PIVNO: Particle Image Velocimetry Neural Operator
Authors: Jie Xu
Xuesong Zhang
Jing Jiang
Qinghua Cui
Venue: NeurIPS 2025 poster ()
Keywords: PIV
Motion Field
Neural Operator
Self-supervised
Primary area: applications

Abstract: Particle Image Velocimetry (PIV) aims to infer underlying velocity fields from time-separated particle images, forming a PDE-constrained inverse problem governed by advection dynamics. Traditional cross-correlation methods and deep learning-based feature matching approaches often struggle with ambiguity, limited resolution, and generalization to real-world conditions. To address these challenges, we propose a PIV Neural Operator (PIVNO) framework that directly approximates the inverse mapping from paired particle images to flow fields within a function space. Leveraging a position informed Galerkin-style attention operator, PIVNO captures global flow structures while supporting resolution-adaptive inference across arbitrary subdomains. Moreover, to enhance real-world adaptability, we introduce a self-supervised fine-tuning scheme based on physical divergence constraints, enabling the model to generalize from synthetic to real experiments without requiring labeled data. Extensive evaluations demonstrate the accuracy, flexibility, and robustness of our approach across both simulated and experimental PIV datasets. Our code is at https://github.com/ZXS-Labs/PIVNO.

PIVNO: Particle Image Velocimetry Neural Operator
Jie Xu*? Xuesong Zhang" t1 Jing Jiang² and Qinghua Cui
Beijing University of Posts and Telecommunications
BBejjing Union University
PPekngg University
WWuhan Sports University
xuesong "xuesong_2hanngbbupp edu.
cuiqinghua@hsc. pku. edu, cn
Abstract
Particle Image Velocimetry (PIV) aims to infer underlying velocity fields from time-separated particle images, forming PDE-constrained inverse problem gov- erned by advection dynamics. Traditional cross-correlation methods and deep learning-based feature matching approaches often struggle with ambiguity, lim- ited resolution, and generalization to real-world conditions. To address these challenges, propose PIV Neural Operator (PIVNO) framework that directly approximates the inverse mapping from paired particle images to flow fields within function space. Leveraging position informed Galerkin-ste -tylee attention operator, PIVNO captures global flow structures while supporting resolution ppolution-adaattve infer- ence across arbitrary subdomains. Moreover, to enhance real-world adaptability, we introduce self-supervised fine-tuning scheme based physical divergence constraints, enabling the model to generalize from synthetic to experiments without requiring labeled data. Extensive evaluations demonstrate the accuracy, flexibility, and robustness of our approach across both simulated and experimental PIV datasets. Our code is at
Introduction
Particle Image Velocimetry (PIV) computer vision-based metrology widely used in various scientific and engincering fields, e.g., physics (133), materials (4-6), life sciences 7 engine designs [1], and locomotion inspection in tissue engineering [2]. By dispersing tracer particles into the fluid under measurement, PIV employs high-frame-rate cameras (typically 10 10yh fps) and high-repetition-rate laser sources (up to 10 Hz) to capture particle image sequences and calculate the particle displacements, providing discrete observation of the underlying motion field for further flow dynamic analysis. The motion of these tracer particles in an incompressible fluid can be modeled by the advection diffusion equation [6]
01 Dr + +V (u/)= DV'),
where T(F, 1) represents the observed scalar field (e.g., particle intensity), is the underlying velocity field, and D is the diffuuson coefficient. Under the assumption no source terms, negligible diffusion (D 0), and divergence-free flow, simplifies to the pure advection equation in the operator form
aT VF E X, where B uu VI, f(ī,t) at (H,t)
tf, f(ī,t),
(2)
'Equal contributions. Conresponding authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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In the context of PIV, One is given [ t1) and [(T, t2) two noisy, discretely sampled particle images at successive time steps-aad and intends infer the latent velocity field that satisfies this the partial differential equation (PDE). This makes PIV fundamentally a PDE-constrained inverse problem, where the forward evolution is governed by physical transport dynamics, but the goal is to invert that process and recover the control variable u from its outcomes. Traditional PIV methods [17] 21 generally rely on local coos-correlatton matching techniques, but suffer from several limitations. First, the information within a single matching window is often insufficient to distinguish completely textureless particle clusters, and this issue worsens when the particle distribution is relatively uniform in the flow. Second, the size of the matching window directly affects both the spatial resolution and the accuracy of the estimated velocity field, leading to an inherent rrad--of that is difficult to balance.
In recent years, deep learning-bat -bsed methods [22] 27 B 28 have been proposed to represent particle image features effectively in high-dimensional latent spaces, thus reducing matching ambiguities. Moreover, the consecutive layers in deep neural networks offer larger receptive fields capable of capturing broader spatial context while preserving spatial resolution and matching accuracy. However, the inverse problem remains fundamentally ill- illppsed:: the observed particle images discrete, noisy, and of limited resolution, hence the numerical solution to the governing PDE is neither unique nor stable. In other words, the inverse operator of B in Eq, 2 may not exist.
Algorithmically, existing PIV approaches [26 25 27 29 30 solve the PDE in (2) via the construction of cost- volume (CV) based on which the displacement array can be inferred. Nevertheless, the resolution of the CV is up to the affordable computational resources and once trained the PIV models' scalability to different measurement requirements is very limited. Inspired by the discretization invariant neural operator (NO) method, recently developed in the field of computational physics [31- 35 as efficient PDE solvers, we inspect the PIV inverse problem through the lens of operator learning. Specifically, we introduce a PIV neural operator(PIVNN) framework that directly approximates the inverse map G (Lt,, + u, bypassing the necessity of Eastiirtiullaryy. we devise a position informed Galerkin-type attention operator, whose approximation properties correspond to the classical Petrov-Galerkin projections [36]. This design enables our model perceive global motion patterns while remaining numerically efficient and copollutionagnostii
The sim2real transferring is another challenge confronting PIV models, let alone obtaining sufficient labeled data for supervision itself is expensive and often impractical in real experiments. Existing synthetic datasets, which, despite offering perfect ground-truth flow, fail to cover all variability of real flow conditions as well as particle image qualities. Consequently, when applied unseen flow regimes or lighting conditions, these models suffer from poor generalization. To address this, we incorporate self-supervised fine-tuning constrained by the physical incompresiibility of the flow, enforcing the divergence-fice conditions, adapt the model to unlabeled real data. This strategy bridges the domain gap between synthetic and real experiments.
methods 37 22 24] are primarily trained
Finally, practical fluid experiments often demand localized flow analysis at varying resolutions [38] However, the region of interest is rarely known a priori, and existing PIV networks are designed for uniform global inference. This limitation prevents adaptive refinement in high- or boundary-layer regions, restricting their utility for fine-scale investigations. In this work, we overcome this challenge by designing our pperator-based architecture support resolution-adaptive inference over arbitrary subdomains, enabling detailed flow field predictions.
In summary, our contributions can be summarized as follows:
We propose the PIV Neural Operator, a neural operator framework that directly maps particle image pairs to flow fields, offering a function space-level approximation of fluid dynamics and improving estimation accuracy.
We incorporate self-supervised mue-tuning mechanism based on physical divergence constraints, enabling domain adaptation from synthetic training to real-world testing without labeled data.
PIVNO enables aesolution-adaptiv flow inference over arbitrary subdomains, allowing fine-grained analysis in critical regions, which is essential for practical experimental fluid mechanics [42 (4)
2 Related Work
Seat-Volum--BBe PIV. Deep learning- PIV has attracted significant attention recently. Early studies [37 24 were primarily based on convolutional neural networks (CNNs) for supervised

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(a) PIVNO
Target coordinate
Feature Encoder
How feoture
RoPE Galerkin
4.0
SR Module
Attention
Feature
Encoder
fold
H1, xWW
Context feature
H, x
(b) Feature Encoder
(c) RoPE
Con33x3 (64)
Unit (64)
(64)
Res. Unit (92)
Res Unit 192)
Res, Unit (128)
Res, Unit (128)
Conv1xa (64)
RUFE-
[d] SR Module
k(x)) Comeorts, Ai
grapi convolution
Clement-wseee addittion
Channel-stton
fully-
depthouise
Feature
MX.))
Rotury
Gaumian
mutipiication
soardhaate
M
AES.)
Figure 1: (a) The PIVNO framework, where each module works collaboratively enable efficient inference from (possible low-resolution) inputs desired high-resolution flow fields. First, Feature Encoder (b) extracts image features from the input pair. Next, the RoPE Galerkin Attention module (c) maps image feature functions to flow field functions via a Galerkin-style projection into the following Conv-GRU modules, where the coarse flow field features undergo iterative refinement. Finally, the SR Module (d) generates high-resolution flow field predictions through Continuous-S Flow Velocimetry, detailed in section 3.2
training to estimate flow fields from images but achieved limited accuracy in flow field estimation
30 proposed cost-volume-basse similarity computation to explicitly model matching relationships between image pairs, effectively improving displacement estimation accuracy. However, the construction of cour-dimensiona cost volume incurs huge computational burden and memory footprint one hand, and more importantly freezes the possible matching accuracy and resolution determined by the grid size and dimensions of the cost volume. Therefore, voluue--auseeee CNNs once trained cannot afford to solve different flow measurement problems with various accuracy requirements.
due to their inefficiency in global motion awareness. Later work 226 25 27
Generalization Challenges in PIV. Supervised learning methods depend on large labeled datasets. Due to the variability in flow conditions and particle image quality, these methods often generalize
poorly to unseen flows lighting setups. To address this, recent studies [3] 28
45 48 have
explored unsupervised optical flow algorithms for fluid flow estimation. These approaches optimize cost function with physical constraints to estimate motion fields, offering better adaptability across diverse scenarios. However, they are highly sensitive to image quality; noise or lighting variations significantly degrade robustness and accuracy.
3 Prticle Image Velocimetry Neural Operator
3.1 Framework of PIVNO
The goal of the proposed PIVNO is mathematically establish a mapping 9 A + U, where A and U/ represent the Hilbert spaces of particle image function pairs and flow fields, respectively. Specifically, we define: A A {a [T, Iz] E RR [1, 1 X + R, (xx) = I2(x - u(x))) to stress that the flow function w(x) relates the input particle grayscale image pair, and U A u X IR2)) represents the function space of output flow fields. X' TR22 the domain of the images. To approximate the mapping operator G, we parameterize it with neural network Go and optimize the

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parameters using supervised training. Given n observations {az, ",),,,1 where aj are sampled from a distribution over A and uj {{a)), the training process minimizes the empirical risk:
min Σ |uy - Go(a;)lli
0 TI
where || llee represents the norm in the space U.
As shown in fig. Taa a). PIVNO depicts structured workflow to transform particle image pairs into accurate flow field estimates. It begins with a feature encoder that extracts localized image features as the foundational representation. The RoPE-GA module implements Petrov- Serro--Galerkin-styyle projection that maps image feature functions to flow field representations. Subsequently, the Conv- GRU module iteratively optimizes the flow field features using the contextual features. Finally, the SR module reconstructs the refined features as a continuous-scale flow field representation, allowing for accurate arbitrarily scaled flow velocity estimates.
3.1.1 Feature Encoder
The core of flow field estimation lies in accurately matching local features; thus, extracting discrimina- tive features is critical for matching precision. As illustrated in fig. [b), we design feature encoder to extract local features from the input image pair (/, I2) and project them into a high-dimensional latent space. To enhance the encoder's representation capability, we integrate six residual units, each implementing a residual convolution operation defined as:
-Jo-=Snn SMLE (()) s(x v)
(())(z) -
I(x),
where <(x - y) denotes the 3 x convolution kernel defined over the local neighborhood (xx) of position v, and (xx) is the residual connection term. After the encoder extracts the spatial features from the image pair [ı and 22, the joint feature 20(x) is obtained through concatenation:
=a)x) Concat(P(Ii), P(12)),
(5)
where Concat merges the two feature maps along the channel dimension; consequently, zu(x) now contains the temporal evolution of the spatial features.
3.1.2 RoPE Galerkin Attention
To construct a operator mapping from the image feature space to the flow field space, we propose the RoPE Galerkin Attention (RoPE-GA) module illustrated in fig. [Tc). [36] presents a general discussion on the parallelism between the finite element methods and its proposed GA module, but
lacks of the positional embedding treatment for practical problems. Since the absolute positions
utmost important for particle tracking, we modify the positional encoding scheme of [49] to adapt to the Galerkin attention mechanism, leading to the enhanced GA matrix with Positional Encoding (PE) and frequency modulations:
A = Re kJ, U3ge (p%(0) e -0 -01)
(6) where P and pK denote the spatial coordinates of position along the and y-axes, Kj and 12 are the components of the key and value along channel dimensions ji and j2, and 05 and 0y are learnable frequency parameters. See Supplementary Material Section A for detailed derivations and discussions.
The operator form of GA reads:
(A[=))<) ~ Σ
Σ
where (vx=) represents the evaluation of the input function x(x) after the linear operator KC at coordinate y&i the index j means the j-th dimention of K(()). The same applies to V and Q and A denotes the integral kernel operator that refines 2<x) through the so-call basis update [50] However, this process is solely linear, lack of nonlinear expression capability.

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In contrast, our RoPE-GA in 6) introduces frequency modulation to each channel j with different learnable frequency parameter 9-- This treatment is equivalent to applying the sinusoidal activation [I to the correlated features K and V after PE.
RoPE-GA in (q. attends to the correlation among the evaluated basis functions, i.c., along the channel dimension. In order to enhance the spatial aggregation capability, we implant a 3 x 3 depthwise convolution between the first fully connected layer and the GELU activation function in the feed-forward network. Previous studies have shown that this operation helps capture finer spatial positional information
|2 55
3.1.3 GRU-based flow refinement
RoPE-GA produces coarse flow features (xr) by mapping along the channel dimension (see Sup- plementary Materials Figure 1), but it does explicitly organize spatial information; this often leads to locally inconsistent estimates, Because spatial continuity is critical for accurate motion estimation, merely stacking additional RoPE-GA layers insufficient. We therefore introduce an iterative refinement mechanism in the spatial domain.
We adopt a convolutional GRU (Conv-GRU) for this purpose. Unlike RAFT [66. which uses image features as the context provider, our method performs recurrent refinement directly on the flow features, as illustrated in fig. Ta a). Specifically, the RoPE-GA features (z) serve two roles: they initialize the hidden state ho and, at every iteration, they are provided as the input feature c to supply local neighborhood information. The Conv-GRU then updates hidden states by convolving over spatial neighborhoods, enabling iterative propagation and correction of local flow evidence. Mechanistically, the update and reset gates regulate information exchange between the current input and the previous hidden state, allowing differential nonlinear mappings across multiple local subspaces. At each iteration, the module fuses the current features with the previous estimate and adjusts both the direction and magnitude of the predicted motion through locally sensitive convolutions and gating. In this way, the Conv-GRi GRU functions not as a temporal recurrence for sequences but as memory- equipped, differentiable spatial transformation that progressively approximates the true flow field within local regions. In practice, we find that small number of refinement steps suffices: five iterations achieve near-optimal performance (see Supplementary Material Section B). The operations of the Conv-GRU module can be expressed as:
3/1 a (Coon3333( We)),
(8)
77 = ([Coonvxxs(/ht111 W,)), h tanbi(oonssxs((r @ h4-1, Wx)),
(9)
(10)
h4 (1 vi @ h4-1 yr @ hr
(11)
where a and tanh denote the sigmoid and hyperbolic tangent activation functions, respectively; Convax3 denotes 3 x convolution; Ws, Wr, and Wh, are the learnable convolutional weights; and C represents element-wise multiplication.
3.2 Continuous-Scale Flow Velocimetry
To achieve accurate super-resolution reconstruction of the flow field, the SR module (fig. (d) is designed by combining random sampling, continuous interpolation, and graph convolution. The first step adapted from SRNO (50), which enhances the scale generalizability of our model:
h(.:)) Concat(s: (81 h4) 11)
(12)
where represents the target coordinate, and E {00, 01, 10, 11) denotes the coordinates of the four neighboring points of x; 81 is the diagonal area of the neighboring grid point's coordinates, and f is the feature vector of the neighboring points. The final feature vector h((x,) is obtained by concatenating the weighted feature vectors hy.
The resulting feature h(xx)) is then input into the subsequent graph convolution layers. Since the features of sampled points already incorporate positional information through the RoPE-GA module, the graph convolution (57.59 can effectively capture the spatial correlations between the randomly sampled points. By supervising on the randomly sampled points, whose results come from the graph convolution performing local information fusion, we are actually enforce PIVNO to attend to the
5

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local correlation in fluids, enhancing the completeness and robustness of the feature representation. Finally, the graph convolution projects the high- -ighdimmessooal features back into the PIV flow field solution space u(x:) via a 3 x 3 convolution:
(x)
n(x y)h(y) dy.
(13)
n(x))
where x(z y) represents the 3 x 3 graph convolution kernel defined over the set of randomly sampled points R(x).
3.3 Self-Supervised Fine-Tuning
We further propose a self-supervised fine-tuning scheme to adapt the simulation based pre-trained models to real experimental data. The fine-tuning strategy employs variational optical flow method consisting three components: a data term, smoothing term, and a divergence regularization term. The supervieee loss function is defined as:
( = La(u) + (,) + AaLav(u).
(14)
where La represents the data term, modeling the similarity of the image pairs, T.g and Tair the spatial smoothing and divergence regularization terms respectively, and X and Xa are their respective weights.
Data Term:
EVEN(W) (T1 (TENN (1) (v))
corr(x) =
(15)
EvEN(E) [14 EVENNO 11)(1))1
La(I,, 12) a Ez(corr)
(16)
where (x:) 12(x u(x)) the warped image compared against J, and u(x) represents the flow
field prediction at the spatial location æ. ((x) denotes the set of all pixel points within the sliding window. (I1));, i, (x)) is the dot product of the real image and the predicted image, and || (z) and [z)]] represent the magnitudes the real image and the predicted image within the window, respectively. The function a=z) = (z² + e²)² is the Charbonnier penalty function, used to smooth the error term 2, where y controls the degree of smoothing, and Ez denotes the expectation over all the positions in the images.
Smoothing Term:
L.(4) (V²u(x))
(17)
where (zu(z) is the second derivative of u(x) (the Laplacian operator), which measures the local variation of the underlying flow field.
Divergence Term:
Law a (V u(x))
(18)
where V - u(x) is the divergence of u(r), which enforces the incompressibility constraint for fluid flow, constraint the divergence of the flow.
Experiments
The experimental evaluation comprises three synthetic datasets and three real-world PIV challenge tasks. Initially, supervised training and benchmark testing are conducted on Synthetic Datasets and 2. Training samples generated by uniformly sampling downsampling factors within the range of 1 to 4x, allowing the model to generalize across varying output resolutions after a single training phase. Subsequently, self-supervised fine-tuning is performed on Synthetic Dataset and on the three real-world PIV benchmarks to enhance cross-domain adaptability. Since ground-trutt, flow fields unavailable for real-world benchmarks, direct quantitative evaluation is inherently infeasible. The selected real- world PIV tasks are chosen for their established and credible evaluation protocols, enabling reliable qualitative comparison.
In addition, ablation experiments are conducted to analyze the contribution of key modules and loss components. Table 3 specifically presents the ablation results of the self-superee self-supervised loss terms and

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Table 1: This table presents the Average Endpoint Error (AEE) on synthetic dataset 1, where the error unit is set to pixels per 100 pixels for easier comparison. All methods are evaluated using fixed input and output resolution of 256 The last four rows show the performance of our method at different input resolutions (e.g.. 64 4 represents an input 64 with 4x upsampling achieve the 256 output). The Params column indicates the number of parameters in millions (M).
JHTDB Channel
DNS turbulence
Methods
Backstep
Cylinder
SQG
Params, (M)
Farneback
660
8.5
8.3
14.1
37.8
33.2
PIV-DCNN
22
4.9
11.7
33.4
47.9
8.40
PIV-LiteFlowNet
23
5.6
8.3
10.4
19.6
20.0
6.25
PIV-LitcFlowNNt-en
3.3
4.9
7.5
12.2
12.6
5.59
UnPwcNet-PIV
13
8.2
7.1
13.4
21.5
25.2
9.37
UnLiteFlowNet-PP
9.4
6,9
8.4
15.0
17.3
5.38
OFVNetS [61]
15.0
1.6
25.0
8.3
22.3
OFV 61
13.7
4.7
32.7
7.0
18.9
PIV-RAFT 25
1.6
13.7
11.7
5.31
ARaft-FlowNet 29
3.1
2.0
8.3
9.6
9.8
PIVNO(256 1)
1.9
0.8
1.7
3.5
2.5
PIVNO(128 2)
0.9
1.1
2.8
4.7
4.1
2.52
PIV x 2)
1.0
1.4
2.9
5.0
4.2
PIVNO(644 X 4)
2.7
3.1
11.3
18.0
18.0
Ground Truth
PIV-RAFT
UnPwcNet-PIV
Farneback
PIVNOlours)
2.0
1.5
AEE:19.2
AEE:19.7
AEE-14.1
AEE:L.1
0.0
3.0
2.0
1.0
AEE-12.5
AEE-1.1
4.0
3.0
2.0
AEE:20.0
AEE:43.9
AEE:46.4
AEE:3.0
Visualization of three flow fields: DNS turbulence (first row), JHTDB channel flow (second row), and SQG (third row). The rectangles indicate over-smoothing even failure artifacts existing in the outputs of comparative methods, while PIVNO faithfully recovers the flow.
Figure 2:
simultancously summarizes the overall performance of the self-supervised strategy on both synthetic and real-world datasets. is therefore presented separately to highlight its central role in validating the effectiveness of the proposed self-superviste strategy. Section 4.3 further investigates the influence of architectural components. Beyond the main results, we also conduct extended analyses, including statistical robustness evaluation (maximum error and standard deviation), cross-domain fine-tuning generalization experiments, and zero-shot resolution generalization studies. of these additional results are provided in the Supplementary Material for completeness. Comprehensive details of the network architecture, dataset configurations, implementation methods, and hyperparameter settings also included therein.

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Table The AEE on synthetic dataset Repre- sentative baselines include the unsupervised Un- LiteFlowNet and the self self-supervised PIV-RAFT.
Table 3: Impact of loss term combinations. The table shows AEE and divergence results on SPID (sim) and solid body rotation (real).
Medaols
Cylnde
Chassel
unisalance
DNS
Ie
Takia
SPID
Sulid Body Rotation Flow
div.
UnLaefioNNs-IV PST
12.
7.9
14.5
22.5
21.5
609
65.9
41.9
61.
40.1
X
1.62
0.25
0.64
455
RIKRART M
53
15.8
19.7
-8036.29
PIYNO 1
4.5
3.4
4.7
4.4
6.6
-12.46
PEYNO x
4.1
3.6
8.5
9,6
13.2
-0.09
5445
PIYNOIS4" x
34
3.1
8,5
10.2
13.
v
x
1.10
-6639 26
0823
-2072.67
PIVNO1G4 x4
63
1966
21.7
21.9
0.60
-13.00
0.17
-1424.37
4.1 Evaluation on Synthetic Datasets
Synthetic Dataset 1:
This PIV dataset [23] contains five classic flow field cases commonly used for training and benchmarking PIV algorithms. As shown in table [ the proposed PIVNO model consistently outperforms state-of-the-arr (SOTA) methods across all evaluation metrics at resolution of 256 Its advantage is especially notable in complex flow regimes such as DNS turbulence, JHTDB channel flow, and SQG sea surface flow. key strength PIVNO lies in its training strategy: the model is trained once on samples uniformly downsampled by factors from 1 x to 4x, enabling robust generalization across multiple upsampling scales during inference. Notably, even with low-resolution inputs (e.g., )4 PIVNO produces high-resolution outputs with accuracy comparable to models using full-resolution inputs. Furthermore, when the output resolution twice that of the input, PIVNO still achieves superior velocity field estimation, demonstrating strong robustness and adaptability in low-resolution settings.
Additionally, 22 presents visual comparisons. The first row displays DNS turbulence, featuring rich small-scale vortical structures and multi-scale turbulence interactions, demonstrating high complexity and dynamic characteristics. Comparatively, only PIVNO effectively captures both local features and global relationships. The second row illustrates JHTDB channel flow, marked by shear effects, stratified velocity gradients, boundary confinement, and turbulent transition behavior. Remarkably, PIVNO accurately captures these boundary layer features. The third row shows SQG sea surface flow, including nonlinear interactions between large-scale background fields and small-scale disturbances, mainly exhibiting wo--dimessional geaut-geostroppic characteristics. In summary, PIVNO handles these complex dynamics with precision.
Synthetic Dataset 2: To evaluate the model's performance under large displacement and high noise conditions, we used synthetic dataset from (25. which simulates large particle displacements, low particle density, and significant noise- boise-providdnng algorithms. As shown in table 2 models like UnL.itelowWNet-PP1 which rely on photometric loss, suffer in high-noise settings due to disrupted motion feature extraction. PIV-RAFT also struggles with large displacements due to limitations in its local correlation-based approach. In contrast, PIVNO maintains stable prediction performance even under such challenging conditions.
suitable testhed for assessing the robustness of PIV
Synthetic Dataset 3: To evaluate the impact of the three loss terms (La. L,, and Ldie) on flow field estimation, we conducted ablation experiments using the SPID dataset [62], which simulates real-world conditions such as noise, particle distribution, and out-of -lane motion. We progressively removed each loss term and evaluated the changes in AEE and divergence metrics. As shown in table 3 we observed that using any single loss function in isolation resulted in a significant performance degradation, with increased AEE and anomalous divergence values, indicating that a single loss term cannot effectively constrain the model's fine-tuning. In contrast, the combination of all three loss terms yielded substantial performance gains, demonstrating their synergistic effect in optimizing flow field estimation. The consistent performance deterioration when omitting any loss function further validates the effectiveness of this fine-tuning strategy.
4.2 Generalizability on Real PIV Challenges
Solid Body Rotation Flow:
To evaluate the generalization capability of the fine-tuning strategy in
real-world scenarios, we selected the solid body rotation flow [63] as a classical benchmark due to its theoretical clarity, uniform vorticity, and strict adherence to solid body rotation. table 3 quantiffis the impact of fine-tuning, showing that combining all three loss terms (La, Lg, and L significantly
8

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Pretrained
Tuncd Mo]
Fine Tuned
Reference
[a] Frame
(b) Frame
Figure 3: Comparison of the flow estimation results in the solid body rotation test.
Particle Prediction
[c] MG-PIV
(d) PIVNO(Ours)
Histogram of Velocity Magnitude
Figure 5: Comparison of turbulent round jet ve- locity fields: (a) and (b) are the original particle images with significant illumination variation; (c) is the fluid vector field estimated using the multigrid PIV method, appearing blurry and dis- continuous: (d) is the fluid vector field estimated using PIVNO, exhibiting fair structures and con- tinuity. The white lines indicate significant mo- tion trajectories.
Velccity Magnitude
Figure 4: (a) Strong vortex particle image; (b) Estimated flow field via proposed method; (c) Ve- locity magnitude histogram. PIVNO accurately estimates flow and avoids "peak locking".
reduces AEE and improves divergence, reinforeing the model's robustness. fig. 3 illustrates this effect by comparing predictions with theoretical values, where the pre-trained model fails to capture rotational characteristics, showing flow discontinuities near the rotation center. While using only La mitigates some errors, it still yields boundary anomalies and lacks smoothness. In contrast, the full fine-tuning strategy enables precise particle displacement prediction, closely aligning with theoretical solution and ensuring high-quality flow estimation across both boundary and rotation center regions. These findings highlight the critical role of the three-loss synergy enhancing model generalization and robustness for flow prediction.
We use PIVNO process the vortex flow field images recorded by the German Acrospace Center in the DNW-LLF large wind tunnel [64]. The motion field contains complex characteristics such as high velocity gradients, particle density loss, size variations and small particles, making it ideal for testing the robustness of our method. Experimental results show that our approach accurately reconstructs intricate velocity distributions even under low particle density and steep velocity gradients, illustrated in fig. 4().. In the experimental images, particle sizes are less than two pixels, causing grayscale distributions suffer from pixelation effects, which hinder precise subpixel-scale localization. During displacement measurement, this blurring effect biases measured values toward integer pixel positions rather than forming a continuous distribution, leading to the so-called peak-locking effeett (65) and reducing accuracy. Our method effectively mitigates this issue.
Strong Vortex:
Turbulent Jet:
We evaluated the turbulent round jet dataset from Delft University of Technology [66]. focusing on high- -raddega regions and flow continuity. As shown in fig. a) and (b), significant lighting changes between the two frames challenged flow field estimation. To obtain the velocity field, the dataset provider uses a multi -iiid PIV method (fig, [(c)). However, due to resolution limits of the grid-based approach, struggles to resolve fine-scale flow structures, causing visible blurring artifacts. In the experiment, fluid is injected from middle-right toward middle-left, while the upper and lower sections are expected flow inward due to pressure differences. Nevertheless, the multi-grid PIV method shows irregular and discontinuous flow in these regions, failing to preserve

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Table 4: Ablation comparisons on Synthetic Dataset 1.
JHTDB Channel
RoPE: Mixed
DWConv
GA
GRU
GCN
Unifirm
Backstep
Cylinder
SQG
turbulence
15.55
8.95
4.04
3.07
0.78
1.88
3.94
2.90
2.30
3.31
2.43
1,08
1.73
3.57
2.61
4.03
2.03
0.84
1.84
3.67
2.73
3.40
2.56
0.87
1.71
3.50
2.57
3.26
1,91
0.79
1.68
3,47
2,54
expected motion coherence. In contrast, our method (fig. d)) captures the central flow trend in both upper and lower regions effectively, yielding structured and continuous velocity field.
4.3 Ablation Studies
conducted ablation studies on Synthetic Dataset to assess the importance of each module in PIVNO. As shown in Table 4 removing any single component degrades performance. Notably, the GA and GRU modules have the most significant impact when being removed, indicating their essential roles in the overall architecture. Other components such RoPE-Mixed and DWConv also contribute consistently. These results validate the necessity of the full model design. More ablation experiments can be found in the supplementary material (Case B).
5 Conclusion
propose PIVNO, a neural operator framework that formulates PIV as a PDE-constrained inverse problem. By leveraging Galerkin-style attention mechanism and self-supervised fine-tuning scheme grounded in physical constraints, PIVNO achieves accurate and resolution addaptive flow esti- mation across synthetic and real-world datasets. The framework demonstrates strong generalization and robustness capabilities, highlighting its potential for high-precision PIV applications.
Limitations. 2D particle velocimetry cannot fully capture the true 3D nature of fluid dynamics. Future work will focus on extending the framework to 3D flow field estimation. Additionally, the current feature extraction module lacks multi-scale feature fusion, which should be taken into consideration when processing high-resolution PIV data.
Acknowledgments and Disclosure of Funding
This work was supported in part by the National Natural Science Foundation of China under Grants 61871055, 62025102, and 82427801.
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Reviews:
[{"review_id": "aVR5r0Q3L3", "reviewer": "Reviewer_gQBY", "summary": "This work proposes PIVNO, a neural method that directly estimates the mapping from paired particle images to flow fields. It addresses three key challenges:\n- introducing the PIV Neural Operator to ensure that the estimated mapping lies within a physically feasible solution space;\n- proposing a self-supervised fine-tuning scheme to bridge the sim-to-real gap;\n- incorporating a super-resolution module to produce high-resolution flow estimations.\n\nExperiments are conducted on three synthetic datasets and three real-world datasets, all of which demonstrate the effectiveness of the proposed approach.", "questions": "1. After fine-tuning on the real dataset, could you also evaluate the fine-tuned model on the synthetic dataset? This analysis would help assess whether the fine-tuning leads to overfitting on real data and a loss of generalization to synthetic scenarios.\n\n2. It would be valuable to involve domain experts (e.g., fluid mechanics researchers) to qualitatively evaluate the correctness of the estimated flow field.\n\n3. In addition to reporting the average endpoint error, could you also provide statistics such as the mean and standard deviation of the error distribution? This would offer a more comprehensive view of the model’s performance and robustness.", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. It is interesting to bypass the traditional cost-volume construction and directly estimate the flow field. The idea of the Galerkin-type attention operator is novel, and the experimental results support its effectiveness.\n\n2. The proposed framework is both sound and innovative.\n\n3. The experiments are comprehensive, evaluating the method on both synthetic and real-world datasets.\n\nWeaknesses:\n\n1. The organization of the experimental section is somewhat confusing. For example, some ablation studies are already presented in Section 4.1 (e.g., Table 3), yet a separate section titled \"Ablation Study\" is also included. While I appreciate the thoroughness, I suggest reorganizing the experimental section for better clarity and coherence.\n\n2. I have concerns regarding the soundness of the evaluation metric used—Average Endpoint Error (AEE). Have the authors considered reporting the maximum error or the per-timestep error? This could help assess error accumulation over time.", "was_revised": true, "final_justification": "All my concerns have been addressed. This is an interesting work. I maintain my score as 'Accept'.", "rebuttal": "Thank you very much for your valuable feedback and suggestions! We greatly appreciate your comments and treat them carefully. For each issue raised, we will make modifications or clarifications as needed. Below, we provide a brief discussion of the concerns raised in the review:\n\n1.Unclear Experimental Structure\n\nYou pointed out that the structure of the experimental section is somewhat confusing, especially the redundancy between Section 4.1 and the separate \"Ablation Study\" section. We fully understand and acknowledge this. The reason for presenting Table 3 separately is that it not only includes ablation experiments on the self-supervised loss components but also summarizes the overall performance of the self-supervised strategy on both synthetic and real datasets. Therefore, we believe it serves as a key indicator of the self-supervised strategy's effectiveness and deserves a separate presentation. The \"Ablation Study\" in Section 4.3 focuses on ablation analysis under other experimental settings. However, we realize that the current structure may not effectively guide readers through the hierarchy of experiments. To address this, we will include additional clarifications in the revised version to explicitly outline the logical structure of the experimental section, thereby improving clarity and coherence.\n\n2.Rationality of Evaluation Metrics\n\nCurrently, in the PIV field, the Average Endpoint Error (AEE) is the mainstream evaluation metric. Moreover, many related works have not released their source code, which makes it less possible to conduct comparisons using alternative metrics. We fully agree with your concern about the rationality of evaluation metrics. To address this, we have additionally reported the maximum error（Max）and standard deviation (Std) of the error distribution on Synthetic Dataset 1. The experimental results are as follows (unit: pixels):\n\n| Metric | Method   | Uniform | Back-Step | Cylinder | JHTDB channel | DNS turbulence | SQG |\n|:------:|:--------:|:-------:|:---------:|:--------:|:-------------:|:--------------:|:---:|\n|  Max   | 64²×4    |  0.662  |   0.660   |  0.745   |     2.653     |      2.564     |2.562|\n|  Max   | 64²×2    |  0.228  |   0.318   |  0.309   |     2.060     |      1.694     |1.491|\n|  Max   | 128²×2   |  0.303  |   0.418   |  0.370   |     1.843     |      1.185     |1.165|\n|  Max   | 256²×1   |  0.660  |   3.457   |  2.328   |     9.604     |      8.995     |2.706|\n|  Std   | 64²×4    |  0.005  |   0.008   |  0.008   |     0.031     |      0.046     |0.027|\n|  Std   | 64²×2    |  0.003  |   0.004   |  0.004   |     0.009     |      0.019     |0.011|\n|  Std   | 128²×2   |  0.003  |   0.003   |  0.004   |     0.009     |      0.016     |0.007|\n|  Std   | 256²×1   |  0.010  |   0.029   |  0.007   |     0.029     |      0.057     |0.011|\n\nIt is particularly noteworthy that under the 256²×1 resolution setting, the model exhibits significantly higher maximum error and standard deviation compared to other settings. Upon further investigation, we found that this is caused by a few extreme error points near the boundary regions, which raise the overall statistics. In contrast, the ×4 and ×2 resolution settings inherently introduce spatial averaging (filtering) effects during the super-resolution reconstruction process, which smoothes out the local noise and thereby reduces both the maximum error and the error variance.\n\n3.Evaluation of Fine-Tuned Models on Synthetic Datasets\n\nOur method primarily focuses on training the model on synthetic datasets to learn a general mapping from input particle images to output motion fields, followed by self-supervised fine-tuning in specific real-world scenarios to enhance performance for the task at hand. As our main goal is to improve performance in these specific real-world tasks, the generalization of the fine-tuned model back to synthetic datasets was not our primary consideration. However, we understand the reviewers’ concern about the generalization ability after fine-tuning and the consistance on its importance. Therefore, we have conducted experiments to evaluate the generalization of the fine-tuned model on the synthetic dataset. The results are as follows (unit: pixels):\n\n| Resolution | Uniform | Back-Step | Cylinder | JHTDB channel | DNS turbulence | SQG   |\n|:----------:|:-------:|:---------:|:--------:|:-------------:|:--------------:|:-----:|\n| 64²×4      | 4.3100  | 2.1492    | 2.3437   | 1.1926        | 1.1089         | 0.9776 |\n| 64²×2      | 6.8775  | 3.5106    | 2.5500   | 1.8067        | 1.0801         | 1.0307 |\n| 128²×2     | 6.9424  | 3.0903    | 3.0121   | 1.8427        | 0.9936         | 1.0770 |\n| 256²×1     | 7.4281  | 3.2023    | 3.7374   | 2.6118        | 2.2822         | 2.4208 |\n\nExperimental results show that after fine-tuning on real-world datasets, the performance of PIVNO on synthetic datasets exhibits  a noticeable decline. This reflects the nature of fine-tuning as a process of adapting a general model to specific scenarios. Given that PIVNO contains only 2.4M parameters, expecting it to maintain strong generalization across both synthetic and real domains without any trade-offs is somewhat overly demanding. Afterall, the core value of our model lies in its adaptability—achieving effective performance in diverse real-world situations through lightweight fine-tuning—rather than aiming for a one-size-fits-all solution through a single training process.\n\n4.Involvement of Domain Experts\n\nWe fully agree with your point that qualitative assessment of flow field estimations by domain experts (e.g., fluid mechanics researchers) would be highly valuable. The datasets and baselines used in our study are widely recognized and scientifically grounded in the field, effectively reflecting the accuracy and applicability of our method. To promote interdisciplinary collaboration and further validate the physical reliability of the model, we will open-source our code and sincerely welcome experts in fluid mechanics and related fields to participate in qualitative evaluations and assesments, jointly advancing the application and development of our method to real-world flow problems."}, {"review_id": "Xrwhs7IGzJ", "reviewer": "Reviewer_7Y2L", "summary": "The paper presents a new method for Particle Image Velocimetry (PIV) using a Neural Operator framework (PIVNO).  The framework leverages a position-informed Galerkin-style attention operator to capture global flow structures while supporting resolution-adaptive inference. It incorporates self-supervised fine-tuning based on physical divergence constraints, helping the model generalize from synthetic to real-world experiments without requiring labeled data.", "questions": "How does PIVNO perform when applied to completely novel fluid dynamics conditions that differ significantly from the training scenarios? Are there specific flow regimes where the model’s performance might degrade.", "limitations": "The author discussed the relevant limitations.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n1. One of the key strengths of PIVNO is its ability to perform resolution-adaptive inference. The model can adapt to varying input resolutions and still provide accurate high-resolution flow field outputs. This makes it highly versatile in real-world applications where the region of interest may have different resolution requirements.\n\n2. The self-supervised fine-tuning mechanism based on physical divergence constraints enables the model to adapt from synthetic datasets to real-world conditions without requiring labeled real data. \n\n**Weaknesses**\n1. The self-supervised fine-tuning allows PIVNO might not be fully reliable for all real-world datasets. The fine-tuning mechanism still relies on certain assumptions, such as the physical divergence-free condition, which may not always hold or may be difficult to enforce in some experimental setups.\n\n2. The current PIVNO framework is designed for 2D flow field estimation. This limitation restricts its ability to fully capture the three-dimensional (3D) nature of many fluid dynamics problems. Extending the model to handle 3D flow fields is a critical next step to broaden its applicability to more complex and realistic fluid systems.\n\n3. Real experimental conditions can vary widely in terms of lighting, particle distribution, and other environmental factors, which may not always be fully covered by the synthetic training datasets.", "was_revised": true, "final_justification": "The author has addressed my concerns and I want to maintain the score.", "rebuttal": "Thank you very much for your valuable feedback and suggestions! We highly appreciate your comments and treat them carefully. For each issue raised, we will make modifications or clarifications as needed. Below, we briefly discuss the points raised:\n\n1.Limitations of 2D Estimation\n\nWe fully agree with your view that \"extending to 3D flow fields is a crucial next step to improve the model’s applicability to more complex and realistic fluid systems.\" Our focus on 2D in this work is based on a careful assessment of existing mainstream 3D methods. Current 3D flow estimation methods mainly adopt particle tracking velocimetry (PTV), which requires a prerequisite  particle detection phase followed by the paticle matching and flow estimation process. This approach is highly sensitive to detection quality and prone to error propagation, and produces sparse Lagrangian motion fields that are difficult to use for dense Eulerian field reconstruction. In contrast, PIVNO follows the PIV paradigm, estimating dense Eulerian flow fields end-to-end directly from particle image pairs, thus avoiding the instability and information loss introduced by intermediate steps. Therefore, we first establish a solid and reproducible baseline in 2D scenarios by integrating physical priors. On this foundation, we have already started developing  tomography-based 3D PIVNO to further extend applicability to complex real-world 3D flows.\n\n2.Physical Assumption Limitations of the Self-Supervised Mechanism\n\nAs you pointed out, we indeed introduce a divergence-free constraint based on the incompressible flow assumption, which may not fully hold in some experimental setups or real applications. It is important to emphasize that the divergence-free constraint is neither inherent nor the only physical constraint usable in the PIVNO framework. It serves as a physical prior tailored to the incompressible flow characteristics of our current datasets. For other fluid types, the PIVNO framework can similarly replace or extend physical constraints to fit new assumptions or scenarios. This constraint is flexible and should be set by the researchers according to the physical context and prior knowledge of the problem.\n\n3.Generalization and Performance Degradation\n\nWe fully recognize that the complexity of fluid dynamics makes it impossible for the model to maintain ideal performance under all conditions. Hence, we propose a self-supervised fine-tuning mechanism specifically to address potential performance degradation in real and complex conditions. In our experiments, the three selected real scenarios differ significantly from the synthetic training data in flow states, particle density, illumination conditions, etc. As shown in Table 3 of the main text, the model’s performance does degrade noticeably when encountering novel flow states significantly different from training. However, self-supervised fine-tuning effectively mitigates this degradation and substantially improves the final estimation accuracy, enabling the model to remain robust and adaptable in complex real-world scenarios."}, {"review_id": "LqHkB3QLji", "reviewer": "Reviewer_PrER", "summary": "This paper addresses the problem of Particle Image Velocimetry (PIV), which aims to infer underlying velocity fields from time-separated particle images. The core idea is to directly learn the inverse mapping from paired particle images to the corresponding flow fields within a function space. The proposed framework utilizes a position-informed Galerkin-style attention operator (RoPE-GA) to capture global flow structures while enabling resolution-adaptive inference. Furthermore, to improve generalization to real-world conditions, the paper introduces a self-supervised fine-tuning scheme based on physical divergence constraints, which allows the model to adapt from synthetic to real experimental data without requiring labels.", "questions": "1/ Regarding the GRU refinement module, beyond its function of iterative optimization, is there a deeper motivation for choosing the Conv-GRU structure specifically? For example, what advantages do the recurrent and gating mechanisms of a GRU offer for the task of flow field refinement compared to simply stacking more RoPE-GA layers or using other types of convolutional networks for feature fusion?\n\n2/ A key claimed advantage of the framework is its resolution adaptability. The experiments in Table 1 demonstrate robustness to low-resolution inputs. To more rigorously validate its properties as a true neural operator, have you considered performing a \"zero-shot resolution generalization\" test? For instance, training the model on a single resolution (e.g., 128x128) and then evaluating its performance on a variety of unseen output resolutions (e.g., 192x192, 256x256, etc.). This would provide stronger evidence of the model's generalization capabilities.\n\n3/ Your ablation in Table 3 shows that using only the divergence loss ($L_{div}$) leads to poor performance. This is an interesting result, especially since other works, such as [1], have successfully used a zero-divergence loss. This suggests the effectiveness of this physical prior is highly context-dependent. Could you provide further discussion on this? For example, under what conditions is the divergence loss a strong regularizer on its own, and when does it require a strong data term to be effective?\n\n[1] Dual-frame Fluid Motion Estimation with Test-time Optimization and Zero-divergence Loss", "limitations": "Please see the section of weaknesses and questions.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\n1/ Novel and Insightful Problem Formulation: Framing the PIV problem as an operator learning task is a strong and modern approach. This conceptual shift from simple feature matching to learning the underlying physical mapping is a significant contribution that is more aligned with the physical nature of the problem and has the potential to overcome the inherent trade-offs of traditional methods.\n\n2/ Innovative and Well-Motivated Architecture: The proposed RoPE-GA module is a key innovation with clear motivation. The authors correctly identify that standard Galerkin Attention lacks the absolute positional information crucial for PIV and creatively integrate Rotary Position Embeddings (RoPE) to address this gap. This results in a well-designed and well-justified architecture.\n\nWeaknesses\n\n1/ Insufficient Motivation for Some Network Components: While the motivation for the core RoPE-GA module is very clear, the justification for other architectural choices could be stronger. For example, the paper states that a Conv-GRU module is used for iterative flow refinement, but does not provide a deep justification for why the GRU architecture is specifically optimal for this task compared to other iterative methods or feature-fusion strategies.\n\n2/ Incomplete Ablation of Self-Supervised Loss Terms: In Table 3, the paper shows that the full three-term loss function performs best. However, this ablation does not fully isolate the individual contributions of each term, particularly the smoothing term (Ls) and the divergence term (Ldiv). The lack of results for combinations like (Ld + Ls) or (Ld + Ldiv) makes it difficult to clearly understand the specific benefit that the physics-based constraint adds over a traditional smoothing regularizer.\n\n3/ Missing Inference Speed Comparison: The paper emphasizes that its operator-based approach avoids the need for computationally expensive cost volumes, implying a potential advantage in inference speed. However, the experimental section lacks a direct comparison of inference times between PIVNO and key baselines like PIV-RAFT. Such a comparison would be essential for a complete assessment of the method's practical advantages.", "was_revised": true, "final_justification": "Please see my cmt below.", "rebuttal": "Thank you very much for your valuable feedback and suggestions! We greatly appreciate your comments and treat them carefully. For each issue raised, we will make modifications or clarifications as needed. Below, we provide a brief discussion of the concerns raised in the review:\n\n1. On the Conv-GRU Structure\n\nWe fully understand your concerns regarding the Conv-GRU structure, and we would like to briefly explain the deeper motivation behind our choice of the Conv-GRU iterative architecture.\n\nBefore applying Conv-GRU, we obtained flow features through the RoPE-GA module (see Supplementary Materials Figure 1). However, the RoPE-GA module only maps function spaces along the channel dimension but does not aggregate or organize the spatial information. This results in flow features that are often coarse, discontinuous, and contain local estimation errors, whereas spatial continuity is critical for accurate flow estimation. Therefore, we need a mechanism that can progressively optimize flow features in the spatial domain, which is why we did not simply stack more RoPE-GA layers.\n\nIn motion estimation fields such as PIV, optical flow, and disparity estimation, the Conv-GRU module has become a mainstream structure for iterative flow refinement, with methods such as RAFT [2] and FoundationStereo [3] adopting similar structures. The effectiveness of this approach has been widely validated, and we did not elaborate on the structural motivation in the main text. Our ablation experiments also confirmed the key role of this module in the final performance. However, we fully recognize your rigorous concern about this issue and are happy to further explain our understanding of its mechanism.\n\nUnlike conventional convolution operations, the hidden state update in Conv-GRU is determined by multiple gated channels (such as the update gate 𝑧, reset gate 𝑟, and candidate state). Convolution essentially projects the input features into a subspace spanned by a set of convolutional kernel functions, and the optimization of these kernels allows for the best possible hidden representation. Comparatively, the convolutions in Conv-GRU operate separately on the current input features and on  the previous hidden state, enabling differential nonlinear mappings across multiple subspaces during the update process. This structure allows the model to understand local motion trends from multiple perspectives and scales, and to iteratively correct the flow states of local regions at each iteration. Additionally, the nonlinear activation functions (e.g., sigmoid, tanh) introduced by the gating mechanism enhance the model's expressiveness, enabling it to adaptively regulate the retention, update, and fusion of information based on the current estimation state, thus overcoming the limitations of single-layer convolutions in modeling capacity.\n\nThroughout the iterative process, Conv-GRU acts as a differentiable spatial transformation module with memory. In each iteration, it fuses the current input features with the previous estimation result (i.e., the hidden state) and continuously adjusts the direction and magnitude of the flow field estimate through locally sensitive convolutional mappings and gating mechanisms. This structure endows the model with the ability to progressively approximate the true flow field within local spatial ranges. Therefore, Conv-GRU in PIVNO is no longer a recursive structure used for handling time sequences but rather an iterative optimization unit with local geometric modeling capabilities that supports differentiated state updates. Its synergistic effect of spatially varying linear combinations and nonlinear gating mechanisms enables the correction and refinement of the local flow field.\n\n2. Self-Supervised Fine-Tuning\n\nThank you for your attention and inquiries regarding the ablation experiments on the self-supervised loss terms in Table 3. We provide a unified response addressing both the theoretical basis and experimental design.\n\nFirst, regarding your observation that “using only the divergence loss term $L_{\\text{div}}$ leads to significant performance degradation,” we explain that this behavior is closely related to the ill-posed inverse problem nature of PIV. PIV maps observed image pairs to fluid motion fields, with inputs containing uncertainties such as repeated patterns, occlusions, and varying lighting conditions, making the inverse problem highly ill-conditioned and solutions non-unique. Under such conditions, a single regularization term is insufficient to constrain the solution space, easily causing the model to converge to physically unreasonable spurious solutions, thus losing estimation effectiveness.\n\nIn contrast, the work you mentioned [1] is based on a PTV framework where inputs are explicitly detected particle spatial coordinates, providing strong, unique observational support that substantially reduces the ill-posedness of the inverse problem. It is important to note that [1] also employs a combined loss of data consistency, smoothness, and divergence constraints during training. The “divergence-only” setting you referred to occurs only during testing, where the backbone network parameters are frozen and local adjustments are made via a residual module. Under such conditions, the “divergence-driven” mechanism does not independently constrain the backbone network, and thus cannot be directly compared to our independent loss configuration during training.\n\nSecond, regarding your concern that the ablation experiments do not fully reveal the individual contributions of each loss term, we acknowledge that not all possible loss combinations were presented in the main paper, but with primary focus on the core experimental questions. Our experimental design emphasizes two aspects: (1) verifying whether a single loss term is sufficient to avoid invalid solutions in PIV scenarios; and (2) assessing whether adding the physics-based divergence constraint to the traditional variational optical flow framework (i.e., data term plus smoothness term) significantly improves model performance. Both of these two points have been verified by our experiments, and we would like to supplement more results in the revised version.\n\n3. Zero-Shot Resolution Generalization\n\nThank you for your recognition of the robustness of our model under low-resolution input conditions. We also agree with your suggestion to test \"zero-shot resolution generalization\" as it would be valuable for further validating the operator properties and scale generalization capabilities of the model. However, we would like to clarify that PIVNO adopts a multi-scale input strategy during training. Specifically, the resolution of input image pairs is randomly sampled within a downsampling range from 1× to 4× (see Supplementary Materials B3). This training approach explicitly exposes the model to a continuous range of resolution distributions, enabling it to adapt to inputs of any scale. In other words, the scale generalization performance of PIVNO is learned progressively through this random scale training strategy. Therefore, the setting you suggested—“train the model on a single resolution (e.g., 128×128) and directly test on higher resolutions”—does not align with the core idea of our current training design. This setup would artificially limit the model's ability to model scale variations and would reduce its true generalization performance. Additionally, we believe that the \"neural operator\" property emphasized by PIVNO is more focused on its ability to map between image pairs (observation space) and fluid motion fields (function space), which differs from purely seeking scale invariance in images.\n\nNevertheless, your suggestion has been highly insightful for us. We are currently conducting a set of \"zero-shot resolution generalization\" experiments, where the model is trained on a single resolution (128×128) and then evaluated at an unseen resolution (256×256). We expect to present the results of this test in the rebuttal in the coming days and further analyze its impact on the model's operator generalization ability. Once again, thank you for raising this thought-provoking question.\n\n4. Inference Speed Comparison\n\nThank you for pointing out the limitations of the inference speed evaluation. We would like to further clarify the statement emphasized in our paper, \"avoiding the computational overhead of cost volume,\" which is intended to highlight the structural advantage in computational complexity of the RoPE-GA module compared to traditional methods that rely on explicit cost volume (CV) construction.\n\nRegarding your suggestion to directly compare the inference time with PIV-RAFT, we believe that, although both methods are comparable in terms of the task, the overall architecture differences (including feature extraction modules and the GRU update mechanism) make a direct runtime comparison potentially unfair and inaccurate in reflecting the actual contribution of the RoPE-GA module to the computational efficiency. Furthermore, it may not fully showcase the performance benefits brought by RoPE-GA.\n\nTo more rigorously evaluate the efficiency advantage of our method's structural design, we have included a controlled ablation experiment in Supplementary Materials A3, where PIVNO-CV follows the RAFT-PIV architecture, employing traditional cost volume construction, while PIVNO represents our proposed method based on the RoPE-GA representation. The results show that the RoPE-GA method significantly reduces inference time while improving accuracy, validating the effectiveness and practical value of our \"cost volume-free\" design strategy.\n\n[1] Dual-frame Fluid Motion Estimation with Test-time Optimization and Zero-divergence Loss, in Neurips 2024.\n\n[2] Recurrent all-pairs field transforms for optical flow, in ECCV 2020.\n\n[3] Foundationstereo: Zero-shot stereo matching, in CVPR 2025."}, {"review_id": "L3F9Cc45Ff", "reviewer": "Reviewer_jXTM", "summary": "The paper presents PIVNO, a neural operator approach for Particle Image Velocimetry that directly maps pairs of particle images to velocity fields. It combines Galerkin-style attention, Conv-GRU refinement, and a self-supervised fine-tuning procedure grounded in physical constraints. Evaluated on both synthetic and experimental data, PIVNO achieves accurate and resolution-adaptive flow estimation.", "questions": "- Could you elaborate on the role and formulation of the smoothing term in the fine-tuning loss function?\n\n- Is it necessary to use graph convolution layer to achieve accurate super-resolution? Have you considered other approaches?\n\n- What are the implications for the model's performance when the inverse problem is ill-posed or lacks a unique solution?\n\n- Could the authors clarify the motivation behind selecting GRU-based refinement and RoPE Galerkin Attention for their architecture, and explain how these choices compare to potential alternatives in terms of performance or inductive bias?", "limitations": "The authors clearly explained the limitations of their approach.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\n\n- The paper evaluates the model on a wide range of benchmarks, including both synthetic and real-world datasets.\n- The model's performance is compared against numerous baselines, including several state-of-the-art methods such as PIV-RAFT and PIV-LiteFlowNet.\n- Although the model involves a complex engineering design, the ablation studies convincingly demonstrate that each component plays a necessary and justified role in achieving the overall performance.\n- The proposed self-supervised fine-tuning strategy, grounded in physical constraints like incompressibility, is a practical and effective solution for adapting the model to real-world data without requiring labeled ground truth.\n\nWeaknesses\n\n- This is a very interesting and well-executed study on Particle Image Velocimetry. However, despite the paper's strengths, its scope is limited to a specific class of PDE-constrained inverse problems and does not readily generalize to broader categories of such problems.\n\n- The paper does not include a comparison between PIVNO and well-established neural operator baselines, such as FNO [1] and CNO [2], or some generic neural operators designed for inverse problems [3],  which could be readily adapted to this class of inverse problems.\n\n- The rationale behind the authors’ choice of GRU-based flow refinement and RoPE Galerkin Attention is not clearly explained and would benefit from further elaboration in the paper.\n\n----\n\n[1] Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., \\& Anandkumar, A. (2020). Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895.\n\n[2] Raonic, B., Molinaro, R., De Ryck, T., Rohner, T., Bartolucci, F., Alaifari, R., ... \\& de Bézenac, E. (2023). Convolutional neural operators for robust and accurate learning of PDEs. Advances in Neural Information Processing Systems, 36, 77187-77200.\n\n[3] Molinaro, R., Yang, Y., Engquist, B., \\& Mishra, S. (2023). Neural inverse operators for solving PDE inverse problems. arXiv preprint arXiv:2301.11167.", "was_revised": true, "final_justification": "The authors have addressed my concerns, and I am confident that this work should be accepted, even though, to the best of my knowledge, this type of work is not typically submitted to NeurIPS.", "rebuttal": "Thank you very much for your valuable feedback and suggestions! We greatly appreciate your comments and treat them carefully. Below, we provide a brief discussion of the concerns raised in the review:\n\n1. GRU-Based Flow Field Refinement and RoPE Galerkin Attention Mechanism\n\nWe sincerely appreciate your specific attention to the RoPE-GA mechanism and the GRU-based flow field refinement module. Regarding the design motivation for the RoPE-GA mechanism, we have provided a detailed explanation of its principles in Section 3.1.2 of the main text and Supplementary Materials A1. The discussion of comparisons with established neural operator benchmarks has been addressed in the response to the next question. As for the design motivation behind the GRU, since several reviewers have shown interest in this aspect, we will provide a more detailed response here.\n\nBefore applying Conv-GRU, we obtained flow features through the RoPE-GA module (see Supplementary Materials Figure 1). However, the RoPE-GA module only maps function spaces along the channel dimension but does not aggregate or organize the spatial information. This results in flow features that are often coarse, discontinuous, and contain local estimation errors, whereas spatial continuity is critical for accurate flow estimation. Therefore, we need a mechanism that can progressively optimize flow features in the spatial domain, which is why we did not simply stack more RoPE-GA layers.\n\nIn motion estimation fields such as PIV, optical flow, and disparity estimation, the Conv-GRU module has become a mainstream structure for iterative flow refinement, with methods such as RAFT and FoundationStereo adopting similar structures. The effectiveness of this approach has been widely validated, and we did not elaborate on the structural motivation in the main text. Our ablation experiments also confirmed the key role of this module in the final performance. However, we fully recognize your rigorous concern about this issue and are happy to further explain our understanding of its mechanism.\n\nUnlike conventional convolution operations, the hidden state update in Conv-GRU is determined by multiple gated channels (such as the update gate 𝑧, reset gate 𝑟, and candidate state). Convolution essentially projects the input features into a subspace spanned by a set of convolutional kernel functions, and the optimization of these kernels allows for the best possible hidden representation. Comparatively, the convolutions in Conv-GRU operate separately on the current input features and on the previous hidden state, enabling differential nonlinear mappings across multiple subspaces during the update process. This structure allows the model to understand local motion trends from multiple perspectives and scales, and to iteratively correct the flow states of local regions at each iteration. Additionally, the nonlinear activation functions (e.g., sigmoid, tanh) introduced by the gating mechanism enhance the model's expressiveness, enabling it to adaptively regulate the retention, update, and fusion of information based on the current estimation state, thus overcoming the limitations of single-layer convolutions in modeling capacity.\n\nThroughout the iterative process, Conv-GRU acts as a differentiable spatial transformation module with memory. In each iteration, it fuses the current input features with the previous estimation result (i.e., the hidden state) and continuously adjusts the direction and magnitude of the flow field estimate through locally sensitive convolutional mappings and gating mechanisms. This structure endows the model with the ability to progressively approximate the true flow field within local spatial ranges. Therefore, Conv-GRU in PIVNO is no longer a recursive structure used for handling time sequences but rather an iterative optimization unit with local geometric modeling capabilities that supports differentiated state updates. Its synergistic effect of spatially varying linear combinations and nonlinear gating mechanisms enables the correction and refinement of the local flow field.\n\n2. Inverse Problems and Neural Operators\n\nWe appreciate the reviewers' questions regarding the applicability of inverse problem categories and comparisons with other neural operator methods. We provide a unified response here and further clarify the fundamental differences between the problem type addressed in this work and existing neural operator approaches.\n\nThe problem we investigate belongs to a special class of PDE-constrained inverse problems: recovering the underlying fluid motion field from particle image pairs. This problem is not only constrained by physical laws but also influenced by the imaging mechanism, making it essentially a visual inverse problem that combines computer vision and physical modeling, rather than a traditional PDE numerical solving task based on function samples (such as field value pairs). Therefore, there are significant differences when compared to methods like FNO [1], CNO [2], or neural operator frameworks designed for general inverse problems [3].\n\nSpecifically, these neural operators typically map between function spaces based on structured grids or a small number of scattered sample points, assuming well-defined problems with finite bandwidth, often suppressing high-frequency information during modeling (such as directly filtering out high-frequency modes in FNO). In contrast, the input for PIV consists of actual image pairs, which involve complex issues such as repeated patterns, occlusions, and uneven particle distributions. These issues manifest in the image space as significant local ambiguities and non-uniqueness, making the problem highly ill-posed. As a result, the solution is not unique. Therefore, PIV requires not only modeling the global mapping structure but also recovering fine-grained local high-frequency motion features, which is difficult to achieve directly using traditional neural operator paradigms.\n\nFor this reason, in PIVNO, we combine an image encoder, a RoPE-GA-based neural operator representation, and a Conv-GRU-based spatial refinement module. Notably, the introduction of Conv-GRU, whose mechanism has been widely validated in computer vision inverse problems such as optical flow and disparity estimation, allows for local state memory and nonlinear optimization within the image space through gating mechanisms, a feature not applicable to traditional PDE numerical problems.\n\nThus, our decision not to directly compare with the above methods is not for dodge but stems from the fundamental differences in problem paradigms, input modalities, and modeling objectives.\n\nAdditionally, regarding the question, \"What are the implications for the model's performance when the inverse problem is ill-posed or lacks a unique solution?\"  we believe that PIV itself is a typical ill-posed inverse problem. Our experimental results under real-world conditions, including significant occlusions, sparse textures, and high shear deformations, already demonstrate the robustness and generalization ability of our framework in dealing with such ill-posed problems.\n\n3. Graph Convolution\n\nThe introduction of the graph convolution layer primarily stems from the cross-resolution training strategy used in the SR Module. This strategy randomly samples the same number of pixels for training from image pairs at different resolutions, enhancing the model's generalization ability under scale variations (see Supplementary Materials B3 for details). However, this sampling method disrupts the regular, evenly spaced arrangement of pixels, making it difficult for traditional convolutions (such as standard 2D convolutions) to directly operate on these non-structured sampling points. Since capturing local features is critical for continuous spatial mapping, graph convolution naturally handles non-uniformly sampled inputs and aggregates local information through adjacency structures, enabling effective continuous mapping and feature fusion in space. Therefore, we believe that graph convolution is a reasonable choice for the task scenario at hand.\n\nTo verify its necessity, we replaced the graph convolution with a 1×1 convolution in an ablation experiment (the \"GCN×\" setting in Table 4). The results showed a significant performance drop, further demonstrating the unique advantages of graph convolution in handling irregular sampling points and capturing local geometric features.\n\n4.Smoothness Term\n\nFor the fluid motion field $ u(\\mathbf{x}) \\in \\mathbb{R}^{C \\times H \\times W} $, the smoothness term in the loss function is defined as follows:\n\n$L_{\\text{smooth}} = \\sum_{c=1}^{C} \\sum_{i=2}^{H-1} \\sum_{j=2}^{W-1} \\frac{1}{(H-2)(W-2)} \\left( \\nabla^2 u_{c,i,j} \\right)^2$\n\nwhere $ \\nabla^2 u_{c,i,j} $ denotes the Laplacian of $ u $ at position $ (i,j) $. Specifically, the Laplacian is defined as:\n\n$\\nabla^2 u_{c,i,j} = u_{c,i+1,j} + u_{c,i-1,j} + u_{c,i,j+1} + u_{c,i,j-1} - 4u_{c,i,j}$\n\nThe smoothing term is designed based on the spatial continuity assumption and regularization theory. The spatial continuity assumption posits that in the physical world, unless there are object boundaries or occlusions, the fluid velocity field is typically continuous and smoothly varying in space, with abrupt changes being rare. The smoothing term aims to explicitly express and reinforce this prior knowledge, encouraging the estimated flow field to remain smooth and consistent spatially. From a regularization perspective, the particle image velocimetry problem is inherently an ill-posed inverse problem, and relying solely on data terms (such as brightness consistency) often struggles to produce a stable and reasonable solution. Introducing the smoothing term as a regularization term effectively constrains the spatial fluctuations of the solution, prevents overfitting to noise, and improves the stability and robustness of the model's estimation."}]

Author rebuttal:

Meta-review: This paper proposes PIVNO, a neural operator approach for Particle Image Velocimetry that directly maps paired particle images to velocity fields. The framework combines a position-informed Galerkin-style attention operator (RoPE-GA) for global flow capture, a Conv-GRU module for iterative refinement, and a self-supervised fine-tuning scheme grounded in physical divergence constraints. PIVNO supports resolution-adaptive inference and incorporates a super-resolution module. Experiments on three synthetic and three real-world datasets demonstrate strong performance compared to multiple baselines.

Reviewers highlighted the paper’s novelty, well-motivated architecture, and comprehensive evaluation. PIVNO effectively bypasses traditional cost-volume approaches, adapts to real-world conditions without labeled data, and shows clear improvements in both accuracy and versatility. Weaknesses include some issues in the experimental section, limited 2D scope, incomplete ablation of certain loss terms, missing inference speed comparisons, and the need for further justification of some architectural choices, such as Conv-GRU refinement.

During rebuttal, the authors clarified the network design, committed to reorganizing experiments for clarity, and agreed to include additional metrics and comparisons in the final version. Based on the novelty, strong empirical results, and satisfactory rebuttal, I recommend ACCEPT (Consensus Reached). The authors should implement the promised improvements in the final version to fully address reviewer concerns.
Decision: Accept (poster)