The Kuramoto Neural Operator: Learning to Solve PDEs via Coupled Oscillator Dynamics

Aleksandr Beznosikov; Dmitry Bylinkin; Leonid Obukhov; Petr Badolia

REVIEW 3 major objections 6 minor 37 references

The Kuramoto Neural Operator: Learning to Solve PDEs via Coupled Oscillator Dynamics

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The Kuramoto Neural Operator learns PDE solution operators as a latent field of coupled spherical oscillators, ranking first or second on most of eight PDE benchmarks and localizing its own prediction error through oscillator incoherence.

desk verdict A carefully run neural-operator paper with a genuinely useful error-localization signal; the main unresolved issue is the fixed, unablated 32×32 canonical grid that the oscillator pathway depends on. read the letter →

arxiv 2608.10234 v1 pith:VVDTXYM4 submitted 2026-08-10 cs.CE cs.LG

Petr Badolia , Leonid Obukhov , Dmitry Bylinkin , Aleksandr Beznosikov This is my paper · ORCID

classification cs.CEcs.LG

keywords neuraloperatorPDEsolvingKuramotomodelcoupledoscillatorslearningerrorlocalizationsphericaldynamicslatent

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 sets out to establish that a PDE solution operator can be learned not as a map in a fixed basis, but as the hidden evolution of a coupled system of spherical oscillators living on the physical domain. The proposed Kuramoto Neural Operator (KNO) updates this oscillator field with learned local couplings, rotations, and a persistent stimulus, then decodes the final oscillator state into the predicted solution. Across eight two-dimensional PDE benchmarks, KNO ranks first or second on seven in-distribution and six out-of-distribution tasks, with the strongest gains on wave, transport, and reaction-diffusion problems. A further claim is that the local incoherence of the latent oscillators, a discrete Dirichlet-energy density, correlates with local prediction error and can localize error regions without any auxiliary uncertainty head. If correct, this gives operator learning a mechanism that is both competitive and partially self-diagnosing.

What carries the argument

The central object is the latent oscillator field: at every point of a fixed $32\times 32$ canonical grid, the network carries $M=16$ oscillators on the unit sphere $\mathbb{S}^{3}$, stored as an $Mn$-channel field. The evolution is governed by the projected Kuramoto update $q^{(r+1)}_{l,t,m} = \mathcal{N}\!\left(q^{(r)}_{l,t,m} + \gamma_{l,t}\,\mathrm{Proj}_{q^{(r)}_{l,t,m}}\!\left(c_{l,t,m} + s_{l,t,m} + \Omega_{l,t,m}q^{(r)}_{l,t,m}\right)\right)$, where $c_{l,t,m}$ is a learned convolutional local message conditioned on the feature field, $s_{l,t,m}$ is a persistent stimulus refreshed by a learned exponential moving average, and $\Omega_{l,t,m}$ is a skew-symmetric rotation. The sphere constraint with tangent projection and normalization retraction is what keeps the dynamics on the manifold for which the vector field is defined; ablations show that retraction by normalization is the most essential operation, while the coupling and rotation contribute most of the accuracy. The oscillator machinery carries the argument because its continuum limit is the paper's stated motivation for why interacting-oscillator dynamics can describe PDEs, and its local incoherence doubles as the error-localization signal.

What would settle it

A decisive test would be to evaluate KNO on a three-dimensional convection- or wave-dominated problem with an out-of-distribution shift that changes both geometry and resolution, for example moving boundaries or a jump from 256 to 1024 grid points. If the relative $\ell^2$ error at the shifted configuration falls below baseline levels, or the final-depth local-incoherence AUROC for the top-10% error pixels drops below roughly 0.8, the claimed broad effectiveness and universality of the error-localization signal would be contradicted.

Watch

Extended reading notes

Core claim

KNO's central claim is that the solution operator $G^\dagger: a \mapsto u$ can be approximated by a learned dynamical system whose latent state is a field of oscillators $q_{l,t,m}(x)\in\mathbb{S}^{n-1}$ evolving directly on a canonical grid over the physical domain. Each layer repeatedly applies a projected Kuramoto-type update: tangent projection of a learned local coupling, a skew-symmetric rotation, and a stimulus term, followed by normalization retraction onto the sphere; the stimulus is then refreshed through a norm-based readout and the decoded oscillator state is added to the feature field as a residual correction. On the Representative PDE Benchmark suite, the method ranks first or second on seven of eight in-distribution tasks and six of eight out-of-distribution tasks, with its clearest advantages on wave-dominated, transport-dominated, and reaction-diffusion problems that have pronounced local spatial structure. The paper further claims that the local incoherence $D_{\mathrm{loc}}(x)$, the mean squared pairwise difference between neighboring oscillators, acts as an architecture-native error-risk map: at final depth its spatial Spearman correlation with local error reaches about 0.9 on continuous translation, about 0.6 on discontinuous translation, and its AUROC for ranking top-10% error pixels stays above 0.8 on translation and airfoil tasks, including under distribution shift. The message is that modeling the mechanism that generates the solution, rather than representing the solution in a fixed basis, yields both accuracy and an interpretable signal about where the network is unsure.

Load-bearing premise

The load-bearing premise is that a fixed $32\times 32$ canonical grid holding only 16 sphere-valued oscillators per point, driven by learned local couplings, is enough to faithfully represent the solution operators of all the benchmark PDEs, including their out-of-distribution variants—an assumption the paper only tests on uniform two-dimensional grids.

Editorial extensions

If this is right

KNO should be the preferred architecture for wave, transport, reaction-diffusion, and heterogeneous elliptic problems on uniform grids, where it ranks first on all five such tasks in both in-distribution and out-of-distribution splits.
Because the dynamics live on a fixed canonical grid, KNO transfers zero-shot to resolutions far from training with less error degradation than the baselines, as demonstrated on Poisson and Airfoil.
The local-incoherence field $D_{\mathrm{loc}}$ provides a probe-free, calibration-free ranking of likely high-error regions that remains informative under distribution shift and across resolutions.
Removing the oscillator constraint and replacing the spherical state with an unconstrained feature field degrades accuracy and stability, so the manifold structure itself is doing work beyond added model capacity.
The architecture is not best in every regime: a convolutional operator wins on the simple linear Poisson map, and KNO loses ground on irregular geometry (Airfoil) and on the Navier-Stokes OOD split that combines geometry and resolution shifts.

Reading between the lines

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

Editorial inference: the same $D_{\mathrm{loc}}$ signal could be used as a spatially weighted training loss, up-weighting regions where the latent dynamics are incoherent, which might further improve accuracy without any architectural change.
Editorial inference: because the error-localization signal survives distribution shift, it could serve as an acquisition function for active learning or as a cheap confidence map for downstream decisions, though the paper does not test these uses.
Editorial inference: the fixed canonical grid makes KNO naturally suited to subgrid or multiscale parameterizations; one could couple the oscillator field at coarse resolution to a fine-resolution decoder and likely extend the zero-shot resolution transfer shown here.
Editorial inference: the drop on Airfoil and Navier-Stokes OOD suggests the next bottleneck is geometry; replacing the Euclidean convolutional coupling with mesh- or graph-based local coupling would be the natural test of whether the oscillator inductive bias, rather than the uniform grid, is what carries the advantage.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

KNO is a real empirical contribution: it adapts Kuramoto oscillators to operator learning, runs a careful multi-benchmark comparison, and shows that local incoherence of the latent oscillator field tracks local error well enough to act as a free error localizer. The ranking claim is actually correct against Table 1: KNO is best or second on 7/8 ID and 6/8 OOD tasks. The main soft spot is not the ranking; it is that the oscillator pathway lives on a fixed 32×32 canonical grid while inputs are 64×64 (or 128×128 for Airfoil and OOD Navier–Stokes). That choice band-limits the oscillator contribution and is never ablated, so the paper's phrasing that the latent state evolves 'directly over the physical domain' overstates what is implemented. The no-oscillator ablation does show the oscillator matters, but it does not separate the low-res oscillator mechanism from the full-res feature pathway. A canonical-grid sweep (S_c = 16/32/64) would fix this.

The experiments are otherwise solid: validation-based selection, three seeds, ID/OOD splits, ablations of M, n, R, T, L, geometric controls, and resolution-transfer studies. The error-localization analysis is post-hoc but carefully done, with Spearman, AUROC, AP, and depthwise formation; the resolution-transfer appendix strengthens it. Missing code and data is annoying and should be fixed before publication. The KNO candidate pool (3) is smaller than the baseline pools (5), which is a minor caveat; if anything it makes the comparison conservative for KNO.

Limitations are honestly stated: uniform grid, 2D only, weak on irregular geometry. Overall, this is a paper worth refereeing, not desk-rejecting. I'd ask the authors for code, a canonical-grid ablation, and a toned-down claim about 'directly over the physical domain' – then it could be a useful contribution to the neural operator literature.

Referee Report

3 major / 6 minor

Summary. The paper proposes the Kuramoto Neural Operator (KNO), a neural operator that represents the latent state as a field of spherical oscillators on a fixed 32x32 canonical grid, evolved by projected Kuramoto-type updates with feature-conditioned local coupling, a persistent stimulus, and residual decoding into the feature field. The architecture is evaluated on eight two-dimensional PDE benchmarks from the Representative PDE Benchmark suite, reporting relative L2 errors for in-distribution and out-of-distribution splits, extensive ablations of dynamic and geometric components, robustness sweeps over oscillator counts and dimensions, zero-shot resolution transfer experiments, and an analysis that connects local oscillator incoherence to prediction-error localization.

Significance. If the reported results hold, KNO is a genuinely different operator architecture: instead of a fixed spectral or convolutional basis, it models the solution operator through learned latent oscillator dynamics, and it demonstrates a distinctive architecture-native signal for localizing prediction error without a trained uncertainty head. The paper has notable strengths: architecture selection is performed on validation data only with predeclared candidate pools, results are averaged over three seeds, ablations are extensive, and the zero-shot resolution transfer experiments in Appendix A.5 are a useful addition. The main performance claim is not circular, because it is evaluated on an external benchmark against standard baselines. The principal concerns are that the fixed low-resolution canonical grid is not ablated, the comparison pool is asymmetric, and the error-localization claim lacks simple baselines.

major comments (3)

[Appendix E.1 and Eq. (5)] The canonical grid Sc=(32,32) is fixed in every reported configuration and is never ablated. Because the oscillator field and stimulus field live on this 32x32 grid, and because the residual update in Eq. (5) is an upsampled 32x32 map, the oscillator pathway can only inject Fourier modes representable at that resolution; on the 128x128 Airfoil and OOD Navier-Stokes tasks this is one quarter of the native resolution. The attribution in Section 6.3 and the Conclusion that local coupling and rotation drive the architecture's local transport and geometry gains is therefore not established. The paper should report a sweep over Sc (for example, Sc=64 or Sc=128) or, at minimum, a control that keeps the canonical grid fixed while varying only the coupling mechanism; otherwise the claim that KNO's latent dynamics evolve 'directly over the physical domain' is unsupported.
[Section 6.1 and Appendix F.1] The comparison is unbalanced across families: each baseline family is searched over five predeclared candidates, KNO over only three, and AKOrN uses a single fixed configuration. Since the central ranking claim in Section 6.2 and Section 7 depends on per-task architecture selection, the paper should either equalize candidate-pool sizes among searched families or report how the ranking changes when additional KNO candidates are included. As reported, the 'first or second on most tasks' claim is not a fully controlled comparison.
[Section 6.5 and Table 6] The error-localization analysis computes Dloc on the 32x32 canonical grid and compares it with error fields at 64x64 or 128x128, but it does not compare against simple baselines such as the gradient magnitude of the predicted solution, a distance-to-interface map, or random spatial rankings. Without such controls, the claim that local incoherence is a distinctive architecture-native risk signal, rather than a quantity that largely inherits geometric smoothness, is not fully supported. Adding these baselines to Table 6 would clarify the added value of Dloc.

minor comments (6)

[Section 2, Eq. (1)] The scalar Kuramoto model uses K for coupling strength, while the generalized model uses K_ij for the coupling kernel; reusing K in the same section is confusing and should be resolved with distinct notation.
[Section 6.2] The sentence 'On all 5 of these tasks' is ambiguous; the five tasks should be listed explicitly so the reader can verify the claim against Table 1.
[Appendix E.1] The statement that all reported configurations use Sc=(32,32) appears only in the appendix; because the canonical grid is central to the architecture, this should also be stated in Section 4.
[References] The Lohe reference is malformed ('MA2539317 Lohe'); the author name and title should be corrected.
[Section 6.1 and Table 1] The paper does not report inference time, parameter counts, or wall-clock training cost for KNO and the baselines, which would help assess the practical trade-offs of the canonical-grid design.
[Table 1, Navier-Stokes OOD row] KNO's OOD error has a very large standard deviation (0.3899 ± 0.1141); the paper should briefly discuss whether this reflects instability across seeds or a particular failure mode, beyond the acknowledged resolution and geometry shift.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: KNO's performance claims are externally benchmarked and the error-localization analysis is a post-hoc empirical correlation, not a fitted input or self-referential derivation.

full rationale

The paper's central claim is empirical: KNO is trained on an external benchmark suite (Representative PDE Benchmark) and compared against nine baseline architectures under an explicit two-stage validation-only selection protocol, with relative L2 error defined in Eq. (6). No fitted parameter is renamed as a prediction, and no equation reduces its output to its input by construction. The error-localization analysis in Sections 6.5 and Appendix B is also not circular: Dloc is a fixed observable of the model's oscillator states, and the reported Spearman correlations, AUROC, and AP are measured post hoc against target gradients and prediction errors. The local-incoherence signal is not used to train the model, to select checkpoints, or to define the loss, so its empirical alignment with error is an independent observation rather than a self-fulfilling construction. The architectural derivation is similarly self-contained: Eqs. (2)-(5) define the oscillator update, stimulus refresh, and residual feature correction, and the ablations in Tables 2 and 3 test components by removal, not by invoking the target result. The closest prior-work citation, AKOrN [Miyato et al., 2025], is not by the present authors and is used only as a baseline and design precedent, not as a load-bearing justification for KNO's performance or for any forbidden uniqueness claim. No self-citation chain, imported uniqueness theorem, or ansatz smuggled in via citation is present. The acknowledged limitations (uniform grid, two-dimensional benchmarks, Airfoil and Navier-Stokes OOD weaknesses) are honest scope statements, not evidence of circularity. Accordingly, no circular step was found and the score is 0.

Assumptions & free parameters 8 free parameters · 5 assumptions · 3 invented entities

The central claim rests on hand-chosen architectural hyperparameters and the standard neural operator training setup, plus the domain assumption that oscillator dynamics are a suitable latent space for PDE solution operators. No truly free continuous parameters are fit to the target result beyond ordinary trained weights.

free parameters (8)

M = 16 (oscillator count) = 16
Chosen by hand; Table 4 sweeps it, with best at 16, showing performance depends on it.
n = 4 (oscillator dimension) = 4
Chosen by hand; Table 5 sweeps it, with best at 4-32, showing dependence.
R = 4 (integration steps per stage) = 4
Chosen by hand; Appendix A.1 sweeps it, with diminishing returns after 4.
c = 64 (feature width) = 64
Chosen by hand; fixed across all experiments (Appendix E).
k = 16 (spectral cutoff) = 16
Chosen by hand; fixed at 16 (Appendix E).
canonical grid 32x32 = 32x32
Chosen by hand; all oscillator dynamics evaluated on this grid (Section 4, Appendix E).
learning rate 1e-3, weight decay 1e-5, batch size 8 = 1e-3, 1e-5, 8
Chosen by hand for all runs (Appendix F).
selected architecture per task (layer count and stage schedule) = varies by task, KNO-1/2/3
Selected from a 3-candidate pool using validation loss; effectively fitted to validation data (Appendix F), so the reported results are post-selection.

assumptions (5)

domain assumption The continuum limit of large ensembles of coupled oscillators can describe a broad class of nonlinear, nonlocal PDEs (Medvedev 2018; Torre 2014; Gupta 2012).
Invoked in Sections 1 and 3 to justify using oscillator dynamics as latent dynamics for PDE operators. It is cited from prior work and not proved in this paper.
domain assumption A neural operator with enough parameters can approximate the solution operator G† from data (Kovachki et al., 2023).
Standard background for all operator learning; used implicitly throughout.
standard math The projected Euler step with normalization preserves the sphere constraint and is a valid discrete integrator for the intended continuous dynamics.
Used in Eq. (3); standard numerical integration assumption, not proved.
domain assumption The benchmark dataset (Raonic et al., 2023) and its ID/OOD splits are representative and correctly implemented.
All results depend on the correctness of the dataset and splits; no independent verification in the paper.
domain assumption Relative L2 error is an appropriate metric for operator quality.
Standard in the field; assumed without discussion.

invented entities (3)

Oscillator field q
purpose: Latent state of M spherical oscillators per grid point; carries the learned dynamics that are decoded into the solution.
Introduced by the paper as a learned latent variable; no external falsifiable handle other than the model's own predictions.
Stimulus field s
purpose: Persistent conditioning signal that carries information from earlier stages and connects the feature and oscillator pathways.
A new architectural state introduced in Section 4; no independent evidence outside the model.
Local incoherence Dloc
purpose: Derived from the oscillator field to act as a local error-risk indicator.
A derived statistic, not a new physical entity, but the paper elevates it to a signal. Its utility is evaluated only on the paper's own test sets.

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Cite this review

Pith. "Pith review of The Kuramoto Neural Operator: Learning to Solve PDEs via Coupled Oscillator Dynamics." pith.science (2026). https://pith.science/paper/VVDTXYM4

@misc{pith2026260810234,
  author       = {Pith},
  title        = {Pith review of: The Kuramoto Neural Operator: Learning to Solve PDEs via Coupled Oscillator Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VVDTXYM4}},
  note         = {Machine review of arXiv:2608.10234}
}

read the original abstract

Operator learning is a rapidly advancing area of computational science. It is particularly well suited to problems where a partial differential equation (PDE) must be solved repeatedly under varying physical configurations. Most existing architectures represent the solution operator in a fixed basis. While this assumption is well aligned with global structures, it is less suitable for phenomena governed by local interactions in physical space. We explore an alternative perspective motivated by the observation that the continuum limit of coupled oscillator systems can describe a broad class of PDEs. Building on this idea, we introduce the Kuramoto Neural Operator (KNO), which represents the solution through the evolution of a latent field of interacting oscillators. Across a diverse collection of PDE benchmarks, KNO achieves strong predictive performance, with improvements over competing approaches. Our experimental evaluation also includes an extensive ablation study that quantifies the contribution of each architectural component incorporated into KNO. Furthermore, we show that the model's prediction error is closely linked to the collective dynamics of the latent oscillators. It varies systematically with their degree of synchronization, providing insights into the underlying mechanisms.

Figures

Figures reproduced from arXiv: 2608.10234 by the authors.

**Figure 1.** KNO architecture. (a) The input is lifted pointwise to a feature field h (blue); the oscillators q (green) are initialized at random on the sphere, and the stimulus s (orange) is initialized from a band-limited projection of h. Next, L KNO layers jointly evolve the triple (h, q, s), and a pointwise projection produces the output. (b) Inside layer l, the feature field hl is resampled to the canonical grid and conditi… view at source ↗

**Figure 2.** Spatial formation of local incoherence across KNO depth. Top/bottom rows show OOD/ID examples chosen closest to the median checkpoint-averaged relative L2 error. Each panel shows the target, the q0-averaged Dloc at initialization and at the indicated layer outputs, and the normalized final error. Translation panels overlay the target u=0.5 contour; Airfoil panels overlay the airfoil boundary. 0 8 16 24 32 0.0 0.2 0.… view at source ↗

**Figure 3.** Depthwise formation of geometry and error alignment. Curves show median within-sample ρ (r) G and ρ (r) E ; lines and bands are the mean and standard deviation over three checkpoints. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Zero-shot resolution transfer on Poisson. Curves show mean relative L2 error and shaded regions show one standard deviation. We display KNO and the strongest competitor. The dashed line marks the training resolution. 32 2 64 2 128 2 256 2 512 2 Evaluation resolution 1…

**Figure 6.** Figure 6: Resolution dependence of final-depth local-incoherence statistics. Curves and bands show the mean and [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: shows representative examples of the target, local absolute error, and final-depth Dloc. 32 2 Target Norm. Error Final Dloc 64 2 (train) 128 2 256 2 512 2 32 2 Target Norm. Error Final Dloc 64 2 (train) 128 2 256 2 512 2 32 2 Target Norm. Error Final Dloc 64 2 128 2 (t…

**Figure 8.** Figure 8: Depthwise formation of error alignment under resolution transfer. Curves show the median within-field [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: Formation of local incoherence at 2562 . Rows show representative samples from Continuous Translation, Discontinuous Translation, and Airfoil. Columns show the target, local incoherence at initialization D (0) loc , selected intermediate layer outputs, final Dloc, and…

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Works this paper leans on

37 extracted references · 28 canonical work pages

[1]

Iterative procedures for nonlinear integral equations

Donald G Anderson. Iterative procedures for nonlinear integral equations. Journal of the ACM (JACM), 12 0 (4): 0 547--560, 1965

work page 1965
[2]

Drivaerml: High-fidelity computational fluid dynamics dataset for road-car external aerodynamics

Neil Ashton, Charles Mockett, Marian Fuchs, Louis Fliessbach, Hendrik Hetmann, Thilo Knacke, Norbert Schonwald, Vangelis Skaperdas, Grigoris Fotiadis, Astrid Walle, et al. Drivaerml: High-fidelity computational fluid dynamics dataset for road-car external aerodynamics. arXiv preprint arXiv:2408.11969, 2024

arXiv 2024
[3]

Spectral methods, volume 285

Claudio Canuto, M Youssuff Hussaini, Alfio Quarteroni, and Thomas A Zang. Spectral methods, volume 285. Springer, 2006

work page 2006
[4]

Laplace neural operator for solving differential equations

Qianying Cao, Somdatta Goswami, and George Em Karniadakis. Laplace neural operator for solving differential equations. Nature Machine Intelligence, 6 0 (6): 0 631--640, 2024

work page 2024
[5]

Continuous versus discontinuous transitions in the d-dimensional generalized kuramoto model: Odd d is different

Sarthak Chandra, Michelle Girvan, and Edward Ott. Continuous versus discontinuous transitions in the d-dimensional generalized kuramoto model: Odd d is different. Physical Review X, 9 0 (1): 0 011002, 2019

work page 2019
[6]

The finite element method for elliptic problems

Philippe G Ciarlet. The finite element method for elliptic problems. SIAM, 2002

work page 2002
[7]

A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized pdes

Stefania Fresca, Luca Dede’, and Andrea Manzoni. A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized pdes. Journal of Scientific Computing, 87 0 (2): 0 61, 2021

work page 2021
[8]

On training implicit models

Zhengyang Geng, Xin-Yu Zhang, Shaojie Bai, Yisen Wang, and Zhouchen Lin. On training implicit models. Advances in neural information processing systems, 34: 0 24247--24260, 2021

work page 2021

Show all 37 references

[9]

One-dimensional lattice of oscillators coupled through power-law interactions: Continuum limit and dynamics of spatial fourier modes

Shamik Gupta, Max Potters, and Stefano Ruffo. One-dimensional lattice of oscillators coupled through power-law interactions: Continuum limit and dynamics of spatial fourier modes. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 85 0 (6): 0 066201, 2012

2012
[10]

Kuramoto oscillators and swarms on manifolds for geometry informed machine learning

Vladimir Ja \'c imovi \'c . Kuramoto oscillators and swarms on manifolds for geometry informed machine learning. Available at SSRN 4865439, 2024

2024
[11]

Rank correlation methods

Maurice George Kendall and Jean Dickinson Gibbons. Rank correlation methods. 1962

1962
[12]

Learning latent field dynamics of pdes

Dmitrii Kochkov, Alvaro Sanchez-Gonzalez, Jamie Alexander Smith, Tobias Joachim Pfaff, Peter Battaglia, and Michael Brenner. Learning latent field dynamics of pdes. In Third Workshop on Machine Learning and the Physical Sciences (NeurIPS 2020), 2020

2020
[13]

Neural operator: Learning maps between function spaces with applications to pdes

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 0 (89): 0 1--97, 2023

2023
[14]

Chemical turbulence

Yoshiki Kuramoto. Chemical turbulence. In Chemical oscillations, waves, and turbulence, pages 111--140. Springer, 1984

1984
[15]

Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems

Randall J LeVeque. Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. SIAM, 2007

2007
[16]

Fourier neural operator for parametric partial differential equations

Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895, 2020

2010 arXiv
[17]

Fourier neural operator with learned deformations for pdes on general geometries

Zongyi Li, Daniel Zhengyu Huang, Burigede Liu, and Anima Anandkumar. Fourier neural operator with learned deformations for pdes on general geometries. Journal of Machine Learning Research, 24 0 (388): 0 1--26, 2023 a

2023
[18]

Geometry-informed neural operator for large-scale 3d pdes

Zongyi Li, Nikola Kovachki, Chris Choy, Boyi Li, Jean Kossaifi, Shourya Otta, Mohammad Amin Nabian, Maximilian Stadler, Christian Hundt, Kamyar Azizzadenesheli, et al. Geometry-informed neural operator for large-scale 3d pdes. Advances in Neural Information Processing Systems,...

2023
[19]

The kuramoto model on a sphere: Explaining its low-dimensional dynamics with group theory and hyperbolic geometry

Max Lipton, Renato Mirollo, and Steven H Strogatz. The kuramoto model on a sphere: Explaining its low-dimensional dynamics with group theory and hyperbolic geometry. Chaos: An Interdisciplinary Journal of Nonlinear Science, 31 0 (9), 2021

2021
[20]

Enhancing fourier neural operators with local spatial features

Chaoyu Liu, Davide Murari, Lihao Liu, Yangming Li, Chris Budd, and Carola-Bibiane Sch \"o nlieb. Enhancing fourier neural operators with local spatial features. arXiv preprint arXiv:2503.17797, 2025

2025 arXiv
[21]

Riesz neural operator for solving partial differential equations

Shouyi Liu, Xiaokang Yang, and Yuntian Chen. Riesz neural operator for solving partial differential equations. In The Fourteenth International Conference on Learning Representations, 2026. URL https://openreview.net/forum?id=Vjw7q1quNt

2026
[22]

Non-abelian kuramoto models and synchronization

MA2539317 Lohe. Non-abelian kuramoto models and synchronization. Journal of Physics A: Mathematical and Theoretical, 42 0 (39): 0 395101, 2009

2009
[23]

Learning nonlinear operators via deeponet based on the universal approximation theorem of operators

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 0 (3): 0 218--229, 2021

2021
[24]

Solving partial differential equations via radon neural operator

Wenbin Lu, Yihan Chen, Junnan Xu, Wei Li, Junwei Zhu, and Jianwei Zheng. Solving partial differential equations via radon neural operator. Advances in Neural Information Processing Systems, 38: 0 157027--157067, 2026

2026
[25]

Almost global convergence to practical synchronization in the generalized kuramoto model on networks over the n-sphere

Johan Markdahl, Daniele Proverbio, La Mi, and Jorge Goncalves. Almost global convergence to practical synchronization in the generalized kuramoto model on networks over the n-sphere. Communications Physics, 4 0 (1): 0 187, 2021

2021
[26]

Deep equilibrium based neural operators for steady-state pdes

Tanya Marwah, Ashwini Pokle, J Zico Kolter, Zachary Lipton, Jianfeng Lu, and Andrej Risteski. Deep equilibrium based neural operators for steady-state pdes. Advances in Neural Information Processing Systems, 36: 0 15716--15737, 2023

2023
[27]

The continuum limit of the kuramoto model on sparse random graphs

Georgi S Medvedev. The continuum limit of the kuramoto model on sparse random graphs. arXiv preprint arXiv:1802.03787, 2018

2018 arXiv
[28]

Artificial kuramoto oscillatory neurons

Takeru Miyato, Sindy L \"o we, Andreas Geiger, and Max Welling. Artificial kuramoto oscillatory neurons. In International Conference on Learning Representations, volume 2025, pages 44278--44322, 2025

2025
[29]

The finite volume method

Fadl Moukalled, Luca Mangani, and Marwan Darwish. The finite volume method. In The finite volume method in computational fluid dynamics: An advanced introduction with OpenFOAM and Matlab , pages 103--135. Springer, 2015

2015
[30]

Convolutional neural operators for robust and accurate learning of pdes

Bogdan Raonic, Roberto Molinaro, Tim De Ryck, Tobias Rohner, Francesca Bartolucci, Rima Alaifari, Siddhartha Mishra, and Emmanuel De B \'e zenac. Convolutional neural operators for robust and accurate learning of pdes. Advances in Neural Information Processing Systems, 36: 0 7...

2023
[31]

U-net: Convolutional networks for biomedical image segmentation

Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computer-assisted intervention, pages 234--241. Springer, 2015

2015
[32]

Derivation of the tight-binding approximation for time-dependent nonlinear schr \"o dinger equations

Andrea Sacchetti. Derivation of the tight-binding approximation for time-dependent nonlinear schr \"o dinger equations. In Annales Henri Poincar \'e , volume 21, pages 627--648. Springer, 2020

2020
[33]

Kuramoto orientation diffusion models

Yue Song, Andy Keller, Sevan Brodjian, Takeru Miyato, Yisong Yue, Pietro Perona, and Max Welling. Kuramoto orientation diffusion models. Advances in Neural Information Processing Systems, 38: 0 4047--4076, 2026

2026
[34]

05 the continuum limit and the wave equation

Charles G Torre. 05 the continuum limit and the wave equation. Continuum, 1: 0 1--2004, 2014

2004
[35]

U-fno—an enhanced fourier neural operator-based deep-learning model for multiphase flow

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: 0 104180, 2022

2022
[36]

Geopt: Scaling physics simulation via lifted geometric pre-training

Haixu Wu, Minghao Guo, Zongyi Li, Zhiyang Dou, Mingsheng Long, Kaiming He, and Wojciech Matusik. Geopt: Scaling physics simulation via lifted geometric pre-training. arXiv preprint arXiv:2602.20399, 2026

2026 arXiv
[37]

Kuramoto oscillatory phase encoding: Neuro-inspired synchronization for improved learning efficiency

Mingqing Xiao, Yansen Wang, Dongqi Han, Caihua Shan, and Dongsheng Li. Kuramoto oscillatory phase encoding: Neuro-inspired synchronization for improved learning efficiency. arXiv preprint arXiv:2604.07904, 2026

2026 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Iterative procedures for nonlinear integral equations

Donald G Anderson. Iterative procedures for nonlinear integral equations. Journal of the ACM (JACM), 12 0 (4): 0 547--560, 1965

work page 1965

[2] [2]

Drivaerml: High-fidelity computational fluid dynamics dataset for road-car external aerodynamics

Neil Ashton, Charles Mockett, Marian Fuchs, Louis Fliessbach, Hendrik Hetmann, Thilo Knacke, Norbert Schonwald, Vangelis Skaperdas, Grigoris Fotiadis, Astrid Walle, et al. Drivaerml: High-fidelity computational fluid dynamics dataset for road-car external aerodynamics. arXiv preprint arXiv:2408.11969, 2024

arXiv 2024

[3] [3]

Spectral methods, volume 285

Claudio Canuto, M Youssuff Hussaini, Alfio Quarteroni, and Thomas A Zang. Spectral methods, volume 285. Springer, 2006

work page 2006

[4] [4]

Laplace neural operator for solving differential equations

Qianying Cao, Somdatta Goswami, and George Em Karniadakis. Laplace neural operator for solving differential equations. Nature Machine Intelligence, 6 0 (6): 0 631--640, 2024

work page 2024

[5] [5]

Continuous versus discontinuous transitions in the d-dimensional generalized kuramoto model: Odd d is different

Sarthak Chandra, Michelle Girvan, and Edward Ott. Continuous versus discontinuous transitions in the d-dimensional generalized kuramoto model: Odd d is different. Physical Review X, 9 0 (1): 0 011002, 2019

work page 2019

[6] [6]

The finite element method for elliptic problems

Philippe G Ciarlet. The finite element method for elliptic problems. SIAM, 2002

work page 2002

[7] [7]

A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized pdes

Stefania Fresca, Luca Dede’, and Andrea Manzoni. A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized pdes. Journal of Scientific Computing, 87 0 (2): 0 61, 2021

work page 2021

[8] [8]

On training implicit models

Zhengyang Geng, Xin-Yu Zhang, Shaojie Bai, Yisen Wang, and Zhouchen Lin. On training implicit models. Advances in neural information processing systems, 34: 0 24247--24260, 2021

work page 2021