Kuramoto Orientation Diffusion Models
Authors: Yue Song
T. Anderson Keller
Sevan Brodjian
Takeru Miyato
Yisong Yue
Pietro Perona
Max Welling
Venue: NeurIPS 2025 poster ()
Keywords: diffusion model
phase synchronization
neural oscillations
Primary area: deep_learning
TL;DR: We introduce a nonlinear score-based generative model operating on periodic domains by leveraging Kuramoto dynamics for orientation-rich data.

Abstract: Orientation-rich images, such as fingerprints and textures, often exhibit coherent angular directional patterns that are challenging to model using standard generative approaches based on isotropic Euclidean diffusion. Motivated by the role of phase synchronization in biological systems, we propose a score-based generative model built on periodic domains by leveraging stochastic Kuramoto dynamics in the diffusion process.  In neural and physical systems, Kuramoto models capture synchronization phenomena across coupled oscillators -- a behavior that we re-purpose here as an inductive bias for structured image generation. In our framework, the forward process performs \textit{synchronization} among phase variables through globally or locally coupled oscillator interactions and attraction to a global reference phase, gradually collapsing the data into a low-entropy von Mises distribution. The reverse process then performs \textit{desynchronization}, generating diverse patterns by reversing the dynamics with a learned score function. This approach enables structured destruction during forward diffusion and a hierarchical generation process that progressively refines global coherence into fine-scale details. We implement wrapped Gaussian transition kernels and periodicity-aware networks to account for the circular geometry. Our method achieves competitive results on general image benchmarks and significantly improves generation quality on orientation-dense datasets like fingerprints and textures. Ultimately, this work demonstrates the promise of biologically inspired synchronization dynamics as structured priors in generative modeling.

Kuramoto Orientation Diffusion Models
Yue Song', T. Anderson Keller", Sevan Brodjian', Takeru Miyato Yisong Yue', Pietro Perona', and Max Welling
Caltech
UUiiversity of Tübingen University of Amsterdam 5 CuspAl
Harvard University
Abstract
Orientation-rich images, such as fingerprints and textures, often exhibit coherent angular directional patterns that are challenging to model using standard generative approaches based on isotropic Euclidean diffusion. Motivated by the role of phase synchronization in biological systems, propose score-based generative model built on periodic domains by leveraging stochastic Kuramoto dynamics in the dif- fusion process. In neural and physical systems, Kuramoto models capture synchro- nization phenomena across coupled oscillators a behavior that we re-purosse here as an inductive bias for structured image generation. In our framework, the forward process performs synchronization among phase variables through globally locally coupled oscillator interactions and attraction collapsing the data into low-entropy von Mises distribution. The reverse process then performs desynchronization, generating diverse patterns by reversing the dy- namics with a learned score function. This approach enables structured destruction during forward diffusion and a hierarchical generation process that progressively refines global coherence into fine-scale details. We implement wrapped Gaussian transition kernels and periodicity -wware aware networks to account for the circular geom- etry. Our method achieves competitive results on general image benchmarks and significantly improves generation quality on orientation-dent datasets like finger- prints and textures. Ultimately, this work demonstrates the promise of biologically inspired synchronization dynamics as structured priors in generative modeling. Code is available at httpp:://gtthub.com http://hhhh..oooonnnnn////ns/o// KingJaneSSongOOieenaioooffffii
global reference phase, gradually
Introduction
Synchronization phenomena, characterized by coordinated rhythmic activity across coupled oscilla- tors, are ubiquitous in nature. Such patterns are fundamental in biological neural networks, where synchronous neural firing supports critical cognitive functions including attention, sensory integration, and memory [9]. Similarly, in physical and engineering systems, synchronization underpins the col- lective behavior coupled oscillatory circuits, chemical reactions, and mechanical structures [36 [2]. Central to understanding these phenomena is the Kuramoto model [36), canonical framework arising from nonlinear dynamics that elegantly describes how global coherence emerges spontaneously from interactions among oscillatory units.
In this work, we explore how these principles of synchronized coherence can be harnessed tackle persistent challenge in generative modeling for orientation -ich data, such as fingerprints, textures, and directional fields. These data appear numerous applications, including fingerprint generation for biometric security [17]. material characterization through crystal orientation analysis [R], and fiber orientation modeling for improved medical diagnostics [7 64). The defining structures of these data types are characterized primarily by the orientations of local features (i.e., piece-wis constant
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Globally coupled Stochastic Kuramoto model with reference phase
d0i
K(t) 0 0:) (te(t) sin 0:) 22
dt
N
Time-reversed SDE
d0;
1
0 (0g)(0.)
(t) @ 0.) + (fe(t) sin
0:) 2D,
22
00;
Locally coupled Stochastic Kuramoto model with reference phase
Standard Diffusion Models
Figure I: Governing stochastic differential equations (SDEs) and representative image samples from our globally and locally coupled Kuramoto orientation diffusion models. Pixels are mapped onto periodic domains angular phase variables. the globally coupled model, each pixel interacts with all other pixels via Kuramoto sinusoidal coupling (highlighted in red). The locally coupled variant corresponds to a similar SDE restricts this sinusoidal coupling to local neighborhood around each pixel. Unlike standard diffusion models, our approach introduces non-isotropic noise dynamics via pulling similar phases together, enabling a more structured destruction process. These dynamics help preserve the global structure in the early stages of diffusion (e.g. the overall shape of the bird), while allowing for faster adaptation noise as the process progresses. The forward SDE synchronizes phase variables through oscillator interactions and global reference phase. The reverse process desynchronizes these variables using learned score functions to synthesize images.
signals with sharp transitions) rather than by raw pixel intensities. Crucially, such orientation patterns exist on periodic domains where angular discontinuities are problematic for conventional models that do not explicitly account for periodicity. Early attempts to handle these issues date back to the seminal work on orientation diffusions for image denoising (48), which highlighted how treating angular data without considering its circular nature can lead to artifacts and loss of coherence. Those findings underscore the need for specialized generative frameworks that explicitly account for the periodic structure inherent in directional data.
In this paper, we introduce a novel nonlinear score-based generative framework that leverages stochastic Kuramoto dynamics to operate directly on periodic domains. Synchronization offers powerful inductive bias here: it encourages local patterns and orientations reinforce one another edges to align, ridges to remain coherent, and flows to stay smooth before noise erodes these features. In our framework, pixel values are first mapped to angular phase variables, which enables natural compatibility with circular geometry and allows the diffusion process to evolve through interactions among phase coupling. We present our governing SDEs and illustrative examples of diffusion dynamics in Fig. Our forward diffusion process performs structured destruction by progressively synchronizing angular phase variables through phase coupling and attraction to a common reference phase, driving the data distribution toward low-entropy von Mises state. The Kuramoto interactions induce non-isotropic dynamics by pulling similar phases together, helping preserve local orientation and making the model particularly suitable for vrientation-dense images.
The reverse generative process then performs desynchronization, leveraging learned periodic score functions to gradually reintroduce variability and reconstruct the image. The generation thus follows a hierarchical process, where the global structure established first, and finer details are subsequently introduced, following coarse-to- to-seeto-fine paradigm. We also propose locally coupled variant of the

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model that aligns with the spatial correlations of image data. Due to the coherence imposed by synchronization, our model converges faster the terminal distribution during the forward process, which in turn enables the reverse process to generate high-quality samples within fewer diffusion steps. handle periodicity robustly, we employ periodicity-awei neural networks with sinusoidal embeddings and define forward transitions using wrapped Gaussian kernels. The score function is then estimated by sampling from these local transitions derived from the forward dynamics.
Experiments on orientation-rich datasets, such as fingerprints, textures, and terrains, demonstrate that our Kuramoto orientation diffusion model consistently produces higher-fidelity images compared to standard diffusion baselines, often requiring fewer diffusion steps. Furthermore, our method remains competitive even on general CIFAR- benchmarks. Beyond images, we also report results on Earth/climate datasets on the 2D sphere and on Navier-Stokes fluid velocity fields, where the synchronization prior aligns with natural periodicity and angular structure, delivering consistent gains. Overall, this work bridges neural oscillation theory and modern score-based generative models, underscoring the potential of biologically inspired synchronization dynamics structured priors.
Kuramoto Orientation Diffusion Models
2.1 Preliminary: Score-based Generative Models
In score-based generative models 3 60 the forward and reverse processes of diffusion models are formulated as pair of coupled SDEs:
Forward-SDE: dæ fxx t) dt +t( dw
(I)
Reverse-SDE. dæ ( 1) g2 Va l(g) di +g(() dw
where f(x, t) represents the vector-valued drift function, g(t) denotes the scalar function of the diffusion coefficient, and w and w are the standard Wiener processes. The boxed term represents the score function, which corresponds to the gradient of the log-density. neural network s(x,, t) is trained to approximate the score function by minimizing the following objective:
C (1,77 Ex, En [g (t) t(x,,)) Vx. log p(2)
(2)
Common choices for the forward SDE include the Variance-Presserving (VP) and Sariance-Explodue (VE) formulations [59 60 23 Typically, the drift function is chosen to be linear, ensuring that the conditional distribution p( a (z)) remains analytically Gaussian and the corresponding score function Zz, log p( p(: (20) can be computed in closed form.
2.2 Stochastic Kuramoto Models with Reference Phase
Fig. jillustrates the forward and reverse processes in our Kuramoto orientation diffusion model. During the forward process (left-to-right), we progressively destroy image information through synchronization, modeled by the following stochastic Kuramoto dynamics [6)::
dop 1 N
Σ
K(t) sin( 0;) KRe((t) sin( 0,) + 22
(3)
dt
where each colored circle represents oscillator with periodic phase 0 [- K(I) is the time-varying coupling strength among oscillators, Kte((t) is the coupling strength to reference phase ther, 2D, denotes the variance of Gaussian noise, and N denotes the number of oscillators. The white rectangle represents the global reference phase whet serving as an attractor to guide the population of oscillators toward a synchronized target. To measure the level of synchronization, compute the complex ordering parameter:
N
((0)
where (t) [- x] denotes the mean phase, and r(t) E [0, 1] measures the coherence of these phases. They correspond to the angle and length of the arrow in Fig. respectively. When 0.

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the system is completely desynchronized. When = 1, the oscillators are perfectly synchronized. For intermediate values 0 < r < the system exhibits partial synchronization, where phases are clustered around the mean phase vb(() with dispersion induced by noise.
Phase Wrapping. After each Ku- ramoto update step, we apply phase wrapping ensure that all phase variables remain within the interval [-n, x]. Specifically, the wrapping is performed as 0 = (0 T mod (2 where mod denotes the modulo oper- ation. This guarantees that the period- icity of the phase variables consis- tently maintained throughout the for- ward process.
Synchvonization
(10)
vI2T) that < rt
von Mises
Swl
Desynchronization
Figure 2: Illustration of Kuramoto orientation diffusion model. In the forward process (left-to-right), angular phase variables (colored circles) synchronize toward low-entropy von Mises distribution, guided by attraction to reference phase (white rectangle). The reverse process (right-to-left) uses learned score functions to desynchronize these phases.
Quasi-equllbrium. In the thermo- dynamic limit (N oo), the collec- tive dynamics of the oscillator popu- lation can be effectively described by
mean-field field dynamics the mean-field evolution of single representative oscillator with phase 0. The mean of Eq. (3) follow the Fokker-Place Planck equation:
Öp(0,t) a (k(t)r(t) sin(((())))))))))))))))))))))))) - 0) + (er((t) sin(Wher - ()))() + D. ppp(0.t) (5)
Öt
0e
002
In our framework, akin to standard diffusion models, the coupling strengths and noise variance gradually increase over time. Consequently, the system does not reach true stationary distribution. Instead, after an initial transient phase, the evolution of p(0. becomes sufficiently slow such that, at each moment, the system can be approximated well by an instantaneous steady state - regime we refer to as quasi-equilibrim, characterized by (p)(0)/Ot ss O.
Under quasi-equilibrium, the phase distribution approximately satisfies:
1
((TTr(T) (T) 0) DT
(T)(T)
Pa (0)
7 exp
cos(t/rer 0)
Dr
where the normalization constant, and denotes the final timestep. The proof is given in the Appendix. In the long-time limit, the average phase (T(") synchronizes toward the reference phase Whet, the distribution further simplifies to a well-known von Mises form (Gaussian on a circle):
von Mises Distribution: Pa(0) R exp (K(T)r(T) + KTT((T) cos(wiie 0)
Dr
Local Coupling. A natural variant the Kuramoto model is to introduce local coupling, where each oscillator interacts only with its neighboring oscillators:
d0p
sin(O) - -0) + -Ker(t) sin(ther 0:) + 2D JEN.
Σ
dl
W
where N₁ denotes the of neighbors for the i-th oscillator. In contrast to the globally coupled case, local interactions introduce spatial inhomogeneity and break the mean-field approximation. As result, the system exhibits diffusion-like behavior, with oscillatory phases progressively smoothing out over space and time. This manifests as blurring effects during the forward process, analogous to phenomena observed in heat dissipation models [5I) and blurring diffusion models [24].
Interestingly, while local coupling prevents analytical simplification of the dynamics, the strong reference phase (via KGr) still guides the system toward global synchronization. After sufficiently long evolution, the terminal distribution concentrates around the reference phase and can be effectively approximated by
Mises distribution, despite the absence of mean-field symmetry.
Coupling Strength and Noise Variance. the forward process, we maintain the relationship Kl)(() D, > (() to balance structure and noise. D, > K (() ensures that the stochastic noise is strong enough to disrupt image details, while Ker((!) > Dr guarantees that the stronger reference coupling dominates over noise and local/global coupling to guide the system toward synchronization.

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Algorithm Training algorithm for Kuramoto orientation diffusion models.
Require: Score prediction network .,, noise variance schedule {2D;) T-1 forward Kuramoto drift f(,,), noise NV I), number of MC samples M.
1: repeat
2: Sample initial phase variable: 0g p(0))
3: Sample timestep: i ~ (),
4: Simulate the forward Markov chain to obtain 0:- p(0t-1100)
5: Initialize sample counter m = 0
6: while m < M do
7: Sample: 0T 02-1 1(0--11
v 2D
8: Wrap phase: 0 = -0mm - T) mod (2T) π
9: Compute local score: V Ver log p(0'" 1 (1)
10: m f- m
11: end while
12:
Compute training loss: L
1 Σm M
-
p(op (
13:
until converged
2.3
Learning the Score Function
In the reverse process (right-to-left in Fig. 2) perform desynchronization guided by learned score functions, progressively restoring image complexity from synchronized von Mises distribution back diverse angular states. Below we illustrate how the score function is learned.
Local Score Matching. our stochastic Kuramoto models, the presence of the nonlinear drift renders the marginal distribution p(0,) intractable, and consequently the score function Vo, log p(0,) also unavailable in closed form. As result, typical score-matching matching algorithms that rely on direct access the marginal density cannot be applied. Nevertheless, we can still train a score network by exploiting the local Markov transition kernel p(O, 00.-) 1), based on the following general identity:
Vo, ((.)) -E 10 (.)) Vo, log (og.(00.0.. 0r-1)
where the expectation is taken over the reverse transition p(O,-1(0.) A detailed derivation of this identity is provided in the Appendix. Although the reverse transition intractable in practice, Denoising Score Matching (DSM) (65) shows that the score function can be learned by sampling from the forward transition p(O, 0.-1) instead. Specifically, the training objective can be written as:
EI,U Eo, Bo. Be. Eo ~pt0, 0.-1) 2D. ()D,)ss(O,,1)) V log ((0,0(0011
(10)
where s0,, t) denotes the score network being optimized. At each training step, we simulate the forward Markov chain to obtain 0;-1, and then sample multiple 0, from the local transition kernel O-)) estimate the training loss via Monte Carlo (MC) approximations.
Wrapped Gaussian Transition. Due to the phase wrapping on periodic domain, the local transition probability 0(0((0,-) follows wrapped Gaussian distribution:
000,0,11) = Ww ( + f(0, 1, 10-1, -1),2D,-1!
K
(0: 10--0:11 -((0 27th)
(11)
Σ
exp
4TD
4D,-1
k=-K
where f(0,,t) t) denotes the forward drift of the Kuramoto model, i.e., the coupling and reference terms weighted by the coupling strength. Since the wrapped Gaussian involves infinite series, neither the transition density nor its score function admits a simple closed-form. We thus approximate small finite number of terms K.
the transition by truncating the summation
Periodicity-aware Networks. To incorporate the inherent periodicity of the phase variables into the score network, we embed the input phases using sinusoidal features [sin(0), cos(O)) input to the
5

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Table 1: FID Scores (4) on SOCOFing fingerprint dataset [5
Diffusion Steps
100
300
1000
SGM (60)
104.92
62.66
23.84
Kuramoto Orientation Diffusion (Globally Coupled)
74.41
47.93
20.64
Kuramoto Orientation Diffusion (Locally Coupled)
67.49
43.57
18.75
Table 2: FID Scores (L) Brodatz texture dataset
8
Steps
100
300
1000
SGM 60
38.33
22.40
20.37
Kuramoto Orientation Diffusion (Globally Coupled)
20.26
18.51
15.42
Kuramoto Orientation Diffusion (Locally Coupled)
18.47
15.93
14.19
network. The network predicts outputs (s1 (0, t), (2 t)] corresponding the two Cartesian components. We then project the output back onto the angular domain via:
s s(0, t) = $1 cos(O) - s2(0,t) sin
(12)
This operation ensures that the score prediction respects the circular geometry of the phase space.
Training Algorithms. We summarize the training algorithms in Alg. [ To adapt non-periodic image data to the periodic phase domain, input pixels the range [-1, are linearly mapped to [-0.9m, 0.9r]. This margin near the boundaries helps distinguish values near -1 and 1, which would otherwise collapse under phase wrapping. Throughout both training and inference, we enforce the periodic geometry by wrapping all phase variables into the interval [-x, .] after each SDE step. Our training procedure relies on estimating local scores using Monte Carlo samples from truncated wrapped Gaussian transitions. We find that using K K and M= samples per step provides good balance between training stability and computational efficiency.
3 Experiments
3.1 Setup
Datasets, Baselines, and Metrics. We evaluate our proposed method across both general- purposs and orientation-rich image generation tasks. For standard benchmarking, we first test on CIFARIO B5 To specifically assess performance orientation-dense data, then apply model the SOCOFing fingerprint dataset [56], the Brodatz texture dataset [1] 8, and the ground terrain dataset 68). The input image resolutions 3232x32 for CIFAR10 10 and xx66 for SOCOFing. The Brodatz
100 steps
300 steps
1000 steps
100 steps
300 steps
1000 steps
Figure 3: Samples generated by our Kuramoto orientation diffusion model on SOCOFing fingerprint and Brodatz texture datasets under varying denoising steps.

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Table 3: FII Scores (1) on the ground terrain dataset 68
Diffusion Steps
100
300
1000
SGM 660
114.90
56.72
33.79
Kuramoto Orientation Diffusion (Globally Coupled)
101.65
54.17
33.56
Kuramoto Orientation Diffusion (Locally Coupled)
92.86
49.68
30.62
100 steps
300 steps
1000 steps
Figure 4: Samples generated by our Kuramoto model on the ground terrain dataset.
texture dataset originally consists of high-resolution samples depicting various textures (e.g., grass, water, sand, wool) with sizes of x12 1x 1024x 1024. Due to the limited number of available images, we increase the dataset size by dividing these high-resolution textures into smaller patches of 3x22 Compared to Brodatz textures, the ground terrain dataset [68] contains broader set of orientation-rich material textures (e.g.. plastic, turf, steel, asphalt, leaves, stone, and brick) at the higher resolution of 128x 128. We primarily compare with the standard variance- preserving (VP) score-based generative model (SGM) [60] To evaluate the quality of generated images,
the Fréchet Inception Distance (FID) [21, widely used metric for assessing visual fidelity and diversity. All models are trained until the FID score converges to ensure fair comparison.
In the Supplementary, on the 2D Sphere [46] 45 [ [8], and (ii) Navier-Stokes fluid velocity fields [6]. The former suite of datasets is defined naturally periodic grids, while the latter one constitutes intrinsically angular data. Together with the image datasets, the benefit of the synchronization inductive bias built into model will be consistently validated across three distinct domains standard iiientationddenss images (with pixels mapped to phases), spherical geophysical fields, and fluid velocity phases.
extend evaluation beyond images to (i) Earth and climate science datasets
Timesteps. Due to the phase coupling in our Kuramoto diffusion model, the forward process leads to faster convergence toward the terminal distribution compared to conventional diffusion models. To explore this advantage, 1000 steps. These comparisons reveal how efficiently our model captures structural information with fewer steps. Details of the noise and coupling schedules are provided in the Supplementary.
evaluate our model under varying diffusion step counts: 100, 300, and
3.2 Results
Fingerprints and Textures. Tables and report FID scores on the SOCOFing fingerprint and Brodatz texture datasets, respectively. Across all diffusion step settings, both globally and locally coupled variants of our Kuramoto orientation diffusion model consistently outperform the standard SGM (60] The improvement is especially notable on the Brodatz texture dataset, where our 100- step Kuramoto model achieves performance comparable to or better than SGM using 1000 steps demonstrating a substantial gain in sampling efficiency. These results underscore the advantage of incorporating structured coupling dynamics for modeling orientation-denss data. The synchronization based inductive bias helps preserve coherent directional patterns, which are critical for the perceptual quality of textures and fingerprints. Notably, the locally coupled variant offers further improvements by aligning with the local spatial correlations. Fig. provides qualitative examples, illustrating that method produces sharp and consistent patterns under varying diffusion steps.
Ground Terrain. Table 3 reports FID on the ground terrain dataset (8), comparing our Kuramoto diffusion model with SGM. The trend mirrors fingerprints and textures: across denoising step counts, our method consistently attains lower FID. This indicates that the synchronization prior continues to help at orientation-dent scenes of higher resolutions. Qualitative results in Fig. iillustrate that,

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Table 4: FID Scores (4) on CIFARIO
35
Diffusion Steps
100
300
1000
SGM [60]
38.04
25.76
3.17
Kuramoto Orientation Diffusion (Globally Coupled)
29.96
25.83
11.58
Kuramoto Orientation Diffusion (Locally Coupled)
28.17
24.86
10.79
100 steps
300 steps
1000 steps
Figure 5: Samples generated by our Kuramoto model on CIFARIO under varying denoising steps.
under fewer denoising steps, our samples exhibit coherent directional structure and material appearance, while additional steps further refine detail realism and texture alignment.
CIFARIO. Table 4 presents the FII) scores for our Kuramoto orientation diffusion models against SGM on CIFAR At 100 diffusion steps, both Kuramoto models substantially outperform SGM, highlighting the effectiveness of structured synchronization an inductive bias in low-step regimes. At 300 steps, Kuramoto models achieve comparable or slightly better performance than SGM, demonstrating their ability to maintain sample quality as diffusion progresses. At 1000 steps, SGM achieves the best overall score, though both Kuramoto models remain competitive.
Fig. displays generated samples under different step counts. As diffusion steps increase, the benefits of structured synchronization become less prominent. We expect that in longer trajectories, the structured reverse dynamics may inhibit the flexibility to fully capture fine-grained details, especially in datasets that lack strong orientation priors. These results suggest that Kuramoto-bass synchronization dynamics are particularly advantageous in generating high-quality images within limited steps, even on general-purpose datasets. However, this structured bias may slightly limit expressiveness under excessive denoising steps on natural images lacking strong orientation patterns.
3.3 Discussions
Structured Destruction. Our Kuramoto forward process introduces very structured destruction process, leveraging either global or local coupling mechanisms to achieve more controlled noising dynamics. Unlike conventional isotropic diffusion which quickly loses object structure, our model preserves the objects in the early stages through synchronized coupling (see also Fig. [ This struc- tured noising progressively aligns similar phase variables while maintaining orientation consistency, allowing the model to better retain structural information. As noise levels increase, the coupling interactions expedite the convergence the noise distribution, enabling faster transition compared to standard diffusion models. This unique dual-phase dynamic- initial structured synchronization followed by rapid noise adaptation allows our model to converge faster while maintaining orienta- tion coherence. The empirical SNR plot in Fig. 6 clearly demonstrates the advantage described above, highlighting how the coupling-driven dynamics allow for efficient and structured noise progression.
Hierarchical Generation Process. Figs. 7 and and depict the hierarchical generation process of our Kuramoto orientation diffusion model. The generation starts from synchronized state sampled from the von Mises distribution, representing a low-entropy configuration with highly aligned phase variables. In the reverse process, the large-scale structure is established first, as global coherence from the initial synchronized state is preserved. This occurs because the forward diffusion inherently maintains global coherence in the later stages, which correspond the early stages in the reverse process. Once the primary structure is set, finer-scale details gradually emerge through successive diffusion steps, driven by anti-coupling dynamies that introduce local variability, allowing the image to evolve into more complex and nuanced patterns, Our model follows a coarse-to-fine paradigm, where the global structure remains consistent while localized, intricate features evolve flexibly.
8

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Figure 7: Hierarchical generation process of our locally coupled Kuramoto diffusion model applied to CIFAR-10. The generation follows coarse-to-fine progression, starting from structured von Mises sample and progressively adding fine-scale details. These results correspond to 100 diffusion steps.
Figure 8: Hierarchical generation of our locally coupled Kuramoto diffusion model on the SOCOFing fingerprint dataset (left) and Brodatz texture dataset (right). The model first establishes large-scale orientation patterns, followed by finer texture details. These results correspond to 300 diffusion steps.
The hierarchical nature of the Kuramoto diffusion enhances interpretability by providing a clear generative pathway from global coherence to local variability. This hierarchical genera- tion approach aligns with the spectral biases observed in coarse- to-fine generation of diffusion models [34] 51]. In typical score- based diffusion models, this hicrarchy arises implicitly from the progressive noise attenuation, preserving frefereuenccy components longer while high-free feeh-frreuency details emerge later. In contrast, Rissanen al. [51] use heat equations for the for- ward process. resulting in isotropic blurring as high-frequency components dissipate. Our Kuramoto model explicitly encodes hierarchical progression through synchronization dynamics. This structured, non-isotropic phase alignment offers more interpretable pathway from global to local patterns.
smpii ca see of L'eea VTCCACAAC SCRAD-9
Figure 6: Empirical SNR over time.
Related Work
Generative Models. Deep generative models have made remarkable progress in the last decade, both in the creation of powerful foundation models [52 and in the design of sample-efficient optimization algorithms [66] 49 Farly advances deep generative modeling, such as Variational Autoencoders (VAEs) (33] and Generative Adversarial Networks (GANs) [19], pushed the frontiers of generative capabilities but faced distinct limitations: VAEs often produced blurry samples, while GANs suffered
from instability training. Recently, diffusion models [7 23
60
58
I3
25 32 have emerged as
powerful, principled alternative. These models learn to reverse noising process by estimating distributions. Building on the view of generative modeling as a continuous transformation from noise to data, Flow Matching [8] and Rectified Flow (39] directly learn velocity fields connecting noise and data without relying on stochastic diffusion, sidestepping the need for score estimation. Stochastic Interpolants 2 further generalize diffusion and flow models by learning stochastic trajectories interpolating between the data and prior, providing unifying perspective across two modeling paradigms. Several recent
the score function V. log p(x;) of intermediate Seausian--coruppe

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efforts extend flow matching and diffusion models to non- non-Euclidean geometries. For example, Riemannian Flow Matching [00] formulates generative models directly on Riemannian manifolds. respecting underlying geometric constraints. Other Riemannian diffusion models account for manifold curvature to enable generative modeling over geometries such as hyperspheres, tori, and hyperbolic spaces 1 (3 26 12 69
14] These advances highlight the growing interest in incorporating geometric inductive biases into generative modeling.
28
Our work contributes to this landscape by exploring nonlinear diffusion models grounded in stochastic Kuramoto dynamics. Instead of linear drifts conventional diffusion models, we introduce structured phase dynamics driven by nonlinear Kuramoto coupling, evolving on the periodic domain. Unlike classical score-based models that assume Gaussian forward process, our framework leads to non- Gaussian, wrapped distributions that evolve according to nonlinear SDEs. This extends recent interest in geometry-aware generative models and interpretable generation process.
Neural Oscillations in Machine Learning. Oscillatory dynamics are central to understanding biological neural systems, where phase-locked rhythms and traveling waves are believed support functions such as sensory binding, working memory, and attention [9 [6]. Motivated by their computational relevance, recent machine learning research has increasingly incorporated oscilla- tory and wave-based dynamics into neural representations, treating them as inductive biases that 31 key focus has been on synchronization, where multiple units align their oscillatory phases to form stable or emergent patterns. This phenomenon has been studied through the lens of the Kuramoto model [36]. a classical framework from nonlinear dynamics that describes how populations of coupled oscillators evolve toward synchronization. In computational neuroscience and machine learning, Kuramoto-inspired models have been applied to tasks such as modeling neural connectivity (7). clustering through emergent synchrony [50]. and mitigating over-smoothing in graph neural networks [44] A recent notable example is the Artificial Kuramoto Oscillatory Neuron (AKOrN) framework [41], which replaces thresholding units with oscillatory ones to bind neurons together through synchronization dynamics. These efforts reflect the growing interest in integrating principles of neural oscillations and synchronization into machine learning frameworks, offering novel perspectives and tools for modeling complex, dynamic systems.
promote generalization and structured behaviors [54 55
D5 61
5 Conclusion
This paper introduces nonlinear score-based generative framework that incorporates stochastic Kuramoto dynamics to model orientation-rich data. By formulating the forward process as syn- chronization and the reverse process desynchronization, our method brings biologically inspired inductive biases into the diffusion process. Through wrapped Gaussian transitions and periodicity- aware networks, the model naturally captures the geometry of angular data. Experiments show that our approach outperforms conventional baselines on orientation-dense datasets and remains competi- tive on general image generation tasks. This work highlights the potential of integrating biologically inspired synchronization dynamics as structured priors in generative modeling of orientation-dena data, paving the way for incorporating nonlinear dynamics into generative models.
Limitations and Future Work. A primary limitation lies in the training efficiency: each training step incurs (TT) time cost due to the need to explicitly simulate the forward Markov chain. Actually, this cost actually can be nearly eliminated with pre-computation & cache: before training, we can the forward SDE on the entire dataset, save all pairs to disk, and then load them directly during training. This makes the simulation cost effectively (()) at each training step. Otherwise if disk space limited, we can simply re-generate and cache one epoch's worth of pairs at the start of each epoch. This still dramatically reduces the simulation overhead.
Beyond computational improvements, applying this framework to neural spiking data offers compelling opportunity to further explore the biological plausibility of synchronization-drinn generative models and to bridge the gap between theoretical neuroscience and machine learning.
Broader Impacts. Our approach may benefit socially relevant domains such as biometric security (e.g,, synthetic fingerprint generation), medical imaging (e.g., fiber orientation modeling in MRI), and scientific visualization (e.g., materials analysis). As with other generative technologies, we advocate for responsible research and deployment in accordance with ethical and societal guidelines.
10

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Acknowledgments
We would like to thank anonymous reviewers for their constructive suggestions and feedback. Yue Song was supported by the gift donation from Cisco. T. Anderson Keller was supported by the Kempner Institute Research Fellowship. Takeru Miyato was supported by the ERC Starting Grant LEGO-3D (850533) and the German Research Foundation (DFG): SFB 1233, Robust Vision: Inference Principles and Neural Mechanisms, project number: 276693517
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Math Derivations and Intuitions
21
A.1
Mean-field Dynamics of Fokker-Plamn Equation for Stochastic Kuramoto Models
21
A.2
Local Score Matching
22
A.3
Fourier Interpretation of Kuramoto Coupling
22
A.4
4.4
Why Kuramoto Diffusion Falls Short on General Natural Images
22
More Experimental Results
22
B.
Implementation Details
22
B.2
Ablation Study and Alternative Metries
23
B.3
Earth and Climate Science Datasets on Spheres
24
4
Navier-Stokes Fluid Velocity Field
25
B.5
More Examples of Generative Samples
26
A
Math Derivations and Intuitions
A.1 Mean-field Dynamics of Fokker-Planck Equation for Stochastic Kuramoto Models
the thermodynamic limit infinite oscillators (N - oo), the collective dynamics of the oscillator population can be effectively described by the mean-field evolution of a single representative oscillator with phase variable 0. The corresponding probability density p(0, I) evolves according to Fokker-Planck equation:
0p(0,t) t) Ot
θ 00
p'0(t,t)
(K(t))(t)snn sin(((t)) 0) Kne(t) sin(Viet - 0)) + Di
002
(13)
where r(t) and (t) denote the magnitude and phase the complex order parameter, respectively. In the long-tim time limit, the qqasi-qquilibrium satisfies (0)/0t = 0. At this point, the stationary solution p(0, T) approximately follows:
8 ((,,T)) 0 (K)())r(T)sin(t)))) sin( - 0) + Kier(T) sin (Diect ()),((,)))
DT 002
00
Dr 8p(0,T)
~ (K(T)r(T)sinn (i 0) + (Te sin( 0(), (), + Ci
00 0p(0,T) T)
(TT)r(T) (in( - 0) + (T) (((T))snn((ii sin(verer 0)
(14)
p(O,T)
00
Dr
logp(0. p(0, T) K(T)r(T) cos(1)(T) - 0) + KTe((T
cos(wher 0) + C2
Dr K(T)r(T) cos(v)(T) 0) KTT((T) cos(tiies 0))
Dr
p(0, T) exp(
Dr
Dr
where C1 C2 are constants, and Z is the normalization constant. We assume C = because the distribution is periodic, and C2 is absorbed into Z. Since the average phase (b() will synchronize the reference thet the quasi-stationary solution is given by:
(f) ~ Z exp ((T)r(T)+ (TT((T cos(Vier - 01)
(15)
Dr
This final form follows von Mises distribution, reflecting the low-entropy steady state induced by synchronized interactions and reference phase attraction.
21

---

A.2 Local Score Matching
At any timestep t of the Markov chain, the marginal distribution p(0,) can be expressed as:
p(O,) / p((0--)p(00(0---)))) d0-
(16)
Vo.p(0,) p(00--)Ve.,(001001111
For any score Vo, (Ot), we have Vo, logr(0.) Vo.r(0.)/p(O.)) Then we can rewrite the above equation leveraging this identity:
Vo, p(,) = p(0r-1)p(0, )))
/
Vo, log p(O.)
log p( 01-1) d0: d0;-1
(17)
Vo, log p(O.) = E0.-1~p(0,-11 110,)
Vo, log po.
The above results indicate that the score can be effectively approximated by the local transition kernel.
A.3 Fourier Interpretation Kuramoto Coupling
From spectral standpoint, local phase coupling behaves like angular low-pass filter. In the small-angle approximation (sin(O, 02) z 0, 02) and without global reference phase, the forward SDE reduces to a stochastic heat equation (or graph Laplacian form equivalently):
d0: K(t V2Dis'
(18)
dt
where 7 denotes the Laplacian operator. Each Fourier component decays like e K(t)k?t where denotes the spatial frequency. Modes with very high spatial frequency (pixel-scale noise) are damped almost instantly, while moderate-frequee much more slowly. Consequently, the Kuramoto drift effectively filters away pixel-scale noise while preserving the smooth, orientation-rich patterns that constitute edges.
modes (coherent, edge-defining structures) decay
A.4
Why Kuramoto Diffusion Falls Short on General Natural Images
Our Kuramoto coupling is foremost synchronization mechanism: it excels pulling similar phases together, thereby preserving edges and repetitive patterns. On orientation-rich data where the data are dominated by the simple outlines and repeating ridge structures, this yields sharp samples in fewer steps. By contrast, natural images (e.g.. CIFAR- 10) demand modeling e-g,, complex, global semantics (object shapes, color gradients, backgrounds), often characterized by higher-order and longer-range correlations. In this setting, the local synchronization bias becomes less relevant and potentially even detrimental; an isotropic diffusion process with a global drift may better suited to capture these large-scale, non-repetitive features. As result, our method trades some global fidelity (reflected higher CIFAR-10 FID) in exchange for strong local coherence.
B More Experimental Results
B.1
Implementation Details
Alg. outlines the inference procedure for model. We adopt the SDE formulation by default, but can be optionally replaced with an ODE solver for improved efficiency by modifying Line to: 0,-1 0, f(0,,t) D, s(0,, t). We AdamW optimizer with learning rate of 1e-4 and apply exponential moving average (EMA) updates with decay rate 0.995. A single NVIDIA A100 GPU is used for all the training and inference processes.
Network Architectures. We use U-Net architecture following the design of 60 equipped with three self-attention layers (63). These are applied at spatial resolutions of 16, 8, and 4 for CIFAR-10
22

---

Algorithm 2 Inference algorithm for Kuramoto orientation diffusion models.
Require: Trained score network s(,, :), noise variance schedule {2Dr forward Kuramoto drift f(:.), noise (N, I).
1: Sample initial phase: 0 p(Or)
2: Initialize timestep counter /
3: while / 0 do
4: Update: 0--1 0, f(0,, 2D; s(0,, + /2Die
5: Wrap phase: 0z-1 = (0:-1 + T) mod (2T)
6: f f
7:
end while
and Brodatz, and 24, 12, and 6 for SOCOFing. Timestep conditioning is implemented via sinusoidal positional embeddings. Each block uses group normalization [67] and GELU activations [20] throughout. The same architecture is shared across both our method and SGM for fair comparison.
Table 5: Linear schedules of noise variance and coupling strength under varying diffusion steps.
Global Coupling
Local Coupling
Steps
2D
K
Kief
2D
K
Kis
100
[
[3e-5,0.03]
[4.5e 5,0.045]
(1e-4.0.1)
[5e 75e-5,000 5,0.05
[5e 5,0.05]
300
[1c-4.0.07]
(3002.0.02]
5,0.03
[10-4.0.07]
[5e 5,0.03
[5e-5,0.00 5,0.033
1000
[1e-4.0.015]
[3c-5,0.0045]
[4.5c (,.5e-5,0.00675
[1e-4,0.025]
[5e-
[5c-5,0.01]
Schedules of Noise and Coupling Strength. variance 2D,, internal coupling strength K, and reference phase coupling K In the globally coupled variant, we maintain the relation Kat 2D K to ensure synchronization toward reference phase while allowing sufficient noise injection. the locally coupled variant, we increase the internal coupling strength K to compensate for its restricted spatial influence, which is limited to a 5 X 5 neighborhood around each pixel. We keep K K in this case, since the reference coupling term inherently has broader impact by acting across the entire image domain. For the SGM baseline, we adopt the VP-SDE formulation with linear variance schedules: [1e-4,0.] for 100 steps, [1e-4, 0.07) for 300 steps, and [1e-4, 0.02) for 1000 steps.
the linear schedules for the noise
B.2 Ablation Study and Alternative Metrics
Table 6: FID Scores (L) on Brodatz texture dataset [1
Steps
100
300
1000
SGM 660
38.33
22.40
20.37
Reference-only Process (K(t) = 0)
33.76
20.54
19.01
Kuramoto Orientation Diffusion (Globally Coupled)
20.26
18.51
15.42
Kuramoto Orientation Diffusion (Locally Coupled)
18.47
15.93
14.19
Ablation of the Kuramoto Coupling. To isolate the impact of Kuramoto coupling, include model variant by setting ((t)= = Eq. Bb. which keeps only the phase-reference drift in the SDE:
d0:
Kte(()) sin 0;) - /2Dzs'
(19)
di
Table presents the evaluation results on Brodatz textures. Removing the non-linear Kuramoto cou- pling raises 100-step FID by over 15 points, which confirms that non-isotropic phase synchronization the key driver of both rapid convergence and thhe performance gain on orientation-rich data.
Alternative Metric. The FID metric is known for several limitations its dependence on Inception embeddings. its Gaussian-momen matching assumption, and occasional misalignment with human judgment. As more robust alternative, we adopt CLIP-MMD (CMMD) 29 which computes nonparametric Maximum Mean Discrepancy the pretrained CLIP feature space, which combines the benefits of distribution-fiee free sample efficiency with the strong embedding power of CLIP.
23

---

Table 7: CMMD Scores (L) on Brodatz texture dataset
Steps
100
300
1000
SGM [60
0,183
0.165
0.141
Kuramoto Orientation Diffusion (Locally Coupled)
0.072
0.045
0.030
Table 7 evaluates our local Kuramoto model versus SGM on the Brodatz. textures dataset using the CMMD score. Our model achieves substantially lower CMMD every step count. Remarkably, the 100-step Kuramoto model outperforms the 1000-step SGM by a large margin. These results mirror and amplify our FID improvements, demonstrating the superior sample fidelity even under distribution-free, CLiP-based evaluation.
B.3 Earth and Climate Science Datasets on Spheres
To further evaluate our Kuramoto orientation diffusion model on naturally periodic data, we consider four real-world datasets of Earth and climate events: significant volcanic eruptions, earthquakes, floods, and wildfires [46 45 5 [8]. These datasets capture empirical spatial distributions of geo- physical events on the surface of the Earth and are inherently defined over a 2D spherical domain. We map longitude to [- s and linearly scale latitude from | n 7 to the same interval to ensure compatibility with the periodic modeling framework. We compare our approach against a suite of Riemannian geometry-ave generative models, including Riemannian CNFs 44) Moser flows [53
CNF matching [4) and Riemannian score-based or flow-based generative models 26 12
Fire
Flood
Quake
Volcano
4.0
Figure 9: Learned density plot of our method on each Earth and climate science dataset. The X-axis denotes the longitude while the Y-axis represents the latitude.
Table 8: Test NLL on Earth and climate science datasets averaged across 5 runs.
Dataset
Volcano
Earthquake
Flood
Fire
Riemannian CNF
40
-6.05-00.1
0.14+:0.3
1.11::0.19
-0.80+00.54
Moser Flow 553
4.21110.17
-0.1611006
0.5710.10
-1.2811.055
CNF Matching
-2.38++0.17
-0.38+0.01
0.25+00.2
-1.40=+0.02
Riemannian score-based
12
-4.22::0.25
-0.19+0.07
0.48+00.17
-1.33+0.06
Riemannian diffusion model
26
-6.6110.96
-0.4010.05
0.43:10.07
-1.3811.05
Riemannian flow matching [10
1..93+1.67
-0.28+0008
0.42+00.05
-1.86++0.11
Our Kuramoto orientation diffusion model
-5.1810.17
-0.1810.06
0.49
-1.4410.05
Fig. visualizes the learned densities produced by our method, capturing both highly concentrated regions (e.g., volcanic and fire clusters) and dispersed patterns. Table presents the test negative log-likelihood (NLL) each dataset. Our method achieves comparable performance against these baselines. We compute NLL using the change-of-variaa
formula under the time-reversed ODE solver: log ((g) = log (0T) + El- Tr(Joco..e) where p(Or) denotes the von Mises prior and b(O,, t) denotes the backward drift, i.e.. b(0,t) t)
f(.. + D. s(,,, t). The Jacobian trace is
estimated via Hutchinson's stochastic estimator
27
While our method achieves competitive NIL. scores across datasets, we note that direct comparison across models can be influenced by the choice of different priors. In particular, von Mises distributions

---

with higher concentration can artificially boost log- log-likelihood scores, whereas broader priors can deflate them. Thus, while our results confirm the effectiveness of our model, qualitative evaluations remain critical for comprehensive assessments.
B.4 Navier-Stokes Fluid Velocity Field
To demonstrate the applicability to real angular data, we evaluate our Kuramoto diffusion model on 2D incompressible Navier-Stokes (NS) velocity fields from [6.. Each velocity frame (Vz, Uy) is converted to polar form:
22 + oy.
0 arctan2(u,, Vz))
(20)
where 2 0 is the amplitude and 0 E (-T, πi is the phase. Since r is positive, we work in log-magnitude z = logr and apply a VP SDE there the log space, while phases evolve via the locally coupled Kuramoto SDE. The forward processes are defined as:
dzj = B, +
1
Amplitude: dt 2
(tt) cos((/ 0g) + VB
20 log To
JEN;
(21)
d0; ri
Phase: dt
Σ
((1) sin(O) - 0:) + Kte((t) sin(wher 03)] + 222
[
JEN;
where the boxed terms couple the two channels: (i) local phase coherence accelerates amplitude growth (via cos in the z-SDE), and (ii) larger amplitude strengthens phase synchronization (amplitude factor ri in the (-SDE). To avoid over-coupling late in diffusion (when noise dominates), enable coupling only in the early stage, i.e., for f < T with a constant a > 1; for 2 T/a: the two SDEs evolve independently under their native log- VP/Kuramoto dynamics.
In the reverse process we train two score models, one in log-magnitude space and one on phases, to solve the corresponding reverse SDEs and jointly synthesize (r, 0).
Unconditional Generation. Following fluid Quid-dynamics practice, we assess realism in the spectral domain at the final time step. From each 2D velocity field, we compute the radial energy spectrum E(k) and then: (i) fit a line to log ((k) vs. log over the mid-wavenumbbe band k [0.1, 0.4] kmaz (with kmna = min( H, W)/2) and report the absolute slope difference, and (ii) compute the 1D Wasserstein distance between the normalized mean spectra real and generated samples.
Table Spectral evaluation generated Navier-Stokes fluid velocity fields.
Metrics
Slope Difference ())
Wasserstein Distance (L)
SGM (Amplitude- Phase Decomposition)
0.7435
0.0015
SGM (Cartesian Coordinates)
0.5590
0.0029
Ours (Naive Kuramoto Phase + Log- VP Amplitude)
0.6954
0,0011
Ours (Coupled Kuramoto Phase Log-VP Amplitude)
0.3343
0.0005
Table shows that the Coupled Kuramoto Phase + Log- VP Amplitude model achieves the best spectral realism among all methods, with the lowest slope error and Wasserstein distance. Relative to the SGM (AmplitudePhaas Decomposition) baseline, this corresponds to a 55% reduction in slope error and 67% reduction in Wasserstein distance. It also improves over SGM (Cartesian Coordinates) by 40% in slope error and 83% in Wasserstein, and over the Naive Kuramoto Log-VP variant by 52% (slope) and 55% (Wasserstein). These gains indicate that explicitly coupling phase synchronization with amplitude evolution yields more physically plausible velocity spectra than treating angle and magnitude independently or modeling them without coupling. Fig. [T0 the average energy spectra with fitted lines, where the coupled model shows a visibly tighter fit in the mid- mid-frequency band and closer overall spectral shape.
Conditional Forecasting. Building on the unconditional results above, we next consider a setting that is arguably even more useful in PDE modeling: forecasting future states from history. Unlike unconditional generation which tests realism in distribution, forecasting probes whether the model has learned the dynamics well enough extrapolate in time.
25

---

Coupled Kuramoto Phase Log-VP Amplitude
SGM (Cartesian Coordinates)
10
10
10
13"0
10"4
Take
colope-rd.00
0dcpp--3.77
107
101
15P
Figure 10: The average energy spectra with fitted lines.
We keep the coupled representations and train two conditioned score models 8, one for phases I and one for log-magnitudes z together with their unconditioned counterparts s, via condition dropout. At inference, each pair fused with classífier-free guidance (CFG) [22]:
Amplitude: A. (z) = 8, w ( (z(()))) - s,)2))
(22)
Phase: س (0) = (0) + w (0)((c))) (0)
where e denotes the two-step Navier-Stokes history, and w is guidance temperature that controls adherence to the condition (larger denotes stronger conditioning). We evaluate accuracy using Mean Squared Error (MSE) between predicted and ground-truth velocity fields.
Table 10: MSE of Navier-Stokes fluid velocity predictions.
Methods
MSE (L)
SGM (Cartesian Coordinates)
0.0260
Ours (Coupled Kuramoto Phase + Log- VP Amplitude)
0.0188
As shown in Table 10 the Coupled Kuramoto Phase Log-VP Amplitude model reduces MSE from 0.0260 0.0188 compared with the SGM (Cartesian Coordinates) baseline, indicating more accurate conditional forecasts. Fig. illlstratee two examples; predictions closely match the ground truth in both horizontal and vertical velocity components.
history
prediction
history
prediction
X velocity
X velocity
Y velocity
Y velocity
Figure 11: Conditional one-step forecasts of method on Savier-Stoker velocity: history (left), predictions (middle), and ground truth (right).
B.5
More Examples of Generative Samples
Figs. 3 [4] and 15 present additional randomly generated samples from 1000-step locally coupled Kuramoto orientation diffusion model on the SOCOFing fingerprint, Brodatz texture, ground terrain, and CIFARIO 10 datasets, respectively.
26

---

Figure 12: Randomly generated samples on SOCOFing fingerprint dataset.
27

---

Figure 13: Randomly generated samples on Brodatz textures dataset.
28

---

Figure 14: Randomly generated samples on the ground terrain dataset.
29

---

Figure 15: Randomly generated samples on CIFAR10 dataset.
30


Reviews:
[{"review_id": "yIK7rKs6Sv", "reviewer": "Reviewer_4DgT", "summary": "This paper proposes a novel approach for building diffusion models, based on so-called Kuramoto dynamics. In this model, each pixel of an image is treated as an oscillator, characterized by a phase, which progressively evolves in the forward noise process towards an base distribution attractor. The model provides a mechanism to couple pixels together in both the noise and denoising process, which the authors say lead to a non-isotropic noise dynamics that is more prone to preserving edges and global features of ‘orientation-rich’ data. The paper provides the derivation of the necessary score matching loss function and provide some illustrations of their model on fingerprint and texture datasets compared to a standard diffusion model approach.", "questions": "- Can the authors illustrate clearly the effect of their modelisation on preserving edges and orientation-rich features? This would help bring the main point of the paper home. I can imagine either showcasing different stages of the denoising process for fingerprint data under different models. Hopefully it should be pretty clear that the fingerprints are pretty salient early on. It might also be possible to compute summary statistics of the images that illustrate the emergence of these features early in the reverse diffusion process. \n\n- Can the authors add additional baselines to their generative modeling evaluation? In particular, it’s possible that the SGM noise schedule differ from the Kuramoto one (e.g. VP-SDE vs VE-SDE or even different noise schedules within the same class of SDE) and that could impact the generation quality without necessarily meaning that the Kuramoto process is better. More generally, I would recommend making the external baselines robust such that the reader can be convinced that the gains are specifically due to the phase synchronisation mechanism, and not because of any other implementation detail.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 1, "originality": 3, "strengths_and_weaknesses": "**Strengths**:\n- The model described here (and by that I mean the specific SDE) is new as far as I know for generative modeling, and the authors go through the derivations necessary for the score matching objective to use, as well as for some analysis of the asymptotic behavior of their SDE.\n- The paper is clear and adequately written\n\n**Weaknesses**:\n- **Unconvincing justification of why this model should be better for ‘orientation-rich’ data**. This is the main selling point suggested by the authors for their method, but it is not clear to me why this approach should be good for that. I understand the concept of having local pixels tending to be attracted to the same phase during the forward process, but that does not necessarily mean preserving edges or orientation because this happens directly in pixel space. If the diffusion was happening in a wavelet or curvelet basis, then I could agree that features such as edges would remain coherent during the noise process.\nI would recommend adding illustrations of that claim, maybe simply by running the forward Kuramoto process without noise, just to illustrate how the features of an image evolve under these dynamics. \n\n- **Limited generative modeling evaluation**: The paper only presents a comparison to SGM on 3 small image datasets.  On SOCOFing and Brodatz the proposed method seems to provide an advantage in terms of FID, but it is not completely clear if a different noise schedule or details of SGM training/sampling could impact these metrics. In general, comparing to a single generative model from 4 years ago is limited. I would recommend adding more thorough experimentations against different diffusion strategies and noise schedules to build a more complete picture. And I would be particularly interested in any illustration of the fact that the model is indeed more performant at orientations (maybe with zoomed-in view of the fingerprints for different models, at different steps of the reverse process).", "was_revised": true, "final_justification": "I appreciate the feedback and additional experiments proposed by the authors. I think the work with the suggested changes and extra evals is now pretty complete.\nI have increased my grade to 4, but I think more results against actual state of the art diffusion models on high resolution datasets would be needed to make it a very compelling 5 paper.", "rebuttal": "We appreciate $\\textcolor{red}{\\text{Reviewer 4DgT}}$’s thoughtful feedback and the recognition of both the novelty and clarity of our Kuramoto diffusion model. Our responses to the comments follow:\n\n--- \n\n**1. Stronger Justification for “Orientation‑rich” Data**\n\nOur non‑isotropic Kuramoto coupling term, $K\\sum \\sin(\\theta_i-\\theta_j)$, encourages similar phases to synchronize during the early stages of diffusion. The regions of consistent orientations, which have similar pixel intensities, will remain coherent, resist randomization, and naturally stay in synchronization during early noise injection. This “phase coherence” pulls similar phases together and prevents the rapid decorrelation that destroys edges and structures. \n\n**Qualitative Evidence:** Fig. 1 of the main paper clearly illustrates the point: samples retain contours and overall shape long after a standard VP‑SDE has already washed them out. \n\n**Quantitative Evidence:** The signal‑to‑noise ratio (SNR) can be considered a good proxy metric of how well the main image structure is preserved. In Fig. 5 we plot the SNR over forward diffusion timesteps and observe that our SNR stays substantially higher for the first 20–30 % of steps, directly reflecting better preservation of edges and structural details.\n\n**Spectral Interpretation:** From a spectral standpoint, **local phase coupling behaves like an angular low‑pass filter**. In the small‑angle approximation ($\\sin(\\theta_i-\\theta_j)\\approx \\theta_i-\\theta_j$) and without a global reference phase, the forward SDE reduces to a heat equation (or a graph Laplacian form equivalently):\n\n$$\\frac{\\partial \\theta}{\\partial t} = K \\nabla^2 \\theta + \\sigma_t\\epsilon$$\n\nEach Fourier component decays like $e^{-K k^2 t}$, where $k$ denotes the spatial frequency. Modes with very high spatial frequency $k$ (pixel‑scale noise) are damped almost instantly, while moderate‑frequency modes (coherent, edge‑defining structures) decay much more slowly. Consequently, the Kuramoto drift effectively filters away pixel‑scale noise while preserving the smooth, orientation‑rich patterns that constitute edges.\n\nWe will add the above discussion in Sec. 3.3 of the revised main paper. \n\n**Note:** Although we cannot update the supplementary material at this stage, we will include forward‑only Kuramoto snapshots on fingerprint and texture data in the camera‑ready version to further demonstrate this structure-preserving behavior. \n\n--- \n\n**2. Expanded Baselines & Noise Schedules**\n\n\nIn the experiments of our paper, all baselines use a linear noise schedule (with “SGM” referring to VP‑SDE) -- please see lines 53–61 of the supplementary for full details. To isolate the benefit of phase synchronization, we extend our comparisons as follows:\n\n\n**Additional Baselines.**  We add VE‑SDE and two blurring diffusion models [35, 16], which exhibit similar smoothing behavior. VE‑SDE underperforms VP‑SDE slightly, and the blurring model [16] matches VP‑SDE’s FID. Our Local Kuramoto model maintains a large lead: \n\nFID on Brodatz Textures\n|Steps|100|300|1000|\n|:---:|:---:|:---:| :---:| \n|Heat Diffusion [35]| 57.18 | 35.41 | 26.19|\n|Blurring Diffusion [16]| 35.14 | 21.29 |19.85 |\n|VE-SDE|40.19| 24.26| 21.53 |\n|VP-SDE|38.33 | 22.40 | 20.37 |\n|**Ours (Local Kuramoto)**|**18.47** |**15.93** |**14.19** |\n\n\n**Cosine Schedule Ablation.** We also evaluate both VP‑SDE and our model under a cosine schedule. Although switching to cosine reduces FID for all methods -- most notably at 100 steps -- our Kuramoto model still outperforms by a wide margin:\n \nFID on Brodatz Textures\n|Steps|100|300|1000|\n|:---:|:---:|:---:| :---:| \n|VP-SDE (linear)|38.33 | 22.40 | 20.37  |\n|VP-SDE (cosine)|30.45| 20.67| 19.84|\n|**Ours (linear)**|**18.47** |**15.93** |**14.19**|\n|**Ours (cosine)**|**15.75** |**14.32** |**13.45**|\n\nThese results confirm that, even with different noise schedules and additional baselines, the **phase synchronization** induced by Kuramoto coupling remains the key driver of improved FID. We will add a succinct discussion of these schedule choices and their quantitative impact in the revised manuscript (Sec. 3.1).\n\n--- \n\n**3. Additional Experiments on High‑Resolution & Vector‑Valued Data** \n\nTo further validate our method’s generality, we evaluated our method on two new datasets:\n\n\n**High-resolution Ground Terrain [A] (3x128x128).** Compared with Brodatz textures, this dataset covers diverse high-resolution materials (e.g., plastic, turf, steel, asphalt, leaves, stone, and brick). We run our Kuramoto diffusion model on this dataset and compare against VP-SDE under the linear schedules: \n\nFID on Ground Terrain Dataset\n|Steps|100|300|1000|\n|:---:|:---:|:---:| :---:|\n|VP-SDE|114.90| 56.72| 33.79|\n|**Ours (Local Kuramoto)** |**92.86** | **49.68** | **30.62** |\n\nThis result demonstrates that our method also supports high-resolution texture dataset.\n\n\n**Navier–Stokes Fluid Vorticity [B] (2x128x128).** We take the PDE dataset from [B] which consists of the fluid vorticity in horizontal and vertical directions at a fixed timestep. We convert the vorticity fields into polar form (amplitude + phase). The amplitude follows standard VP‑SDE, while the phase evolution is driven by our Kuramoto model. We assess generation quality in the spectral domain via the energy‑spectrum mid-frequency slope difference and normalized Wasserstein distance:\n\n|        Metric        | VP‑SDE | **Ours (Phase Kuramoto + Amplitude Diffusion)** |\n| :------------------: | :----: | :---------------------------------------------: |\n|   Slope Difference   | 0.7435 |                    **0.6954**                   |\n| Wasserstein Distance | 0.0015 |                    **0.0011**                   |\n\n\nThese results demonstrate the applicability of our method on real **angular data**. Furthermore, our joint amplitude–phase diffusion generates more realistic physical spectra in vector‑valued scientific data, which opens pathways for combined amplitude‑phase generative modeling. We will detail the experimental setup in the supplementary.\n\n\nTogether with the core experiments in the paper, we have now validated this synchronization bias across three distinct domains: \n\n**Orientation‑Rich Images (fingerprints, textures mapped as phases)**, where we demonstrate sharper sample generation with fewer steps, \n\n**Earth & Climate Science Data (naturally periodic grids)**, where we show competitively faithful reconstruction of large‐scale geospatial patterns,  \n\n**Navier–Stokes Fluid Vorticity (real angular data)**, where we yield closer matches to the physical energy spectrum when phase coupling is applied. \n\nAcross all settings, this synchronization bias proves to be a powerful, domain‑agnostic mechanism for modeling data where **local coherence** is critical.\n\n--- \n\n**4. Richer Qualitative Comparisons** \n\n\nAlthough we cannot update the current supplementary for now, in the camera‑ready version we will add side‑by‑side, zoomed‐in comparisons of fingerprint and texture samples that clearly show crisper ridge patterns under our Kuramoto diffusion versus the standard diffusion baseline.\n\n\n\n> [A] Deep texture manifold for ground terrain recognition. CVPR 2018.\n>\n> [B] Clifford Neural Layers for PDE Modeling. ICLR 2023.\n\n--- \n\nWe appreciate $\\textcolor{red}{\\text{Reviewer 4DgT}}$’s insightful feedback and hope these clarifications and extended evaluations will convincingly demonstrate the advantages of our Kuramoto diffusion model. Any further comments are more than welcome!"}, {"review_id": "X0C71WynTa", "reviewer": "Reviewer_3GqZ", "summary": "The authors introduce Kuramoto orientation diffusion, which uses stochastic Kuramoto dynamics to evolve the pixels of an image, modeled as coupled oscillators, to a state of phase synchronization sampled from a von Mises distribution, analogous to the evolution of isotropic Euclidean diffusion to a Gaussian distribution. Because the Kuramoto drift function is nonlinear, the authors take care to learn the score function using wrapped Gaussian transition functions from one timestep to the next, rather than relying on a closed form expression. They also ensure that periodicity is enforced by mapping between angular and phase variables and ensuring the phase variables are \"wrapped\" to an appropriate value within the domain $[-\\pi, \\pi]$. The authors show that the phase-synchronization inductive bias built into Kuramoto orientation diffusion allows it to generate sharper images (improved FID scores) from datasets that are dominated by periodic patterns, such as textures and fingerprints, more efficiently (i.e. with fewer diffusion steps). The authors also show that this diffusion can generate natural scene images like those from CIFAR10, though this diffusion becomes less efficient than standard SGM after a large number of diffusion steps.", "questions": "1. Is this method limited to low-resolution images? Why weren't higher-resolution datasets chosen? If the limited number of samples in the Brodatz was an issue, why wasn't a different dataset chosen, e.g. cellular imaging datasets like RxRx3 or JUMPCP, which contain many highly-textured objects per image?\n1. How important were the choices of noise schedules to the overall results? The schedules are presented but there is no discussion of how the particular choices were made, or what effect other choices had on the FID scores, generated image quality, etc.\n1. Please explain in more detail the claim made throughout the paper, and in particular in the Structured Destruction paragraph of Section 3.3, that states that objects within the image are preserved for more diffusion steps due to synchronized coupling. How do we know that this is true? How does Figure 5 support this claim?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. Novel diffusion method based on a well-known stochastic phase-synchronization process\n1. Clear performance gains on the SOCOFing fingerprint and Brodatz texture datasets over standard score-based generative models (SGMs), and respectable performance on natural images (CIFAR10).\n1. Well-written paper, easy to follow.\n\nWeaknesses:\n1. Only low resolution images were considered, with SOCOFing being the highest resolution at 1x96x96.\n1. No discussion on the choice of $\\psi_{ref}$ and its effect on results. I would imagine that a well-chosen $\\psi_{ref}$ could improve FID convergence.\n1. Some weakly supported claims about this diffusion (see Questions below).", "was_revised": true, "final_justification": "The authors demonstrated applicability and usefulness of the approach across and wide variety of datasets and provided both quantitive and qualitative evidence for what they call structured destruction.", "rebuttal": "We are grateful to $\\textcolor{purple}{\\text{Reviewer 3GqZ}}$ for highlighting the innovative aspects of our Kuramoto framework and for careful reading and insightful recommendations. We respond to each of the comments below:\n\n--- \n\n**1. Scaling to High‑Resolution & Vector‑Valued Data**\n\nOur initial evaluation on SOCOFing and Brodatz benchmarks was designed to highlight clear gains on canonical orientation‑rich datasets while controlling for compute budget. However, the approach itself is not inherently limited to low resolution. In our ongoing experiments, here we scale the Kuramoto model to higher-resolution textures dataset and real angular data. \n\n**High-resolution Ground Terrain [A] (3x128x128).** Compared with Broadatz textures, this dataset spans diverse high-resolution materials (e.g., plastic, turf, steel, asphalt, leaves, stone, and brick). We run our Kuramoto diffusion model on this dataset and compare against VP-SDE under the linear schedules: \n\nFID on Ground Terrain Dataset\n|Steps|100|300|1000|\n|:---:|:---:|:---:| :---:|\n|VP-SDE|114.90| 56.72| 33.79|\n|**Ours (Local Kuramoto)** |**92.86** | **49.68** | **30.62** |\n\n\n**Navier–Stokes Fluid Vorticity [B] (2x128x128).** We take the PDE dataset from [B] which consists of the fluid vorticity in horizontal and vertical directions at a fixed timestep. We convert the vorticity fields into polar form (amplitude + phase). The amplitude follows standard VP‑SDE, while the phase evolution is driven by our Kuramoto model. We assess generation quality in the spectral domain via the energy‑spectrum mid-frequency slope difference and normalized Wasserstein distance:\n\n|        Metric        | VP‑SDE | **Ours (Phase Kuramoto + Amplitude Diffusion)** |\n| :------------------: | :----: | :---------------------------------------------: |\n|   Slope Difference   | 0.7435 |                    **0.6954**                   |\n| Wasserstein Distance | 0.0015 |                    **0.0011**                   |\n\n\nThese experiments confirm that our Kuramoto diffusion (i) supports **higher‑resolution textures** and (ii) effectively models real physical **angular data**. Furthermore, the combination with amplitude diffusion paves the way for integrated amplitude–phase generative modeling. Details of the experimental setup will be provided in the supplementary material.\n\n**Future Work.** We plan to extend evaluations to large‑scale medical/biological texture collections (e.g. cellular imaging like JUMPCP [C]) in the future work.\n\n---\n\n**2. Choice of $\\psi_{ref}$, Noise Schedules, and Additional Baselines**\n\n**Choice of $\\psi_{ref}$.** For the reference phase $\\psi_{ref}$, we simply fix it to 0 by default, as it is the middle point of $[-\\pi, \\pi]$. $\\textcolor{purple}{\\text{Reviewer 3GqZ}}$ is right that this value can be tuned to improve alignment with data statistics. For example, for the Brodatz textures images, choosing a slightly negative $\\psi_{ref}$  which matches the data distribution can slightly improve the performance.\n\nFID on Brodatz Textures in 100 Steps\n|$\\psi_{ref}$|$0$|$- \\pi /4$|$- \\pi /2$|\n|:---:|:---:|:---:| :---:| \n|**Ours (linear)**| 18.47 | **17.95**| 18.36|\n\n**Noise Schedules.** For the noise and coupling strength, we keep the relation $K_{\\text{ref}}(t) > D_t > K(t)$ to ensure that the image information is destroyed in the forward process. In the experiments of our paper, all baselines use a **linear** noise schedule (with “SGM” = VP‑SDE) -- please see lines 53–61 of the Supplementary for full details. Here we evaluate both VP‑SDE and our model under a **cosine** schedule. Although switching to cosine reduces FID for all methods -- most notably at 100 steps -- our Kuramoto model still outperforms by a wide margin:\n \nFID on Brodatz Textures\n|Steps|100|300|1000|\n|:---:|:---:|:---:| :---:| \n|VP-SDE (linear)|38.33 | 22.40 | 20.37  |\n|VP-SDE (cosine)|30.45| 20.67| 19.84|\n|**Ours (linear)**|**18.47** |**15.93** |**14.19**|\n|**Ours (cosine)**|**15.75** |**14.32** |**13.45**|\n\n**Additional Baselines.** We add VE‑SDE and two blurring diffusion models [35, 16], which exhibit similar smoothing behavior. VE‑SDE underperforms VP‑SDE slightly, and the blurring model [16] matches VP‑SDE’s FID. Our Local Kuramoto model maintains a large lead: \n\nFID on Brodatz Textures\n|Steps|100|300|1000|\n|:---:|:---:|:---:| :---:| \n|Heat Diffusion [35]| 57.18 | 35.41 | 26.19|\n|Blurring Diffusion [16]| 35.14 | 21.29 |19.85 |\n|VE-SDE|40.19| 24.26| 21.53 |\n|VP-SDE|38.33 | 22.40 | 20.37 |\n|**Ours (Local Kuramoto)**|**18.47** |**15.93** |**14.19**|\n\n\nThese results confirm that, even with different noise schedules and additional baselines, the **phase synchronization** induced by Kuramoto coupling remains the key driver of improved FID. We will add a succinct discussion of these schedule choices and their quantitative impact in the revised manuscript (Sec. 3.1).\n\n---\n\n**3. Structured Destruction**\n\n\nBy “structured destruction”, we refer to the two-stage **structured noising process** induced by our Kuraomoto drift: (1) early preservation of coherent features via local phase coupling, and (2) accelerated convergence to the noise distribution at later timesteps. Since non-isotropic Kuramoto coupling enforces local phase alignment, coherent image features (edges, ridges) will stay in synchronization and resist randomization during early noise injection. In later stages, these structures keep synchronized and jointly evolve to noise distribution at a faster pace.\n\n\n**Quantitative Evidence (Fig. 5):** The signal‑to‑noise ratio (SNR) can be considered a good proxy metric of how well the main image structure is preserved. Formally, it is defined as: \n\n$$SNR(t) = \\frac{||x_0 - \\mathbb{E}[x_t]||^2}{Var[x_t]}$$\n\nwhich tracks how much of the original structure remains at time $t$. Our Kuramoto diffusion exhibits substantially higher SNR than VP‑SDE over the first 20–30 % of steps, directly quantifying that more “signal” (i.e., structured features) survives before noise overwhelms. After this window, our SNR decays more quickly, reflecting the faster convergence to the noise distribution.\n\n**Qualitative Evidence (Fig. 1):** The quantitative evidence also correlates with the empirical observation in Fig. 1. Under our Kuramoto model, the samples retain contours and overall shape long after a standard VP‑SDE has already washed them out.\n\n**Spectral Interpretation:** From a spectral standpoint, **local phase coupling behaves like an angular low‑pass filter**. In the small‑angle approximation ($\\sin(\\theta_i-\\theta_j)\\approx \\theta_i-\\theta_j$) and without a global reference phase, the forward SDE reduces to a heat‑equation form (or a graph Laplacian equation equivalently):\n\n$$\\frac{\\partial \\theta}{\\partial t} = K \\nabla^2 \\theta + \\sigma_t\\epsilon$$\n\nEach Fourier component decays like $e^{-K k^2 t}$, where $k$ denotes the spatial frequency. Modes with very high spatial frequency $k$ (pixel‑scale noise) are damped almost instantly, while moderate‑frequency modes (coherent, edge‑defining structures) decay much more slowly. Consequently, the Kuramoto drift effectively filters away pixel‑scale noise while preserving the smooth, orientation‑rich patterns that constitute edges.\n\nWe will add the above discussion of structured destruction in Sec. 3.3 of the revised main paper. \n\n\n> [A] Deep texture manifold for ground terrain recognition. CVPR 2018.\n>\n> [B] Clifford Neural Layers for PDE Modeling. ICLR 2023.\n> \n> [C] JUMP Cell Painting dataset: morphological impact of 136,000 chemical and genetic perturbations. bioRxiv 2023.\n\n--- \n\nThank you again for your insightful feedback! We hope these revisions will greatly enhance the clarity, and we are more than happy to have further discussions."}, {"review_id": "CF8Y5wPE5o", "reviewer": "Reviewer_X1gf", "summary": "This paper proposes a method inspired by biology, Kuramoto Orientation Diffusion Models. The data is represented in phase domain, and the diffusion process follows explicit, non-linear, locally or globally coupled phase dynamics. This leads to an approximate von Mises distribution at equilibrium. The method is experimentally validated and compared with a baseline diffusion model on CIFAR10, fingerprint, Brodatz texture datasets, and in the appendix on earth and climate science datasets.", "questions": "Q1: As explained in Weaknesses, I would like to see ablations on the different aspects of your model. At the very least, for instance, train a model where $K(t)=0$, i.e., without coupling. This would then show us what the effect is of representing the data in phase coordinates, and of the specific diffusion process, sampling and training of your model.\n\nQ2: I assume $\\psi_{ref}$ is sampled uniformly from $[-\\pi, \\pi]$? And is thus sampled together with the starting von-Mises noise when denoising a sample?\n\nQ3: Am I correct in understanding that for the image datasets (CIFAR10, fingerprints and textures), you represent the images by mapping the pixel intensities to angles? If so, how exactly are rotation invariant datasets (fingerprints and textures) more relevant for your method?\n\nQ4: In deep learning, (hyper)spherical representation have already rich literature. See e.g.\n- https://arxiv.org/abs/1804.00891\n- https://arxiv.org/abs/2006.04437\n\nMore recently, there are many works defining diffusion models on (at least) SO(2), e.g.:\n- https://openreview.net/forum?id=BY88eBbkpe5\n- https://arxiv.org/abs/2312.11707\n\nYou might also be interested in this work where SDEs are defined on spherical manifolds, with a linear SDE: https://neurips.cc/virtual/2023/poster/71379\n\nHow do these relate to your work?\n\nQ5: Do you think representing the angles using 2-d coordinates could also be a (simpler) way to achieve your goals? Maybe then the forward conditional process could still be Gaussian? Related to this, can you comment on the choice of wrapped Gaussian distributions compared to von Mises for the diffusion process?\n\nQ6: How exactly is the Kuramoto concept relevant to your model, and to the experiments presented in the paper?", "limitations": "Yes, but in appendix. This should be in the main paper.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strenghts:\n- The paper is well written and pleasant to read.\n- The paper explores in a creative way novel, fundamental ways to define generative diffusion models, inspired by biology.\n- The experiments are on multiple datasets.\n\nWeaknesses:\n- Concepts from biology can be a source of inspiration for many of us, however, I have trouble seeing really a profound or fundamental connection with the method and applications. Rather, the connection is rather vague and on an abstract level, not concrete. I will make this point more concretely in the weaknesses and questions listed below.\n- A crucial aspect in this line of work, is how exactly the data is represented. It is clear from the start that the data should be in the form of phase angles $\\theta$. Only on line 167-168 it is briefly explained that for image data (which means for all experiments in the main paper), *pixel intensities* are mapped to angles ([-1, 1] -> [-.9 pi, .9 pi]. Thus *not* angles in the spatial dimensions, for instance by representing the pixels in polar coordinates. This is very confusing to me and in my opinion troubles much of the motivation and context of this method.\n- Since the forward process is non-linear, and thus the conditional non-Gaussian, the method requires calculating the Markov chain (solving the forward SDE) during training, and also the score model is modelling the local transition $p(\\theta_t | \\theta_{t-1})$. These are all considerate changes compared to the baseline diffusion model. There are no ablations on these aspects, that could identify exactly why the model trains 'faster'. I would like to see more evidence on this.\n- The limitations and broader impact sections are in the appendix. Please put them in the main paper.", "was_revised": true, "final_justification": "The authors provided extra experiments that convinced me on one of my questions", "rebuttal": "We thank $\\textcolor{brown}{\\text{Reviewer X1gf}}$ for their thoughtful critique and recognition of our novelty. Below, we address each question point by point:\n\n---\n\n**1. Phase Mapping & Validation on True Angular Data**\n\nFor image datasets, we do linearly map the normalized pixel intensities to phase angles. We agree with $\\textcolor{brown}{\\text{Reviewer X1gf}}$ that this mapping does not encode spatial relations of pixels. However, it provides a simple, stable, one‑dimensional circular representation: in orientation‑rich images (e.g., fingerprints, textures), coherent ridges and edges manifest as locally consistent intensity patterns, which become locally aligned phases under this mapping.\n\nWe considered more sophisticated alternatives but found none that reliably preserved both periodicity and semantic structure:\n\n**HSV mapping:** We tried mapping RGB images to the HSV color space. However, only the Hue channel is periodic; the other two channels are non-periodic and thus remain untreated.\n\n**Spherical Autoencoder:** We also tried using a Spherical Autoencoder to encode the pixel intensity into a periodic latent code. However, this approach collapses semantic structure in multi‑class datasets like CIFAR‑10 -- images of different categories fall into similar angles, which greatly hurts sample fidelity.\n\nBy contrast, our linear mapping is data-agnostic, easy to implement, and most importantly, lets the Kuramoto drift operate directly on $S^1$, naturally preserving local coherence.\n\nWhat we really aim to demonstrate in the paper is that the Kuramoto-based diffusion in the circular domain yields a synchronization inductive bias, destroying signals faster and effectively modeling local coherence. Although this phase embedding serves as a good proxy for natural images, this synchronization mechanism is ideally suited for truly angular data. To demonstrate the scalability beyond synthetic phase embeddings, we apply the Kuramoto model to genuinely periodic scientific data:\n\n**Navier–Stokes Fluid Vorticity [A] (2x128x128).** We take the PDE dataset from [B] which consists of the fluid vorticity in horizontal and vertical directions at a fixed timestep. We convert the vorticity fields into polar form (amplitude + phase). The amplitude follows standard VP‑SDE, while the phase evolution is driven by our Kuramoto model. We assess generation quality in the spectral domain via the energy‑spectrum mid-frequency slope difference and normalized Wasserstein distance:\n\n|        Metric        | VP‑SDE | **Ours (Phase Kuramoto + Amplitude Diffusion)** |\n| :------------------: | :----: | :---------------------------------------------: |\n|   Slope Difference   | 0.7435 |                    **0.6954**                   |\n| Wasserstein Distance | 0.0015 |                    **0.0011**                   |\n\nThese experiments confirm that our Kuramoto diffusion can effectively model **real physical angular data**, complementing the image‑based experiments. Moreover, the joint amplitude–phase framework opens the door to polar diffusion models on vector‑valued fields. Detailed setup will be included in the supplementary.\n\n---\n\n**2. Isotropic Baselines & No‑Coupling Variant** \n\nTo isolate the impact of Kuramoto coupling, we include a model variant with $K(t)=0$, which keeps only the phase-reference drift in the SDE, as well as blurring diffusion models [16,35] with isotropic linear drifts.\n\nFID on Brodatz Textures\n|Steps|100|300|1000|\n|:---:|:---:|:---:| :---:| \n|Heat Diffusion [35]| 57.18 | 35.41 | 26.19|\n|Blurring Diffusion [16]| 35.14 | 21.29 |19.85 |\n|Reference-only Process| 33.76  | 20.54  | 19.01  |\n|**Ours (Local Kuramoto)**|**18.47** |**15.93** |**14.19**|\n\n**Improved Quality in Fewer Steps.**  Removing Kuramoto coupling raises 100‑step FID by over 15 points, and isotropic baselines [16,35] also incur much higher FIDs. These results confirm that **non‑isotropic phase synchronization** is the key driver of both rapid convergence and structure‑preserving diffusion on orientation‑rich data.\n\n---\n\n**3. Choice of $\\psi_{ref}$** \n\nFor the reference phase $\\psi_{ref}$, we simply fix it to 0 by default, as it is the middle point of $[-\\pi, \\pi]$. This value can be tuned to improve alignment with data statistics. For example, for the Brodatz texture images, choosing a slightly negative $\\psi_{ref}$ which matches the data distribution can slightly improve the performance.\n\nFID on Brodatz Textures in 100 Steps\n|$\\psi_{ref}$|$0$|$- \\pi /4$|$- \\pi /2$|\n|:---:|:---:|:---:| :---:| \n|**Ours (linear)**| 18.47 | **17.95**| 18.36|\n\n---\n\n**4. 2‑D Embedding vs. Wrapped Gaussians** \n\nOne can indeed embed the phase via $x,y = \\cos(\\theta), \\sin(\\theta) \\in \\mathbb{R}^2$ and then derive an equivalent SDE in Cartesian coordinates. However, this route introduces several complications. Applying Itô’s lemma with $x,y$ and substituting $d\\theta$ yields:\n\n$$d x =  f_x(x,y)dt + \\sqrt{2 D_t} (-y) d W_t, d y = f_y(x,y)dt + \\sqrt{2 D_t} (x) d W_t $$\n\nwhere the non-linear drifts $f_x$ and $f_y$ are inherited from Kuramoto coupling in the phase space. The difficulties lie in 3 aspects: (1) The terms in $f_{x}$ and $f_{y}$ involve products like $x_i y_j - y_i x_j$. The mean update is no longer an affine functions of $(x,y)$, (2) the multiplicative noise acts only in the tangent direction, making the covariance dependent on the state $(x,y)$, (3) To ensure $(x,y)$ stay in the unit circle, a projection to $S^1$ has to be performed after every SDE step, which destroys the Gaussian‐increment structure.\n\nTogether, these factors mean the exact finite‐step transition in $(x,y)$ is neither Gaussian nor available in closed form. By contrast, we use a wrapped‑Gaussian kernel directly in the angular domain, which offers exact, closed‑form transitions that reside intrinsically on $S^1$, leading to a simple and tractable score‐matching objective.\n\nWe will add the full derivations and explanation in the supplementary.\n\n---\n\n**5. Related Work of Hypersphere Representations & Manifold SDEs**\n\nThank you for highlighting these important lines of research. We will integrate the following into our Related Work section:\n\n**Hyperspherical Latent Models:** [B] and [C] introduce VAEs with latent spaces on $S^d$, using von Mises–Fisher or power‐spherical priors. These priors generalize the circular von Mises distribution we use on $S^1$. \n\n**Diffusion on Rotation & Spherical Manifolds:** [D] and [E] extend score‐based diffusion to the 3D rotation group SO(3), deriving trackable marginal kernels and SDE/ODE samplers on these manifolds. [F] presents a linear, homogeneous SDE in a spherical latent space. None of these methods introduces local oscillator‐coupling; their drifts are global and linear.\n\nWe differ from previous manifold diffusion methods in the following aspects:\n\n**(1) Non-isotropic, Nonlinear, Local Drift.** We apply non-isotropic and non-linear Kuramoto coupling that synchronizes local/global phases. In contrast, prior manifold SDEs ([D,E,F]) rely on **linear** drift fields over the entire manifold.\n\n**(2) Intrinsic Wrapped–Gaussian Transitions:** We derive exact local transition densities on $S^1$ via wrapped Gaussians and directly learn their scores. Other methods typically embed into $\\mathbb{R}^n$ and perform manifold diffusion via manifold projection or auxiliary parametrizations.\n\n**(3) Local Coherence Inductive Bias:** Crucially, our biologically inspired Kuramoto coupling explicitly aligns neighboring phases, enforcing **local coherence (preserving edges, textures)** during diffusion. Prior SO(3) and hyperspherical approaches instead prioritize the **global symmetry preservation (e.g., rotational invariance)** without any mechanism for neighborhood synchronization. \n\nAlthough both our method and prior diffusion models [D,E,F] leverage circular or hyperspherical geometry, our focus on **local phase synchronization** is fundamentally different from their emphasis on **maintaining geometric invariances**. We will cite [B–F] and emphasize these distinctions in the revised manuscript.\n\n---\n\n**6. Role of the Kuramoto Concept** \n\nThe Kuramoto concept is the heart of our approach: we borrow the **biological phase‐synchronization mechanism** -- each oscillator (a phase variable) being pulled toward synchronization with its connected neighbors. By embedding this coupling directly into the forward diffusion SDE, we introduce an inductive bias that actively enforces **local alignment**, preserving fine‐scale structures (e.g., edges, ridges) as noise is injected.\n\nWe validate this synchronization bias across three distinct domains: \n\n**Orientation‑Rich Images (fingerprints, textures mapped as phases)**, where we demonstrate sharper sample generation with fewer steps, \n\n**Earth & Climate Science Data (naturally periodic grids)**, where we show competitively faithful reconstruction of large‐scale geospatial patterns,  \n\n**Navier–Stokes Fluid Vorticity (real angular data)**, where we yield closer matches to the physical energy spectrum when phase coupling is applied. \n\nAcross all settings, this synchronization bias proves to be a powerful, domain‑agnostic mechanism for modeling data where **local coherence** is critical.\n\n---\n\n**7. Limitations**\n\nWe thank the reviewer for the reminder. We’ll relocate the Limitations and Broader Impact sections into the main text for the camera‑ready version.\n\n\n> [A] Clifford Neural Layers for PDE Modeling. ICLR 2023.\n> \n> [B] Hyperspherical Variational Auto-Encoders. UAI 2018.\n> \n> [C] The Power Spherical distribution. ICML Workshop 2020.\n> \n> [D] Denoising Diffusion Probabilistic Models on SO(3) for Rotational Alignment. ICLR Workshop 2022.\n> \n> [E] Unified framework for diffusion generative models in SO(3): applications in computer vision and astrophysics. AAAI 2024.\n> \n> [F] Latent SDEs on Homogeneous Spaces. ICML 2023.\n\n---\n\nThank you for your valuable insights -- we hope this clarifies our contributions and welcome any further questions!"}, {"review_id": "aDsIw8Ciei", "reviewer": "Reviewer_ZpUX", "summary": "Standard generative approaches based on isotropic Euclidean diffusion struggle with orientation rich inputs, which often exhibit coherent angular directional patterns. To tackle this problem, the work proposes to make use of stochastic Kuramoto dynamics in the diffusion process, borrowing from the idea that Kuramoto models capture synchronization phenomena across coupled oscillators. The reformulated forward process gradually collapse the data into a low-entropy von Mises distribution, and generate diverse patterns by reversing the dynamics with a learned score function.  The approach presented in this work enables structured destruction during forward diffusion and a hierarchical generation process that progressively refines global coherence into fine-scale details.", "questions": "1. The generated CIFAR10 images, in my eyes, look good, but the FID scores do not reflect that. Are there any intuition why this diffusion process does not work well? Also, it would be nice for you to contrast and highlight the theoretical benefits of the Kuramoto model in contrast of the von Mises distribution as done in this [work](https://arxiv.org/pdf/2506.10576v1). FYI, I do not think you need to perform experimental comparisons to it --- just a brief explanation to build your reasoning for Kuramato model is fine.\n\n2. What if you scale your timesteps to higher than 1000? For example, you could try it with 4000 timesteps to see if the trend of FID decreasing is still maintained. \n\n3. You mentioned about the relation to blurring diffusion process in your text. Why don't you compare to them and perhaps extend this relationship as well?", "limitations": "Yes, see Appendix A.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "## Strengths\n\n1. The work introduces a new diffusion process, which preserves angular information, based on the mathematical Kuramoto model which describes synchronization. The transformed distribution (from the targeted distribution) behaves like a von Mises distribution at large timesteps. Nonetheless, these aspects make the work very interesting as the new diffusion process is based on a biological-plausible process. \n\n2. According to the work's experiment results, the new diffusion process demonstrates much better performance than the SDE-based Gaussian diffusion model in [Song et al. (2021)](https://arxiv.org/abs/2011.13456) on fingerprint and texture datasets.\n\n3. An advantage of the introduced diffusion process is the enabling of hierarchical generative process, which allows for a more coherent generative process resembling coarse-to-fine generation.\n\n## Weaknesses\n\n1. The work has limited results and the introduced approach underforms in terms of generating CIFAR10 images. Moreover, for NLL experiments on Earth and climate science datasets, the introduced approach only performs decently.\n\n2. The presence of the nonlinear drift in the stochastic Kuramoto model renders the marginal distribution intractable, while at each training step, it requires $O(T)$ operations to simulate the forward Markov chain.\n\n3. It is still unclear on the exact benefits of the new diffusion process over Gaussian and other processes. \n\n## Comment \n\nOverall, I think the work is interesting, but it has limited results and can be presented better to highlight the approach and also the problem it is trying to tackle. \n\nBy the way, there is an error in your supplementary regarding $\\nabla_{\\theta_t} \\log p(\\theta_t)$ on line 38 of section B.2, which should be $\\nabla_{\\theta_t} \\log p(\\theta_t) = (p(\\theta_t))^{-1} ~ \\nabla_{\\theta_t} p(\\theta_t)$.", "was_revised": true, "final_justification": "Overall, I think it's a nice work. It has utilized a niche biological-inspired forward process and made it work at best. Indeed, the evaluations can seem limited but I think the authors have done their best to make a complicated forward process to work decently. There are some benefits which they have shown. Thus, I do think it's a good work to be published (after some revisions according to my and other reviewers' comments).", "rebuttal": "We appreciate $\\textcolor{blue}{\\text{Reviewer ZpUX}}$’s recognition of our method’s novelty and thank you for the thorough feedback and constructive suggestions. Below, we address each of their points in detail:\n\n--- \n\n**1. Reducing the O(T) Simulation Cost**\n\nAlthough our local score matching requires sample pairs $(\\theta_t,\\theta_{t+1})$ along the forward chain, this cost actually can be nearly eliminated with precomputation:\n\n**Pre‐simulate & cache:** Before training, we can run the forward SDE on the entire dataset, save all $(\\theta_t,\\theta_{t+1})$ pairs to disk, and then load them directly during training. This makes the simulation cost effectively O(0) at each training step.\n\n**Epoch‐wise precompute:** Otherwise, if disk space is limited, we can simply re‐generate and cache one epoch’s worth of pairs at the start of each epoch. This still dramatically reduces the simulation overhead.\n\nWe will detail the above discussion of reducing training costs in the limitations section.\n\n---\n\n**2. Intuition for General vs. Orientation‑Rich Images**\n\nOur Kuramoto coupling is foremost a synchronization mechanism: it excels at pulling similar phases together, thereby preserving edges and repetitive patterns (see qualitative example in Fig. 1 and the SNR over time in Fig. 5). On orientation‑rich data (fingerprints, textures) where the data are dominated by the **simple outlines and repeating ridge structures**, this yields sharp samples in few steps. \n\nBy contrast, natural images (e.g., CIFAR‑10) demand modeling **complex, global semantics (object shapes, color gradients, backgrounds), often characterized by higher order longer-range correlations**. In this setting, the local synchronization bias becomes less relevant and potentially even detrimental; and an isotropic diffusion process with a global drift may be better suited to capture these large‑scale, non‑repetitive features. As a result, our method trades some global fidelity (reflected in higher CIFAR‑10 FID) in exchange for strong local coherence.\n\n---\n\n**3. Comparisons: Blurring Diffusion & Reference-only Process**\n\nWe thank the reviewer for the constructive suggestion of comparing with blurring diffusion models. Below, we compare against blurring diffusion models [16,35] with linear isotropic drifts, and a reference-only variant (setting $K(t)=0$) on Brodatz textures, to isolate the benefit of nonlinear, non‑isotropic coupling:\n\nFID on Brodatz Textures\n|Steps|100|300|1000|\n|:---:|:---:|:---:| :---:| \n|Heat Diffusion [35]| 57.18 | 35.41 | 26.19|\n|Blurring Diffusion [16]| 35.14 | 21.29 |19.85 |\n|Reference-only Process| 33.76  | 20.54  | 19.01  |\n|**Ours (Local Kuramoto)**|**18.47** |**15.93** |**14.19**|\n\n**Reference-only Process.** Setting $K(t)=0$ yields an **isotropic drift** toward a single reference phase, completely ignoring any neighbor information.  Also, even with only the isotropic reference drift, the marginal transitions are still intrackable due to the phase wrapping, forcing us to rely on the local score matching trick again. The FID worsens by around 15 points at 100 steps,\n\n**Blurring Diffusion Models.** Although these two models [35,16] exhibit blurring behavior, they actually apply **linear isotropic drift** in the SDE. The spatially uniform drift cannot selectively preserve orientation patterns. Therefore, they fall far short of our model at 100 steps.\n\nTogether, these comparisons demonstrate that **nonlinear, non-isotropic phase coupling** is the key ingredient for rapid, structure‐preserving diffusion on orientation‑rich data. \n\nIn the Related Work section, we will also cite [C], which develops spherical von Mises–Fisher diffusion on $S^d$, which generalizes the circular von Mises we use on $S^1$. \n\n---\n\n**4. Longer Sampling Chains**\n\nWe agree that more sampling steps can indeed further refine results. In preliminary runs on Brodatz:\n\nFID at 1,000 steps: 14.19\n\nFID at 2,000 steps: 13.52\n\nFID at 4,000 steps: 12.89\n\nWe see continued improvement as the denoising step increases. This suggests that in the CIFAR‑10 case, extending to 4,000 steps may likewise narrow the FID gap. \n\n---\n\n**5. Corrections of $\\nabla \\log p(\\theta)$**\n\nThanks for catching the typo. The derivation of Eq. (5) is correct, but line 38 is indeed wrong. We have corrected it in the revision.\n\n---\n\n**6. Additional Experiments on High‑Resolution & Vector‑Valued Data** \n\nTo further validate our method’s generality, we evaluated our model on two new datasets:\n\n**High-resolution Ground Terrain [A] (3x128x128).** Compared with Brodatz textures, this dataset covers diverse high-resolution materials (e.g., plastic, turf, steel, asphalt, leaves, stone, and brick). We run our Kuramoto diffusion model on this dataset and compare against VP-SDE under the linear schedules: \n\nFID on Ground Terrain Dataset\n|Steps|100|300|1000|\n|:---:|:---:|:---:| :---:|\n|VP-SDE|114.90| 56.72| 33.79|\n|**Ours (Local Kuramoto)** |**92.86** | **49.68** | **30.62** |\n\nThis result demonstrates that our method also supports high-resolution texture dataset.\n\n\n**Navier–Stokes Fluid Vorticity [B] (2x128x128).** We take the PDE dataset from [B] which consists of the fluid vorticity in horizontal and vertical directions at a fixed timestep. We convert the vorticity fields into polar form (amplitude + phase). The amplitude follows standard VP‑SDE, while the phase evolution is driven by our Kuramoto model. We assess generation quality in the spectral domain via the energy‑spectrum mid-frequency slope difference and normalized Wasserstein distance:\n\n|        Metric        | VP‑SDE | **Ours (Phase Kuramoto + Amplitude Diffusion)** |\n| :------------------: | :----: | :---------------------------------------------: |\n|   Slope Difference   | 0.7435 |                    **0.6954**                   |\n| Wasserstein Distance | 0.0015 |                    **0.0011**                   |\n\n\nThese results demonstrate the applicability of our method on real **angular data**. Furthermore, our joint amplitude–phase diffusion generates more realistic physical spectra in vector‑valued scientific data, which opens pathways for combined amplitude‑phase generative modeling. We will detail the experimental setup in the supplementary.\n\nTogether with the core experiments in the paper, we have now validated this synchronization bias across three distinct domains: \n\n**Orientation‑Rich Images (fingerprints, textures mapped as phases)**, where we demonstrate sharper sample generation with fewer steps, \n\n**Earth & Climate Science Data (naturally periodic grids)**, where we show competitively faithful reconstruction of large‐scale geospatial patterns,  \n\n**Navier–Stokes Fluid Vorticity (real angular data)**, where we yield closer matches to the physical energy spectrum when phase coupling is applied. \n\nAcross all settings, this synchronization bias proves to be a powerful, domain‑agnostic mechanism for modeling data where **local coherence** is critical.\n\n\n> [A] Clifford Neural Layers for PDE Modeling. ICLR 2023.\n> \n> [B] Hyperspherical Variational Auto-Encoders. UAI 2018.\n> \n> [C] Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres. ICML 2025.\n\n--- \n\nWe hope these additional results and explanations clarify the versatility and strengths of Kuramoto‑based diffusion, and we are more than happy to answer any further questions!"}]

Author rebuttal:

Meta-review: * Summary 
Authors describe a new type of diffusion model where the typical Brownian motion does not describe the dynamics but instead by a dynamical process generated by the Kuramoto potential, which is a type of nonlinear pairwise potential.  The distinct advantage of this type of potential is that, since it's a rotational invariant potential, it inspired the use of this potential to correct for generative processes that struggle with generating Orientation-rich images. This new method can then be crafted as a form of inductive bias for a network to learn such a representation 

* Strengths and reasons to accept
The paper's biggest strength is in its innovative idea of using dynamical processes that are not analytically tractable to reverse, and require a numerical calculation of the transition probability. In general, the paper is full of novel approaches that can open up a whole new class of ways to create and train diffusion models. The paper itself is also extremely well written and clear. The derivation in the supplementary material is also novel and significantly strengthens the paper since it gives rigorous results on the model.
Authors also engaged extensively during the rebuttal and gave a lot more rational arguments, and added experiments to prove their point. 
Authors also show their method empirically over several different datasets. 

* Weakness 
The biggest weakness is the small datasets and low-resolution images. Furthermore, the justification for the method is slightly lacking in the paper, but has been addressed in the rebuttal, which means we need to see if they incorporate the feedback from the discussion
Decision: Accept (poster)