Pith. sign in

REVIEW 3 major objections 5 minor 59 references

MoNo: Multiscale Optimal Transport Neural Operator for Solving PDEs on General Geometries

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read MoNo claims that replacing softmax-based latent-token projection with an optimal-transport coupling, CoTAP, yields balanced stable multiscale latent spaces that beat prior neural operators on seven PDE benchmarks at lower computational…

desk verdict Solid architecture paper with a real AirfRANS confound; deserves serious refereeing, not desk rejection. read the letter →

arxiv 2608.09764 v1 pith:63KHTLFF submitted 2026-08-10 cs.LG cs.AI

Zijiang Yang , Xiaomeng Wu , Dongmei Fu This is my paper · ORCID
classification cs.LGcs.AI MSC 68T0749Q22
keywords neuraloperatoroptimaltransportSinkhorniterationlatenttokenassignmenttransformerattentionpartialdifferentialequationsmultiscalearchitecturegeneralgeometries
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

Transformer-based neural operators compress the spatial observations of a partial differential equation into a small set of latent tokens, but the softmax projection used to build those tokens does not guarantee they are used evenly; in a hierarchy of latent spaces the imbalance is inherited and worsens until deeper tokens collapse and carry almost no information. This paper proposes MoNo, a neural operator that replaces softmax projection with CoTAP, a cross-scale assignment formed as an entropy-regularized optimal transport problem with uniform marginal constraints: every source point must send the same total mass and every latent token must receive the same total mass. The balanced, bidirectional projections that CoTAP derives from one shared transport plan make a four-level progressive multiscale hierarchy stable, which the authors argue is what lets the model learn long-range physical interactions cheaply on arbitrary geometries. On seven PDE benchmarks, including the real-world AirfRANS dataset, the paper reports lower relative L2 errors than existing state-of-the-art neural operators while using far fewer FLOPs at large point counts. If the claim is right, balanced optimal-transport assignment is a practical alternative to softmax projection for building latent spaces in Transformer neural operators.

What carries the argument

The central object is CoTAP (Cross-scale Optimal Transport Assignment and Projection), the latent-space construction method that carries the whole argument. Between adjacent spaces $\Omega^{l-1}$ and $\Omega^{l}$, an MLP projector builds an initial score matrix $S_{\mathrm{init}}$, and CoTAP solves the entropy-regularized optimal transport problem $\arg\max_{S\in C^{l-1,l}}\langle S, S_{\mathrm{init}}\rangle + \tau H(S)$ over the transport polytope with uniform marginal constraints, using log-domain Sinkhorn iterations (alternating row and column scalings toward the uniform totals). The solution $S_{\mathrm{ot}}$ has every row summing to $1/N_{l-1}$ and every column to $1/N_l$, so the encoding projection $S_{\mathrm{enc}} = N_l S_{\mathrm{ot}}$ and the decoding projection $S_{\mathrm{dec}} = N_{l-1} S_{\mathrm{ot}}$ are balanced bidirectional projections derived from one shared plan: no latent token can be over- or under-assigned, and the two directions transfer consistent mass rather than arbitrary learnable mixing. Reusing the same construction at each of the four latent scales is what makes progressive compression stable, because imbalance is never inherited from a previous level. A Triton-based fused Sinkhorn solver (CoTAP Iteration) with an explicitly derived reverse-mode gradient keeps the OT solve cheap, and it is this solver that carries the efficiency claim at large point counts.

What would settle it

Retrain the strongest baseline, LinearNO, on AirfRANS under MoNo's exact training protocol (each sampled case contains all surface points plus randomly sampled volume points, batch size 4, the same number of optimization steps), and recompute the surrounding-field, surface-field, and lift-coefficient errors; if the gaps over MoNo shrink below the claimed 18.2% and 83.1% improvements, the state-of-the-art claim is not robust to the protocol difference.

Watch

Extended reading notes

Core claim

The paper's central claim is that the instability and imbalance of softmax-based latent projection, rather than the attention mechanism itself, is what limits the accuracy and efficiency of Transformer neural operators, and that this limit can be lifted by constructing latent spaces with optimal transport. MoNo diagnoses a specific failure mode: unconstrained learnable assignment matrices, normalized only row- or column-wise by softmax, concentrate assignment mass on a few latent tokens, and when such projections are stacked into a hierarchy the imbalance is inherited and amplified until deeper latent spaces suffer token collapse. CoTAP replaces that projection with a shared transport plan: an MLP produces an initial score matrix, which is projected onto the transport polytope with uniform row and column marginals via entropy-regularized Sinkhorn iterations, and the normalized plan is rescaled once to give the encoding matrix and once to give the decoding matrix, so both directions preserve the same mass-transfer relation and each latent token receives equal total mass. On top of this, MoNo stacks four progressively compressed latent spaces with token counts halving at each level, and proves that (i) one CoTAP projection followed by one latent self-attention layer is an input-dependent integral kernel operator on the original physical domain, and (ii) the full hierarchy defines a neural operator. Empirically the paper reports the best relative L2 errors on all seven benchmarks, including the AirfRANS surrounding-field, surface-field, and lift-coefficient metrics, and reports 23.46 GFLOPs for MoNo-light at 65,536 points versus 132.83 GFLOPs for LinearNO at a comparable parameter count.

Load-bearing premise

The load-bearing premise is that the AirfRANS state-of-the-art comparison is fair even though MoNo is trained under a modified sampling scheme (each sampled case contains all surface points plus randomly sampled volume points, with batch size 4) that the published baselines did not use; if the baselines were retrained under the same scheme, their errors could differ and the claimed margins could shrink.

Editorial extensions

If this is right

  • Replacing softmax-based projection with the balanced OT coupling lowers error even at matched model scale: MoNo-light, with a parameter count comparable to prior methods, improves relative L2 error on Airfoil, Plasticity, NS2D, and Elasticity by at least 2.0%, 9.1%, 3.7%, and 16.0% respectively.
  • The multiscale hierarchy only pays off when assignments are stable: in the ablation, adding progressive multiscale modeling without CoTAP provides only limited improvements, while CoTAP and the hierarchy together cut Elasticity error from 0.0077 to 0.0042.
  • Balanced assignment converts extra observation points into accuracy: on Darcy, MoNo's error falls steadily as resolution rises from 85×85 to 211×211, whereas Transolver and LinearNO degrade with resolution, and at 211×211 MoNo leads by at least 13.2%.
  • The same latent construction transfers out of distribution: on AirfRANS OOD Reynolds and OOD angle-of-attack extrapolation, MoNo reports the best surrounding- and surface-field errors and the best lift-coefficient Spearman correlation.
  • Efficiency is not sacrificed: at 65,536 observation points MoNo-light computes 23.46 GFLOPs versus 132.83 GFLOPs for LinearNO at comparable parameter count, and full MoNo at 97.26 GFLOPs still undercuts Transolver++ (121.17).

Reading between the lines

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

  • A natural extension the paper does not pursue is replacing uniform marginals with physically informed non-uniform ones (for example, allocating more mass to high-gradient or boundary regions); whether that beats uniform marginals would test whether balance itself, rather than entropy regularization, is the source of the gains.
  • The paper reports that its AirfRANS training protocol differs from the baselines' published one (all surface points plus randomly sampled volume points, batch size 4, versus fully random sampling with batch size 1); whether that difference changes baseline errors is untested, so the claimed margins are the load-bearing unknown.
  • The resolution-stability of CoTAP assignments suggests the learned coupling behaves like a discretization-invariant transport map; if so, a trained MoNo could be evaluated zero-shot on re-meshed or upsampled versions of the same geometry, which the paper does not test.
  • Because Theorem 1 recasts CoTAP-plus-attention as an input-dependent integral kernel on the original domain, the learned hierarchy can be compared spectrally with Fourier-type operators; examining the kernel's effective rank across scales would show whether the multiscale compression is genuinely exploiting long-range structure rather than just reducing point count.
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

3 major / 5 minor

Summary. The paper proposes MoNo, a multiscale Transformer-based neural operator for PDEs on general geometries. The central component is CoTAP, which constructs cross-scale assignments between adjacent latent spaces by solving an entropy-regularized optimal transport problem with uniform marginals, thereby producing balanced bidirectional projection matrices. The authors argue that this avoids token imbalance and collapse that arise with softmax-based projections, and enables stable deep multiscale hierarchies. The method is evaluated on six standard benchmarks plus AirfRANS, with claims of improved accuracy and computational efficiency; a fused Sinkhorn implementation is described, and theoretical appendices provide integral-operator and neural-operator interpretations.

Significance. If the empirical claims hold under controlled comparison, the paper makes a useful contribution: it identifies a concrete failure mode of softmax latent-space construction (token imbalance and collapse), offers an OT-based alternative with uniform marginal constraints, and demonstrates a practical fused Sinkhorn implementation. The multiscale hierarchy is a natural and potentially efficient design. The ablations in Table 6 support the qualitative importance of CoTAP and multiscale modeling, and the visualizations in Figures 2, 4, and 5 provide intuitive evidence of more balanced assignment patterns. The main caveat is that the headline 'outperforms SOTA' claim, especially on AirfRANS, is currently supported by a comparison protocol that differs between MoNo and the baselines. The theoretical representation as an integral operator is a useful framing, though the stated theorems are stronger than the assumptions used in their proofs.

major comments (3)
  1. [Appendix F, 'Training'] The AirfRANS comparison in Table 2 is confounded by a training-protocol mismatch. The authors state that 'the fully random sampling strategy with a batch size of 1 adopted in previous works causes severe training instability' and therefore use 'all surface points together with randomly sampled volume points' with a batch size of 4. The baseline numbers in Table 2 are taken from prior papers that used the fully random protocol. Because surface points are a tiny fraction of the 32,000 sampled points under random sampling, MoNo's guaranteed surface coverage in every batch provides a supervision advantage that is independent of CoTAP. This directly affects the largest claimed gains: surface error 0.0013 vs. 0.0077 (an 83.1% relative improvement) and lift-coefficient error 0.0415 vs. 0.0491. To support the 'outperforms SOTA' claim, the baselines should be retrained under the same sampling and batching protocol, or a matched protocol should be used, and the resulting numbers reported.
  2. [Appendix F, 'Training'; all experimental sections] All experiments are run with a single random seed (seed 0), and no error bars or variance estimates are reported anywhere. Several close margins in Tables 1 and 3 (e.g., Airfoil MoNo 0.0048 vs. LinearNO 0.0049; Darcy 141x141 MoNo 0.0049 vs. LinearNO 0.0053) may not be statistically meaningful without multiple seeds. The authors should report means and standard deviations over at least three to five seeds for the main tables, or at minimum for the closest comparisons. This is essential for evaluating the claimed state-of-the-art accuracy.
  3. [Appendix B, Theorem 1 and Eq. (8); Appendix D] Theorem 1 is stated for any bounded measurable physical state field and any continuous anchor token field, but the proof begins by assuming that the continuous CoTAP coupling 'admits a nonnegative measurable density' with respect to the product measure, and the kernel is defined in terms of that density. The existence and regularity of such a density for arbitrary continuous G0 is not established. Therefore the theorem as stated is stronger than what is proven; it should be restated as a conditional result that assumes the existence of the CoTAP coupling density. The same issue applies to Theorem 2, which assumes such densities at every level. This does not invalidate the empirical method, but it overstates the theoretical contribution.
minor comments (5)
  1. [Figure 1] The label 'LinerNO' in Figure 1(b) is a typo and should read 'LinearNO'.
  2. [Evaluation at Multiple Resolutions] The text states that 'Both MoNo and MoNo-light effectively utilize the additional information provided by higher-resolution observations', but in Table 3 MoNo-light's error at 211x211 (0.0059) is higher than at 141x141 (0.0057). The sentence should be qualified to acknowledge this non-monotonic behavior.
  3. [Abstract and Conclusion] The claim that MoNo 'outperforms existing state-of-the-art neural operators' is repeated in the abstract and conclusion without qualification. Given the protocol mismatch on AirfRANS and the lack of multiple seeds, this statement should be softened or explicitly conditioned on the comparisons performed.
  4. [Table 2 and Table 12] The citation style for Transolver++ is inconsistent: Table 2 uses 'Transolver++ (2025)' while the text and reference list use '(Luo et al. 2025a)'.
  5. [Appendix B, Eq. (12)-(13)] The attention kernel alpha is stated to satisfy alpha >= 0 and the integral normalization in Eq. (12) only after it is used in Eq. (13) to define beta. Moving the assumptions forward would make the proof easier to follow.

Circularity Check

1 steps flagged · score 4.0 of 10

Empirical SOTA/efficiency claims are independent; Theorem 1's integral-kernel representation is a definitional identity (the kernel is defined as the composition), but it is not load-bearing for the benchmark results.

  1. self definitional [Appendix B, Eqs. (13), (17), (18) (proof of Theorem 1)]
    "The inner integral in Equation (16) defines the effective original-space kernel: κ(p,ξ) := Z Ω1 β(p,η)r(ξ,η)dµ1(η), (17). It follows that, for µ0-almost every p ∈ Ω0, we can get: T(G0,F0enc)(p) = Z Ω0 κ(p,ξ)F0enc(ξ)Wv dµ0(ξ), (18), which proves the integral-operator representation on Ω0."

    Theorem 1 asserts that the composite operator T admits an integral kernel κ on Ω0. The proof does not derive κ from an independent condition; it defines κ(p,ξ) as exactly the composition of the CoTAP decoding kernel r(p,ζ), the attention kernel α(ζ,η), and the CoTAP encoding kernel r(ξ,η), after marginalizing over latent variables (Eqs. 13 and 17). Substituting that definition into Eq. (16) and applying Fubini makes Eq. (18) true by construction. The integral-operator representation is therefore a definitional identity (a composition of kernels is a kernel), not a derived prediction. This step is not load-bearing for the empirical benchmarking and efficiency claims, which rest on Tables 1-5, but as a stated first-principles contribution it reduces to its own definition.

full rationale

MoNo's central claim is empirical: lower relative L2 errors on standard and AirfRANS benchmarks (Tables 1-4) and lower GFLOPs at 65,536 points (Table 5), produced by the concrete MoNo architecture with CoTAP. Those results do not depend on Theorem 1 or Theorem 2, and Table 6 provides an internal ablation control. The only definitional reduction I can exhibit is Theorem 1's kernel construction in Appendix B, where the kernel kappa is defined as the composition of the CoTAP and attention kernels, making the conclusion a tautology. The AirfRANS comparison in Table 2 uses a modified sampling protocol (Appendix F: each sampled case contains all surface points together with randomly sampled volume points, batch size 4) while borrowing baseline numbers from prior work trained with fully random sampling; that is a comparison-fairness confound, not a circularity, so I do not score it here. The paper's self-citations (Yang, Qiu, and Fu 2023; Wang, Yang, Fu, and Liu 2025) appear only in related-work context and are not load-bearing. No uniqueness theorem, ansatz-smuggling citation, or fitted-parameter-as-prediction was found. Overall, the central empirical claims are self-contained, with one non-load-bearing theoretical statement that is circular by definition, giving a score of 4.

Assumptions & free parameters 6 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physical entities. The free parameters are standard model and training hyperparameters, selected by validation on the same benchmarks that are later reported. The key assumptions are regularity of the transport coupling for the operator-theoretic interpretation and the comparability of benchmark data across papers.

free parameters (6)
  • OT temperature tau = 1.0
    Controls assignment sharpness; chosen by validation as shown in Appendix I Table 14, where values 0.5, 1.0, and 4.0 were tested.
  • Sinkhorn iterations T = 8
    Number of iterations in CoTAP; chosen by validation, with T=8 giving better Elasticity results than T=12 (Appendix I Table 14).
  • First latent token count N1 = 512 (MoNo-light), 1024 (MoNo)
    Architectural capacity choice, ablated in Appendix I Table 13; larger N1 improves until saturation.
  • Token dimensions Dg, Df = 96 (MoNo-light), 192 (MoNo)
    Capacity choice, ablated in Appendix I Table 13.
  • Token reduction factor = 2
    Compression between adjacent latent spaces; ablated in Appendix I Table 15, where x4 loses information and x1 lacks hierarchy.
  • Learnable scale for P1 weight normalization = 3.0 (initial)
    Introduced to prevent MLP weight-norm growth during training; not otherwise justified.
assumptions (3)
  • domain assumption Standard self-attention admits an integral-operator representation with an input-dependent attention kernel.
    Invoked in Appendix B, Eq. (11), citing Kovachki et al. and the neural operator literature. This is the basis for interpreting latent attention as a nonlocal integral operator on the latent space.
  • ad hoc to paper The continuous CoTAP coupling between adjacent spaces admits a nonnegative measurable density with respect to the product measure.
    Stated in Appendix B, Eq. (8) and repeated in Appendix D. The discrete construction is defined first, and then a continuous density is assumed to exist, which is not guaranteed by the architecture or by the training procedure.
  • domain assumption The benchmark datasets, data splits, and ground-truth physical fields are accurate and match prior work.
    All reported errors depend on the correctness of the benchmark implementations and the claim that 'we mainly adopt benchmark configurations and data splits as in previous works', stated in Appendix F.

how reviews work

0 comments
Cite this review

Pith. "Pith review of MoNo: Multiscale Optimal Transport Neural Operator for Solving PDEs on General Geometries." pith.science (2026). https://pith.science/paper/63KHTLFF

@misc{pith2026260809764,
  author       = {Pith},
  title        = {Pith review of: MoNo: Multiscale Optimal Transport Neural Operator for Solving PDEs on General Geometries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/63KHTLFF}},
  note         = {Machine review of arXiv:2608.09764}
}
read the original abstract

Transformer-based neural operators have achieved substantial progress in solving Partial Differential Equations (PDEs) by projecting spatial observations into compact latent tokens and learning physical interactions in latent spaces. However, we reveal that existing learnable projection mechanisms cannot ensure stable and balanced assignments from observation points to latent tokens, causing some latent tokens to be over-assigned while others remain underutilized. This limitation further restricts the design of hierarchical architectures, as assignment imbalance is continuously inherited and amplified across latent spaces, eventually causing severe token collapse in deeper spaces. To address these issues, we propose MoNo (Multiscale Optimal Transport Neural Operator), a progressive multiscale neural operator that efficiently solves PDEs on general geometries through stable latent-space construction. At its core is CoTAP (Cross-scale Optimal Transport Assignment and Projection), a novel latent-space construction method that formulates cross-space assignment between adjacent spaces as an entropy-regularized optimal transport problem, thereby constructing balanced bidirectional projections and stable latent spaces. CoTAP also ensures stable information transfer across multiple latent spaces, further enabling multiscale architectures on general geometries, which in turn support more efficient learning of long-range physical interactions. Extensive experiments demonstrate that MoNo outperforms existing state-of-the-art neural operators in both prediction performance and computational efficiency. Code is available at https://github.com/ZijiangY1116/MoNo.

Figures

Figures reproduced from arXiv: 2608.09764 by the authors.

Figure 1
Figure 1. Comparison of MoNo with State-Of-The-Art [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Comparison of assignment patterns on Darcy at [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Illustration of MoNo. (a) Given a physical system, MoNo first embeds the spatial positions and input observations into [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Visualization of assignment patterns and prediction errors. (a) The assignment patterns learned by MoNo on NS2D. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Visualization of assignments across different spa [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 18 canonical work pages

  1. [1]

    Communication, Simulation, and Intelligent Agents: Implications of Personal Intelligent Machines for Medical Education

    Clancey, William J. Communication, Simulation, and Intelligent Agents: Implications of Personal Intelligent Machines for Medical Education. Proceedings of the Eighth International Joint Conference on Artificial Intelligence (IJCAI-83)

  2. [2]

    Classification Problem Solving

    Clancey, William J. Classification Problem Solving. Proceedings of the Fourth National Conference on Artificial Intelligence

  3. [3]

    , title =

    Robinson, Arthur L. , title =. 1980 , doi =. https://science.sciencemag.org/content/208/4447/1019.full.pdf , journal =

    1980

  4. [4]

    New Ways to Make Microcircuits Smaller---Duplicate Entry

    Robinson, Arthur L. New Ways to Make Microcircuits Smaller---Duplicate Entry. Science

  5. [5]

    Clancey and Glenn Rennels , abstract =

    Diane Warner Hasling and William J. Clancey and Glenn Rennels , abstract =. Strategic explanations for a diagnostic consultation system , journal =. 1984 , issn =. doi:https://doi.org/10.1016/S0020-7373(84)80003-6 , url =

    doi:10.1016/s0020-7373(84)80003-6 1984

  6. [6]

    and Rennels, Glenn R

    Hasling, Diane Warner and Clancey, William J. and Rennels, Glenn R. and Test, Thomas. Strategic Explanations in Consultation---Duplicate. The International Journal of Man-Machine Studies

  7. [7]

    Poligon: A System for Parallel Problem Solving

    Rice, James. Poligon: A System for Parallel Problem Solving

  8. [8]

    Transfer of Rule-Based Expertise through a Tutorial Dialogue

    Clancey, William J. Transfer of Rule-Based Expertise through a Tutorial Dialogue

Show all 59 references
  1. [9]

    The Engineering of Qualitative Models

    Clancey, William J. The Engineering of Qualitative Models

  2. [10]

    2023 , eprint=

    Attention Is All You Need , author=. 2023 , eprint=

    2023

  3. [11]

    Pluto: The 'Other' Red Planet

    NASA. Pluto: The 'Other' Red Planet

  4. [12]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    Transolver is a linear transformer: Revisiting physics-attention through the lens of linear attention , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  5. [13]

    arXiv preprint arXiv:2502.02414 , year=

    Transolver++: An accurate neural solver for pdes on million-scale geometries , author=. arXiv preprint arXiv:2502.02414 , year=

    arXiv

  6. [14]

    arXiv preprint arXiv:2402.02366 , year=

    Transolver: A fast transformer solver for pdes on general geometries , author=. arXiv preprint arXiv:2402.02366 , year=

    arXiv

  7. [15]

    Advances in Neural Information Processing Systems , volume=

    Latent neural operator for solving forward and inverse pde problems , author=. Advances in Neural Information Processing Systems , volume=

  8. [16]

    arXiv preprint arXiv:2301.12664 , year=

    Solving high-dimensional pdes with latent spectral models , author=. arXiv preprint arXiv:2301.12664 , year=

    arXiv

  9. [17]

    arXiv preprint arXiv:2310.12487 , year=

    Improved operator learning by orthogonal attention , author=. arXiv preprint arXiv:2310.12487 , year=

    arXiv

  10. [18]

    Advances in Neural Information Processing Systems , volume=

    Scalable transformer for pde surrogate modeling , author=. Advances in Neural Information Processing Systems , volume=

  11. [19]

    International conference on machine learning , pages=

    Gnot: A general neural operator transformer for operator learning , author=. International conference on machine learning , pages=. 2023 , organization=

    2023

  12. [20]

    arXiv preprint arXiv:2205.13671 , year=

    Transformer for partial differential equations' operator learning , author=. arXiv preprint arXiv:2205.13671 , year=

    arXiv

  13. [21]

    Advances in neural information processing systems , volume=

    Choose a transformer: Fourier or galerkin , author=. Advances in neural information processing systems , volume=

  14. [22]

    arXiv preprint arXiv:2111.13802 , year=

    Factorized fourier neural operators , author=. arXiv preprint arXiv:2111.13802 , year=

    arXiv

  15. [23]

    Journal of Machine Learning Research , volume=

    Fourier neural operator with learned deformations for pdes on general geometries , author=. Journal of Machine Learning Research , volume=

  16. [24]

    Advances in Water Resources , volume=

    U-FNO—An enhanced Fourier neural operator-based deep-learning model for multiphase flow , author=. Advances in Water Resources , volume=. 2022 , publisher=

    2022

  17. [25]

    arXiv preprint arXiv:2010.08895 , year=

    Fourier neural operator for parametric partial differential equations , author=. arXiv preprint arXiv:2010.08895 , year=

    2010 arXiv

  18. [26]

    Advances in Neural Information Processing Systems , volume=

    Geometry-informed neural operator for large-scale 3d pdes , author=. Advances in Neural Information Processing Systems , volume=

  19. [27]

    arXiv preprint arXiv:2003.03485 , year=

    Neural operator: Graph kernel network for partial differential equations , author=. arXiv preprint arXiv:2003.03485 , year=

    2003 arXiv

  20. [28]

    International conference on learning representations , year=

    Learning mesh-based simulation with graph networks , author=. International conference on learning representations , year=

  21. [29]

    arXiv preprint arXiv:1611.08097 , year=

    Geometric deep learning: going beyond euclidean data , author=. arXiv preprint arXiv:1611.08097 , year=

    arXiv

  22. [30]

    Proceedings of the IEEE conference on computer vision and pattern recognition , pages=

    Pointnet: Deep learning on point sets for 3d classification and segmentation , author=. Proceedings of the IEEE conference on computer vision and pattern recognition , pages=

  23. [31]

    Thirty-sixth Conference on Neural Information Processing Systems Datasets and Benchmarks Track , year=

    Airfrans: High fidelity computational fluid dynamics dataset for approximating reynolds-averaged navier--stokes solutions , author=. Thirty-sixth Conference on Neural Information Processing Systems Datasets and Benchmarks Track , year=

  24. [32]

    ACM Transactions on Graphics (TOG) , volume=

    Learning three-dimensional flow for interactive aerodynamic design , author=. ACM Transactions on Graphics (TOG) , volume=. 2018 , publisher=

    2018

  25. [33]

    Nature Reviews Physics , volume=

    Physics-informed machine learning , author=. Nature Reviews Physics , volume=. 2021 , publisher=

    2021

  26. [34]

    Science , volume=

    Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations , author=. Science , volume=. 2020 , publisher=

    2020

  27. [35]

    Nature communications , volume=

    Physically informed artificial neural networks for atomistic modeling of materials , author=. Nature communications , volume=. 2019 , publisher=

    2019

  28. [36]

    Science advances , volume=

    Analyses of internal structures and defects in materials using physics-informed neural networks , author=. Science advances , volume=. 2022 , publisher=

    2022

  29. [37]

    International conference on computational science , pages=

    NVIDIA SimNet™: An AI-accelerated multi-physics simulation framework , author=. International conference on computational science , pages=. 2021 , organization=

    2021

  30. [38]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    A multi-objective optimization framework for adaptive weighting in physics-informed machine learning , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  31. [39]

    Artificial Intelligence Review , volume=

    Physics-informed neural networks for PDE problems: A comprehensive review , author=. Artificial Intelligence Review , volume=. 2025 , publisher=

    2025

  32. [40]

    Computer Methods in Applied Mechanics and Engineering , volume=

    Physics-informed neural network for modelling the thermochemical curing process of composite-tool systems during manufacture , author=. Computer Methods in Applied Mechanics and Engineering , volume=. 2021 , publisher=

    2021

  33. [41]

    Proceedings of the AAAI conference on artificial intelligence , volume=

    Spline-pinn: Approaching pdes without data using fast, physics-informed hermite-spline cnns , author=. Proceedings of the AAAI conference on artificial intelligence , volume=

  34. [42]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    DMIS: Dynamic mesh-based importance sampling for training physics-informed neural networks , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  35. [43]

    arXiv preprint arXiv:1910.03193 , year=

    Deeponet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators , author=. arXiv preprint arXiv:1910.03193 , year=

    1910 arXiv

  36. [44]

    Advances in neural information processing systems , volume=

    Attention is all you need , author=. Advances in neural information processing systems , volume=

  37. [45]

    Journal of Machine Learning Research , volume=

    Neural operator: Learning maps between function spaces with applications to pdes , author=. Journal of Machine Learning Research , volume=

  38. [46]

    arXiv preprint arXiv:2406.02176 , year=

    Aroma: Preserving spatial structure for latent pde modeling with local neural fields , author=. arXiv preprint arXiv:2406.02176 , year=

    arXiv

  39. [47]

    The Thirty-eighth Annual Conference on Neural Information Processing Systems , year=

    Universal physics transformers: A framework for efficiently scaling neural operators , author=. The Thirty-eighth Annual Conference on Neural Information Processing Systems , year=

  40. [48]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    Aerogto: An efficient graph-transformer operator for learning large-scale aerodynamics of 3d vehicle geometries , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  41. [49]

    Advances in Neural Information Processing Systems , volume=

    Geometry aware operator transformer as an efficient and accurate neural surrogate for pdes on arbitrary domains , author=. Advances in Neural Information Processing Systems , volume=

  42. [50]

    Advances in Neural Information Processing Systems , volume=

    Poseidon: Efficient foundation models for pdes , author=. Advances in Neural Information Processing Systems , volume=

  43. [51]

    Advances in Neural Information Processing Systems , volume=

    Multiple physics pretraining for spatiotemporal surrogate models , author=. Advances in Neural Information Processing Systems , volume=

  44. [52]

    arXiv preprint arXiv:2405.17527 , year=

    Unisolver: Pde-conditional transformers towards universal neural pde solvers , author=. arXiv preprint arXiv:2405.17527 , year=

    arXiv

  45. [53]

    The American Mathematical Monthly , volume=

    Diagonal equivalence to matrices with prescribed row and column sums , author=. The American Mathematical Monthly , volume=. 1967 , publisher=

    1967

  46. [54]

    2009 , publisher=

    Optimal transport: old and new , author=. 2009 , publisher=

    2009

  47. [55]

    arXiv preprint arXiv:2505.01281 , year=

    A Physics-preserved Transfer Learning Method for Differential Equations , author=. arXiv preprint arXiv:2505.01281 , year=

    arXiv

  48. [56]

    Advances in Neural Information Processing Systems , volume=

    Physense: Sensor placement optimization for accurate physics sensing , author=. Advances in Neural Information Processing Systems , volume=

  49. [57]

    Advances in Neural Information Processing Systems , volume=

    SpiderSolver: A Geometry-Aware Transformer for Solving PDEs on Complex Geometries , author=. Advances in Neural Information Processing Systems , volume=

  50. [58]

    arXiv preprint arXiv:2507.20065 , year=

    Geometric operator learning with optimal transport , author=. arXiv preprint arXiv:2507.20065 , year=

    arXiv

  51. [59]

    International Journal of Applied Mechanics , volume=

    Learning Material-Geometry-Aware Fourier Neural Operator for Stress-Strain Prediction and Parameter Inversion , author=. International Journal of Applied Mechanics , volume=

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.