. D2no: Efficient handling of heterogeneous input function spaces with distributed deep neural operators.Computer Methods in Applied Mechanics and Engineering, 428:117084, 2024. 18 Multi-Input DeepONets in Ph...

    2024

  22. [30]

    Gnot: A general neural operator transformer for operator learning

    Zhongkai Hao, Zhengyi Wang, Hang Su, Chengyang Ying, Yinpeng Dong, Songming Liu, Ze Cheng, Jian Song, and Jun Zhu. Gnot: A general neural operator transformer for operator learning. InInternational Conference on Machine Learning, pages 12556–12569. PMLR, 2023

    2023

  23. [31]

    Vito: Vision transformer-operator.Computer Methods in Applied Mechanics and Engineering, 428:117109, 2024

    Oded Ovadia, Adar Kahana, Panos Stinis, Eli Turkel, Dan Givoli, and George Em Karniadakis. Vito: Vision transformer-operator.Computer Methods in Applied Mechanics and Engineering, 428:117109, 2024

    2024

  24. [32]

    Learning nonlinear operators via deeponet based on the universal approximation theorem of operators.Nature machine intelligence, 3(3):218–229, 2021

    Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators.Nature machine intelligence, 3(3):218–229, 2021

    2021

  25. [33]

    Tianping Chen and Hong Chen. Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems.IEEE transactions on neural networks, 6(4):911–917, 1995

    1995

  26. [34]

    Learning the solution operator of parametric partial differential equations with physics-informed deeponets.Science advances, 7(40):eabi8605, 2021

    Sifan Wang, Hanwen Wang, and Paris Perdikaris. Learning the solution operator of parametric partial differential equations with physics-informed deeponets.Science advances, 7(40):eabi8605, 2021

    2021

  27. [35]

    Luis Mandl, Somdatta Goswami, Lena Lambers, and Tim Ricken. Separable physics-informed deeponet: Breaking the curse of dimensionality in physics-informed machine learning.Computer Methods in Applied Mechanics and Engineering, 434:117586, 2025

    2025

  28. [36]

    Sequential deep operator networks (s-deeponet) for predicting full-field solutions under time-dependent loads.Engineering Applications of Artificial Intelligence, 127:107258, 2024

    Junyan He, Shashank Kushwaha, Jaewan Park, Seid Koric, Diab Abueidda, and Iwona Jasiuk. Sequential deep operator networks (s-deeponet) for predicting full-field solutions under time-dependent loads.Engineering Applications of Artificial Intelligence, 127:107258, 2024

    2024

  29. [37]

    Deep learning operator network for plastic deformation with variable loads and material properties.Engineering with Computers, 40(2):917–929, 2024

    Seid Koric, Asha Viswantah, Diab W Abueidda, Nahil A Sobh, and Kamran Khan. Deep learning operator network for plastic deformation with variable loads and material properties.Engineering with Computers, 40(2):917–929, 2024

    2024

  30. [38]

    A physics-informed variational deeponet for predicting crack path in quasi-brittle materials.Computer Methods in Applied Mechanics and Engineering, 391:114587, 2022

    Somdatta Goswami, Minglang Yin, Yue Yu, and George Em Karniadakis. A physics-informed variational deeponet for predicting crack path in quasi-brittle materials.Computer Methods in Applied Mechanics and Engineering, 391:114587, 2022

    2022

  31. [39]

    Learning mappings from iced airfoils to aerodynamic coefficients using a deep operator network.Journal of Aerospace Engineering, 36(5):04023035, 2023

    Tun Zhao, Weiqi Qian, Jie Lin, Hai Chen, Houjun Ao, Gong Chen, and Lei He. Learning mappings from iced airfoils to aerodynamic coefficients using a deep operator network.Journal of Aerospace Engineering, 36(5):04023035, 2023

    2023

  32. [40]

    Training a deep operator network as a surrogate solver for two-dimensional parabolic-equation models.The Journal of the Acoustical Society of America, 154(5):3276–3284, 2023

    Liang Xu, Haigang Zhang, and Minghui Zhang. Training a deep operator network as a surrogate solver for two-dimensional parabolic-equation models.The Journal of the Acoustical Society of America, 154(5):3276–3284, 2023

    2023

  33. [41]

    Ehsan Haghighat, Umair bin Waheed, and George Karniadakis. En-deeponet: An enrichment approach for enhancing the expressivity of neural operators with applications to seismology.Computer Methods in Applied Mechanics and Engineering, 420:116681, 2024

    2024

  34. [42]

    Improved generalization with deep neural operators for engineering systems: Path towards digital twin.Engineering Applications of Artificial Intelligence, 131:107844, 2024

    Kazuma Kobayashi, James Daniell, and Syed Bahauddin Alam. Improved generalization with deep neural operators for engineering systems: Path towards digital twin.Engineering Applications of Artificial Intelligence, 131:107844, 2024

    2024

  35. [43]

    From proxies to fields: Spatiotemporal reconstruction of global radiation from sparse sensor sequences.arXiv preprint arXiv:2506.12045, 2025

    Kazuma Kobayashi, Samrendra Roy, Seid Koric, Diab Abueidda, and Syed Bahauddin Alam. From proxies to fields: Spatiotemporal reconstruction of global radiation from sparse sensor sequences.arXiv preprint arXiv:2506.12045, 2025

    2025 arXiv

  36. [44]

    Deep neural operator-driven real-time inference to enable digital twin solutions for nuclear energy systems.Nature Scientific reports, 14(1):2101, 2024

    Kazuma Kobayashi et al. Deep neural operator-driven real-time inference to enable digital twin solutions for nuclear energy systems.Nature Scientific reports, 14(1):2101, 2024

    2024

  37. [45]

    Virtual sensing-enabled digital twin framework for real-time monitoring of nuclear systems leveraging deep neural operators.npj Materials Degradation, 9(1):21, 2025

    Raisa Hossain, Farid Ahmed, Kazuma Kobayashi, Seid Koric, Diab Abueidda, and Syed Bahauddin Alam. Virtual sensing-enabled digital twin framework for real-time monitoring of nuclear systems leveraging deep neural operators.npj Materials Degradation, 9(1):21, 2025

    2025

  38. [46]

    Deep operator learning- based surrogate models with uncertainty quantification for optimizing internal cooling channel rib profiles

    Izzet Sahin, Christian Moya, Amirhossein Mollaali, Guang Lin, and Guillermo Paniagua. Deep operator learning- based surrogate models with uncertainty quantification for optimizing internal cooling channel rib profiles. International Journal of Heat and Mass Transfer, 219:124813, 2024

    2024

  39. [47]

    Seid Koric and Diab W Abueidda. Data-driven and physics-informed deep learning operators for solution of heat conduction equation with parametric heat source.International Journal of Heat and Mass Transfer, 203:123809, 2023

    2023

  40. [48]

    On the training and generalization of deep operator networks.SIAM Journal on Scientific Computing, 46(4):C273–C296, 2024

    Sanghyun Lee and Yeonjong Shin. On the training and generalization of deep operator networks.SIAM Journal on Scientific Computing, 46(4):C273–C296, 2024. 19 Multi-Input DeepONets in Physics and Multiphysics

    2024

  41. [49]

    Multifidelity deep operator networks for data-driven and physics-informed problems.Journal of Computational Physics, 493:112462, 2023

    Amanda A Howard, Mauro Perego, George Em Karniadakis, and Panos Stinis. Multifidelity deep operator networks for data-driven and physics-informed problems.Journal of Computational Physics, 493:112462, 2023

    2023

  42. [50]

    Deepm&mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks.Journal of Computational Physics, 436:110296, 2021

    Shengze Cai, Zhicheng Wang, Lu Lu, Tamer A Zaki, and George Em Karniadakis. Deepm&mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks.Journal of Computational Physics, 436:110296, 2021

    2021

  43. [51]

    Zhiping Mao, Lu Lu, Olaf Marxen, Tamer A Zaki, and George Em Karniadakis. Deepm&mnet for hypersonics: Predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators.Journal of computational physics, 447:110698, 2021

    2021

  44. [52]

    Zhongyi Jiang, Min Zhu, and Lu Lu. Fourier-mionet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration.Reliability Engineering & System Safety, 251:110392, 2024

    2024

  45. [53]

    Physics-informed neural operator for learning partial differential equations.ACM/JMS Journal of Data Science, 1(3):1–27, 2024

    Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadenesheli, and Anima Anandkumar. Physics-informed neural operator for learning partial differential equations.ACM/JMS Journal of Data Science, 1(3):1–27, 2024

    2024

  46. [54]

    High-fidelity multiphysics modelling for rapid predictions using physics-informed parallel neural operator.arXiv preprint arXiv:2502.19543, 2025

    Biao Yuan, He Wang, Yanjie Song, Ana Heitor, and Xiaohui Chen. High-fidelity multiphysics modelling for rapid predictions using physics-informed parallel neural operator.arXiv preprint arXiv:2502.19543, 2025

    2025 arXiv

  47. [55]

    Pretraining codomain attention neural operators for solving multiphysics pdes.Advances in Neural Information Processing Systems, 37:104035–104064, 2024

    Md Ashiqur Rahman, Robert Joseph George, Mogab Elleithy, Daniel Leibovici, Zongyi Li, Boris Bonev, Colin White, Julius Berner, Raymond A Yeh, Jean Kossaifi, et al. Pretraining codomain attention neural operators for solving multiphysics pdes.Advances in Neural Information Proc...

    2024

  48. [56]

    Diffeomorphism neural operator for various domains and parameters of partial differential equations.Communications Physics, 8(1):15, 2025

    Zhiwei Zhao, Changqing Liu, Yingguang Li, Zhibin Chen, and Xu Liu. Diffeomorphism neural operator for various domains and parameters of partial differential equations.Communications Physics, 8(1):15, 2025

    2025

  49. [57]

    Deep neural operator enabled digital twin modeling for additive manufacturing.Advances in Computational Science and Engineering, 2(3):174–201, 2024

    Ning Liu, Xuxiao Li, Manoj R Rajanna, Edward W Reutzel, Brady Sawyer, Prahalada Rao, Jim Lua, Nam Phan, and Yue Yu. Deep neural operator enabled digital twin modeling for additive manufacturing.Advances in Computational Science and Engineering, 2(3):174–201, 2024

    2024

  50. [58]

    Kazuma Kobayashi et al. Explainable, interpretable, and trustworthy AI for an intelligent digital twin: A case study on remaining useful life.Engineering Applications of Artificial Intelligence, 129:107620, 2024

    2024

  51. [59]

    Digital twin-centered hybrid data-driven multi-stage deep learning framework for enhanced nuclear reactor power prediction.Energy and AI, 19:100450, 2025

    James Daniell et al. Digital twin-centered hybrid data-driven multi-stage deep learning framework for enhanced nuclear reactor power prediction.Energy and AI, 19:100450, 2025

    2025

  52. [60]

    Adversarial vulnerabilities in neural operator digital twins: Gradient-free attacks on nuclear thermal-hydraulic surrogates.arXiv preprint arXiv:2603.22525, 2026

    Samrendra Roy et al. Adversarial vulnerabilities in neural operator digital twins: Gradient-free attacks on nuclear thermal-hydraulic surrogates.arXiv preprint arXiv:2603.22525, 2026

    2026

  53. [61]

    Graph neural operator towards edge deployability and portability for sparse-to-dense, real-time virtual sensing on irregular grids.arXiv preprint arXiv:2604.01802, 2026

    William Howes et al. Graph neural operator towards edge deployability and portability for sparse-to-dense, real-time virtual sensing on irregular grids.arXiv preprint arXiv:2604.01802, 2026

    2026 arXiv

  54. [62]

    Sensing without colocation: Operator-based virtual instrumentation for domains beyond physical reach.arXiv preprint arXiv:2510.18041, 2025

    Jay Phil Yoo et al. Sensing without colocation: Operator-based virtual instrumentation for domains beyond physical reach.arXiv preprint arXiv:2510.18041, 2025

    2025

  55. [63]

    Alam et al

    S. Alam et al. Neutronic investigation of alternative & composite burnable poisons for the soluble-boron-free and long life civil marine small modular reactor cores.Scientific reports, 9(1):19591, 2019

    2019

  56. [64]

    Alam et al

    S. Alam et al. Small modular reactor core design for civil marine propulsion using micro-heterogeneous duplex fuel. part i: Assembly-level analysis.Nuclear Engineering and Design, 346:157–175, 2019

    2019

  57. [65]

    Alam et al

    S. Alam et al. Small modular reactor core design for civil marine propulsion using micro-heterogeneous duplex fuel. part ii: Whole-core analysis.Nuclear Engineering and Design, 346:176–191, 2019

    2019

  58. [66]

    Farid Ahmed et al. Numerical investigation of the thermo-hydraulic performance of water-based nanofluids in a dimpled channel flow using al2o3, cuo, and hybrid al2o3–cuo as nanoparticles.Heat transfer, 50(5):5080–5105, 2021

    2021

  59. [67]

    Multiphysics bench: Benchmarking and investigating scientific machine learning for multiphysics pdes.arXiv preprint arXiv:2505.17575, 2025

    Changfan Yang, Lichen Bai, Yinpeng Wang, Shufei Zhang, and Zeke Xie. Multiphysics bench: Benchmarking and investigating scientific machine learning for multiphysics pdes.arXiv preprint arXiv:2505.17575, 2025

    2025 arXiv

  60. [68]

    Mionet: Learning multiple-input operators via tensor product.SIAM Journal on Scientific Computing, 44(6):A3490–A3514, 2022

    Pengzhan Jin, Shuai Meng, and Lu Lu. Mionet: Learning multiple-input operators via tensor product.SIAM Journal on Scientific Computing, 44(6):A3490–A3514, 2022

    2022

  61. [69]

    Gstools v1

    Sebastian Müller, Lennart Schüler, Alraune Zech, and Falk Heße. Gstools v1. 3: a toolbox for geostatistical modelling in python.Geoscientific Model Development, 15(7):3161–3182, 2022

    2022

  62. [70]

    Dolfinx: the next generation fenics problem solving environment.Preprint, 2023

    Igor A Baratta, Joseph P Dean, Jørgen S Dokken, Michal Habera, Jack HALE, Chris N Richardson, Marie E Rognes, Matthew W Scroggs, Nathan Sime, and Garth N Wells. Dolfinx: the next generation fenics problem solving environment.Preprint, 2023. 20 Multi-Input DeepONets in Physics ...

    2023

  63. [71]

    Foivos Antoulinakis, Daniel Chernin, Peng Zhang, and Y . Y . Lau. Effects of temperature dependence of electrical and thermal conductivities on the joule heating of a one dimensional conductor.Journal of Applied Physics, 120(13):135105, 2016

    2016

  64. [72]

    Electrochemical–thermal model of a lithium-ion battery.Energies, 18(7):1764, 2025

    Paul Kalungi and James Menart. Electrochemical–thermal model of a lithium-ion battery.Energies, 18(7):1764, 2025

    2025

  65. [73]

    J. A. Lopez Molina, M. J. Rivera, and Enrique J. Berjano. Electrical-thermal analytical modeling of monopolar rf thermal ablation of biological tissues: determining the circumstances under which tissue temperature reaches a steady state.Mathematical Biosciences and Engineering...

    2016

  66. [74]

    Coupled electro-thermal field in a high current electrolysis cell or liquid metal batteries.Royal Society Open Science, 5(2):171309, 2018

    Ze Sun, Liwei Cai, Haiou Ni, Gui-Min Lu, and Jian-Guo Yu. Coupled electro-thermal field in a high current electrolysis cell or liquid metal batteries.Royal Society Open Science, 5(2):171309, 2018

    2018

  67. [75]

    Feulvarch, V

    E. Feulvarch, V . Robin, and J.-M. Bergheau. Resistance spot welding simulation: a general finite element formulation of electrothermal contact conditions.Journal of Materials Processing Technology, 153–154:436–441, 2004

    2004

  68. [76]

    Review on modeling and simulation of continuous casting.steel research international, 89(1):1700312, 2018

    Brian G Thomas. Review on modeling and simulation of continuous casting.steel research international, 89(1):1700312, 2018

    2018

  69. [77]

    Continuous casting of steel

    Seppo Louhenkilpi. Continuous casting of steel. In Seshadri Seetharaman, editor,Treatise on Process Metallurgy, pages 373–434. Elsevier, Boston, 2014

    2014

  70. [78]

    Multiphysics model of metal solidification on the continuum level.Numerical Heat Transfer, Part B: Fundamentals, 58(6):371–392, 2010

    Seid Koric, Lance C Hibbeler, Rui Liu, and Brian G Thomas. Multiphysics model of metal solidification on the continuum level.Numerical Heat Transfer, Part B: Fundamentals, 58(6):371–392, 2010

    2010

  71. [79]

    Simple constitutive equations for steel at high temperature.Metallurgical Transactions A, 23:903–918, 1992

    Patrick F Kozlowski, Brian G Thomas, Jean A Azzi, and Hao Wang. Simple constitutive equations for steel at high temperature.Metallurgical Transactions A, 23:903–918, 1992

    1992

  72. [80]

    Coupled thermo-mechanical finite-element model with application to initial solidification.The University of Illinois at Urbana-Champaign, 1996

    Hong Zhu. Coupled thermo-mechanical finite-element model with application to initial solidification.The University of Illinois at Urbana-Champaign, 1996

    1996

  73. [81]

    Efficient thermo-mechanical model for solidification processes.International journal for numerical methods in engineering, 66(12):1955–1989, 2006

    Seid Koric and Brian G Thomas. Efficient thermo-mechanical model for solidification processes.International journal for numerical methods in engineering, 66(12):1955–1989, 2006

    1955

  74. [82]

    Dassault Systèmes Simulia Corp., Providence, RI, 2022

    Dassault Systèmes.Abaqus/Standard User’s Manual, Version 2022. Dassault Systèmes Simulia Corp., Providence, RI, 2022. Accessed via licensed software documentation

    2022

  75. [83]

    Multiphysics modeling of continuous casting of stainless steel.Journal of Materials Processing Technology, 278:116469, 2020

    Matthew LS Zappulla, Seong-Mook Cho, Seid Koric, Hyoung-Jun Lee, Seon-Hyo Kim, and Brian G Thomas. Multiphysics modeling of continuous casting of stainless steel.Journal of Materials Processing Technology, 278:116469, 2020

    2020

  76. [84]

    Qibang Liu, Weiheng Zhong, Seid Koric, and Hadi Meidani. Sequential neural operator transformer for high- fidelity surrogates of time-dependent non-linear partial differential equations.Engineering Applications of Artificial Intelligence, 172:114428, 2026

    2026

  77. [85]

    Sequential deep operator neural networks for plastic and thermo-viscoplastic transient material behavior.International Journal of Plasticity, 201:104689, 2026

    Jaewan Park, Kazuma Kobayashi, Qibang Liu, Amar Alem Koric, Diab W Abueidda, Syed Bahauddin Alam, and Seid Koric. Sequential deep operator neural networks for plastic and thermo-viscoplastic transient material behavior.International Journal of Plasticity, 201:104689, 2026

    2026

  78. [86]

    Junyan He, Seid Koric, Diab Abueidda, Ali Najafi, and Iwona Jasiuk. Geom-deeponet: A point-cloud-based deep operator network for field predictions on 3d parameterized geometries.Computer Methods in Applied Mechanics and Engineering, 429:117130, 2024

    2024

  79. [87]

    Sensor degradation in nuclear reactor pressure vessels: the overlooked factor in remaining useful life prediction.npj Materials Degradation, 8(1):71, 2024

    R Hossain et al. Sensor degradation in nuclear reactor pressure vessels: the overlooked factor in remaining useful life prediction.npj Materials Degradation, 8(1):71, 2024

    2024

  80. [88]

    Deep learning for plasticity and thermo- viscoplasticity.International Journal of Plasticity, 136:102852, 2021

    Diab W Abueidda, Seid Koric, Nahil A Sobh, and Huseyin Sehitoglu. Deep learning for plasticity and thermo- viscoplasticity.International Journal of Plasticity, 136:102852, 2021

    2021

  81. [89]

    Single-branch

    National Center for Supercomputing Applications (NCSA). Delta user documentation. https://docs.ncsa. illinois.edu/systems/delta/en/latest/index.html, 2025. Accessed: January 25, 2025. Acknowledgments The authors thank the National Center for Supercomputing Applications (NCSA),...

    2025

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.