Dual-frame Fluid Motion Estimation with Test-time Optimization and Zero-divergence Loss
Authors: Yifei Zhang
Huan-ang Gao
Zhou Jiang
Hao Zhao
Venue: NeurIPS 2024 poster (2024)
Keywords: Dual-frame Fluid Motion Estimation
Test-time Optimization
Self-supervision
Data Efficiency
Cross-domain Robustness
Primary area: machine_learning_for_physical_sciences
TL;DR: A new dual-frame fluid motion estimation method that is completely self-supervised and notably outperforms its fully-supervised counterparts while requiring only 1% of the training samples (without labels) used by previous methods.

Abstract: 3D particle tracking velocimetry (PTV) is a key technique for analyzing turbulent flow, one of the most challenging computational problems of our century. At the core of 3D PTV is the dual-frame fluid motion estimation algorithm, which tracks particles across two consecutive frames. Recently, deep learning-based methods have achieved impressive accuracy in dual-frame fluid motion estimation; however, they heavily depend on large volumes of labeled data. In this paper, we introduce a new method that is **completely self-supervised and notably outperforms its fully-supervised counterparts while requiring only 1\% of the training samples (without labels) used by previous methods.** Our method features a novel zero-divergence loss that is specific to the domain of turbulent flow. Inspired by the success of splat operation in high-dimensional filtering and random fields, we propose a splat-based implementation for this loss which is both efficient and effective. The self-supervised nature of our method naturally supports test-time optimization, leading to the development of a tailored Dynamic Velocimetry Enhancer (DVE) module. We demonstrate that strong cross-domain robustness is achieved through test-time optimization on unseen leave-one-out synthetic domains and real physical/biological domains. Code, data and models are available at [https://github.com/Forrest-110/FluidMotionNet](https://github.com/Forrest-110/FluidMotionNet).

Dual-frame Fluid Motion Estimation with Test-time Optimization and Zero-divergence Loss
Yifei Zhang
Huan-ang Gao AIR, Tsinghua University
University of Chinese Academy of Sciences zhangyifei2la@mai11 ucas.ac ac.
gha240mails tsinghua. edu. cn
Zhou Jiang
Hao Zhao
Beijing Institute of Technology jzianbbit. edu. en
AIR, Tsinghua University
Beijing Academy of Artificial Intelligence zhachao@air. taaahaaaair..ssinhuuu..a. edu. en
Abstract
3D particle tracking velocimetry (PTV) is key technique for analyzing turbulent flow, one of the most challenging computational problems of our century. At the core of 3D PTV is the dual-frame fluid motion estimation algorithm, which tracks particles across two consecutive frames. Recently, deep learning-ban bassed methods have achieved impressive accuracy in dual-frame fluid motion estimation; however, they exploit supervised scheme that heavily depends on large volumes of labeled data. In this paper, introduce new method that is completely self-supervised and notably outperforms its supervised counterparts while requiring only of the training samples (without labels) used by previous methods. Our method features a novel zero-divergence loss that is specific to the domain of turbulent flow. Inspired by the success of splat operation in nigh-dimensional filtering and random fields, we propose a splat-based implementation for this loss which is both efficient and effective. The self-supervised nature of our method naturally supports test-time optimization, leading to the development of a tailored Dynamic Velocimetry Enhancer (DVE) module. We demonstrate that strong cross-domain robustness is achieved through test-time optimization on unseen leave-one-our synthetic domains and real physical/biological domains. Code, data and models are available at https://github.com//orreee hatp://uoooooootoooooooooooaaatt 10/luidMotion
Introduction
Measuring and understanding turbulent fluid flow a crucial problem as is ubiquitous in various
aspects of our lives, both in nature [27] 37 20 and within engineered society 28 81 35
26 69
68 93]. Throughout history. flow visualization techniques have played a vital role in quantifying and analyzing turbulent flow [4] 660 [6] 76). Among existing flow visualization techniques, 3D
particle tracking velocimetry (3D PTV),, which tracks individual particles between consecutive frames, distinguishes itself with high spatial resolution and precise measurement of velocity vectors [32 3
Before the advent of deep learning, PTV algorithms primarily focus on designing hand-crafted
features for particle matching 62 12 99 With the onset of deep learning, deep neural networks
(like DeepPTV (50) and GotFlow 3D (53)) have been introduced to solve this task. The core algorithm of 3D PTV dual-frame fluid motion estimation, illustrated in Fig.
However, it is important to note that existing state-of-the-ard (SOTA) dual-frame fluid motion estima- tion algorithms (shown in left two panels of Fig. I have limitation: they require large amount of fully annotated data. known that deep learning generally requires a substantial amount of
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

---

TRAIN
Segerviadd
TRAIN
Sugervised
TRAIN
Self Sepe-Spee
Berative
X,
Correspondenee Learning
Flary
X
Correspondence Learning
X
Flow
Pakestan
Traits Flney
Residual Learning
Flow
TEST
fsst-tmme Optlmicattio
TEST
fest-timm Uptiriizaaion
1
TEST
Isnd lime Opttiitzatioon
Iieralive Upelate
Correspondence Learning
X
Corresgundence
1 110m
Xusi
X
Learning
Erimalio
Residwal P Learning
Flow
DVE
Flow
DeepPTV
GotFlown1
Ours
Figure Paradigm Shift: Given two frames of flow particles X, and X4+46, DeepPTV [50] adopts two-stage network for large- and small-scale motion refinement. GotFlow3D (53) trains correspondence learning network and an RNN-based residual prediction network. They are trained
in fully supervised manner with annotated data and do not support test-time optimization. Our
purely self-supervised method diverges from these approaches and employs DVE (see Sec. 3.3 for on-the-fly test-time optimization. The "Snowflake" denotes frozen weights.
in-domain data for optimal results. This demand poses a significant challenge to the Al4Science field, especially in 3D PTV where collecting suitable data is complicated due the need for precisely se- lected tracer particles, tailored illumination, and camera settings [63]. Additionally, certain scenarios like flow fields under unique geometric conditions or cytoplasmic flows disease contexts are rare, making it nearly impossible to compile a comprehensive dataset.
To alleviate the aforementioned challenge, in this paper, we introduce novel purely self-supervised framework with test-time optimization designed specifically for dual-frame fluid motion estimation in the 3D PTV process, as highlighted in the right panel of Fig.[] Concerning the intrinsic difficulties
associated with PTV data collection, especially in specialized contexts [6] 74, 644, we consider working under limited size of dataset, little as 1 1% typically used by existing fully-supervised methods (notably without accessing labels). Fluid particles have special physical properties, for which resort to the inherent divereieeeeece principle of incompressible fluid velocity fields and design a novel aero-divergence eeff-suprriiee loss tailored for fluid. Per implementation, we introduce the successful idea of splat in high-dimensional filtering (2) and random fields 38 and design a splat-based zero-divergence loss that is both efficient and effective.
Moreover, since our method is self-supervised, it naturally supports test-time optimization. Thus we introduce a module termed Dynamic Velocimetry Enhancer (DVE), shown in the right panel of Fig. M which optimizes the initial predicted flow during test-time based on the specific input data on the fly, ensuring an improved level of accuracy across various testing scenarios. This is critical for cross-domain robustness. The difficulty in collecting diverse PTV data leads to the common practice of using synthetic datasets. However, since synthetic data is generated based on hand-crafted priors, it cannot accurately represent specific real-world distributions, resulting in models that lack the necessary cross-domain robustness for practical applications.
Through comprehensive experiments, we demonstrate that our purely self-supervised framework (right panel of Fig. [ significantly outperforms its fully-supervised counterparts (left two panels of Fig. even under cata-constrained conditions (using as low as 1% data). Additionally, cross-domain robustness analyses confirm the framework's intrinsic ability to generalize to unseen domains, including leave-one-out synthetic domains and real-world physical/biological domains, underscoring the practical utility of our approach for real-world 3D PTV applications.
To summarize, our main contributions are: 1. A novel self-supervised framework with test-time optimization for dual-frame fluid motion estimation, surpassing fully-supervised methods with minimal samples (as low as 1 1%). 2. A splat-based zero-divergence self-supervised loss for fluid dynamics, which is both efficient and effective. 3. A test-time optimization module named Dynamic Velocimetry Enhancer (DVE) that significantly improves cross-domain robustness.

---

2 Related Work
2.1 Test-time Optimization and Test-time Domain Adaptation
Test-time optimization, also known as test-time refinement (TTR), exploits the inherent structure of
data in self- supervised manner without requiring ground truth labels [14
80
24 21] Applications
of TTR include point cloud registration [25], depth estimation 7 9 object recognition [80] 89],
human motion capture [86), and segmentation with user feedback 73 77 31. Test-time domain
adaptation (TTA), a specific form of TTR, adapts a model trained on a source domain to a new target domain using unsupervised loss function based on the target distribution [90; [54] 104]. One
significant challenge in TTR is achieving per-sample adaptation at test time without compromising inference efficiency. Recent studies [85] 67 have explored using generative models enable efficient
test-time adaptation. In this work, we introduce our DVE module (Sec. 3.3). which conducts test-time optimization but maintains efficiency when compared with prior methods.
2.2
Learning-based Scene Flow Estimation
We include this section because our research is closely related to point-based, learning-driven scene flow estimation from point clouds a key component in understanding scenes through point clouds
43 22]. Both areas of study concentrate on learning flows or correspondences from two frames of data 977 102, 103
[ 84
T00: 55). Advances in scene flow estimation have been driven by (59] and DlyiiggThings3D (58). Drawing from the related
benchmarks such as KITTI Scene Flow
field of optical flow [5] 30 82 recent developments in scene flow estimation utilize methods
including encoder-decoder architectures [23] 57), multi-scale representations
94), recurrent
modules 36 83 92], and other strategies 44 70)
Self-supervision and Test-time Optimization for Scene Flow. Self-supervised learning has received attention for scene flow estimation from point cloud data 95 61
46
75 45 and monocular
images [29] I0: [OS]. PointPweNet (95) introduces cycle consistency loss, inspiring Mittal et al. [6]] to incorporate it with nearest neighbor loss for establishing point cloud correspondence. This method also employs Chamfer Distance [19]. smoothness constraints, and Laplacian regularization for self- supervision. SLIM (5] addresses self-supervised scene flow estimation and motion segmentation simultaneously. Flowstep3d (36] uses a soft point matching module for pairwise point correspondence. Self-supervision naturally supports test-time optimization. Pontes et al. [66] eschew model training for real-time optimization by minimizing the graph Laplacian over source points to enforce rigid flow. Li et al. 48 replace the explicit graph with a neural prior using coordinate-baser MLP to implicitly regularize the flow field. SCOOP 40 combines pre-training
subset of data to learn soft correspondences and secures initial flows with optimization based refinement steps.
Our work is distinct from these scene flow methods as our data source is a specific domain: flow particles. Fluid particles differ from typical scene flow point clouds due to their disordered local distribution [5I) (see Sec. (.2.1] and unique physical properties. We propose graph-based feature extractor and divereiveeee regularization to leverage these properties (see See. B.2p.
2.3 Particle Tracking Velocimetry (PTV)
Particle tracking is fundamental tool in turbulence analysis, progressing from traditional methods like streak photography (18 to advanced techniques such as Laser Speckle Photography (LSV) 665] This evolution establishes the foundation for Particle Tracking Velocimetry (PTV). PTV gains prominence with the development of automatic tracking algorithms (1, which represents a significant advancement manual methods [13]. Modern PTV calculates velocities by matching particle pairs between frames (1 and has been applied widely across various fields such as materials science,
hydrodynamics, biomedical research, and environmental science (28 35 101 27
Deep Learning Methods of Dual-frame Fluid Motion Estimation in PTV. Before the advent of deep learning, PTV algorithms primarily focused on improving particle matching by considering group particle movement
(62 using multiple time step data 12]. or conducting spatial area segmen-
tation [99]. With the onset of deep learning, deep neural networks have been designed for particle
motion estimation from point cloud pairs (57] 50 53 70 96, 92). Among them, DeepPTV (50) and
GotFlow3D [53] are specifically tailored for fluid flow learning and in fully supervised manner, as
3

---

Self-supervised Learning
Graph based Feature Extractor
Divergunce
Splatting
Gmph Construction
Shared
Weights
Optimal Transport
Smnoolh
Graph- based Extractor
Recosstruction
Inital Flow Estimate
Input Particles XA, Xt+At
Frozen
Test Phase
Correspondence Learning
Initial Fuow Pstimate
Reconstruction
Trainable
Residual
Final Flow
Gradlnn
Flow
Dynamic Velocimetry Enhancer
Figure 2: Upper: Training Phase. First, we (a) use input point clouds to construct graphs, which are then passed through trainable (b) feature extractor, and we solve (c) optimal transport problem using self-supervised loss terms including (g) reconstruction loss, (f) smooth loss, and (e) zero- divergence loss for initial flow estimation. Lower: (h) Test- eime DVE. With the initial flow estimate Finit, we optimize a residual R to generate the final flow F using another reconstruction loss (g*).
demonstrated in Fig. Our work follows these prior efforts, aiming to develop data-efficient and cross-domain robust motion estimation technique through self-supervision and test-time optimization.
Methods
3.1 Problem Formulation
To elucidate the architecture and functionality of our proposed method for dual-frame fluid motion estimation, outline the problem follows: The method processes two consecutive, unstructured sets 3D particles, X, E R x3 and Xt+ E g3*x3 recorded at times and t + At. It outputs the predicted flow motion F E R mapping each particle x; from X, vector f, that indicates its movement between the two frames, capturing the flow dynamics the turbulent 3D environment.
3.2 Training with Fewer Samples
In the training phase. aim to learn the patterns of fluid flow using considerably fewer samples, as low as 1% of what conventional approaches require, given the inherent difficulties in gathering data for specific scientific domains. We design the network as depicted at the top of Fig. 2 to train a based feature extractor (Fig. (2) that extracts per-point features for the following soft point matching. These features initialize the flow between the point clouds using the optimal transport module (Fig. (2),, and we employ self-supervision losses, as shown Fig. ee,f.g). for training. However, fluid particles exhibit complex motion features compared to typical LiDAR point clouds (Sec. 3.2.T). which complicates feature learning under self-supervision with limited data. Consequently, we employ strong graph-based feature extractor (Sec. (.2.2) and propose novel zero-divergence loss (Sec. 3.2.4.3.1 tailored to address these challenges.
3.2.1 Complexity of Fluid Flow. A common assumption in LiDAR scene flow estimation is the smoothness of flow. However, this is not enough for fluid particles due to their unique geometric distribution, shown in the left of Fig. The fluid velocity field is smooth only at a coarse scale but remains complex at a fine local scale. Therefore, we need strong relation- based graph feature extractor and more specific regularization to capture the intricate properties of fluid particles.
3.2.2 Graph-based Feature Extractor. Point cloud-based extractors, including PointNet 77 and PointNet++ 72 are commonly used in LiDAR scene flow estimation 44) While these extractors effectively discern broader spatial structures, their capability to grasp intricate local relationships, which is vital for analyzing fluid dynamics, can be inadequate. In contrast, graph-based feature

---

Splat-based Zero-divergence Loss
Inttal Eutmare
Dr Spars: Ploee
Sglat INGRT
Usrpec Grid Flor
Figure 3: LEFT: A visualization of fluid flow in Fluidflow data. RIGHT: The divergence loss in our training phase is obtained by splatting the original sparse flow to grid points and then minimizing the divergence loss on the resulting grid points.
extractors excel at capturing local patterns by considering the relationships between proximate nodes, in our context, particles. Hence, drawing inspiration from GotFlow3D (53), opt for a graph-based feature extraction backhone, as depicted in Fig. L Initially, we construct a static nearest-neighhor graph from the input point cloud. This graph then processed through several GeoSetConv layers [72] to form a high-dimensionnl geometric local feature. To further enrich the feature, we construct a dynamic graph using EdgeConv [9]] based the high-dimensional feature, forming GNN that outputs static-dynami features. The dynamic graph expands the receptive field and focuses on geometric feature properties. Further details can be found in Appendix A...1
3.2.3 Solving Optimal Transport for Soft Correspondence. With the static-dynamic feature from the feature extractor (Sec. 3.2.2). formulate the correspondence linking problem through the framework of optimal transport (87) where a higher transport cost between two points indicates lower similarity within the extracted feature space. The optimal transport plan yields the soft correspondence weight between X, and Xt+ as shown in Fig. 2 which can be used to formulate an initial flow estimate init. This follows the common scene flow method, thus we leave the details to Appendix A.2
3.2.4 Self-supervised Losses. Since manually linking particles between sets is notably intricate, advocate for the adoption of self-supervised losses (Fig-2p-g)
3.2.4.1. Reconstruction Loss. A core principle guiding self-supervised flow learning is the fact that X, F and X should be similar. The Chamfer distance (CD) is standard metric used to measure the shape dissimilarity between point clouds in point cloud completion. Therefore, we adopt as our reconstruction loss. We also add a regularization term [40 to prevent degeneration:
Lapcon
1
P. min Y!E!"
Σ
- y,,1 - + Acont
(1-p.)
D
K,EY
Y/E
Here, Y" represents the estimated point cloud formed by X, Finit and y is the target point cloud formed by Xt+ Pt denotes the confidence of matching in the Optimal Transport (Sec. 3.2.3).
which is the weighted sum of transport costs. Aconf term is used to avoid the trivial solution p; = 0.
3.2.4.2. Smooth Loss. Given the infinitely differentiable characteristic of the velocity field, it is postulated that the field should exhibit a certain level of continuous and smooth transitions (at coarse scale). In light of this theoretical underpinning, we introduce smooth regularization loss to enforce and maintain this continuous behavior in the velocity field, which is defined as,
lff - fulli
Lymooth
Σ
Σ
(2)
X;EX (EN
Here, A' represents the point cloud formed by X, X. Nx();) represents the index set of the / closest points x,. f, and fç denote the estimated flow vectors at points x; and x respectively.
3.2.4.3.1 Zero-divergence Loss. Smooth Loss not enough for fluid particles, as mentioned in Sec. 3.2. Concerning the intrinsic properties of the velocity field, we note that incompressible fluids exhibit zero divergence by definition. Moreover, compressible fluids can also be approximated as incompressible under conditions like low Mach numbers, justifying this in many engineering contexts. Hence, we introduce a zero-divergence regularization, which also compensates for the shortcomings of Smooth Loss, we will show later.
3.2.4.3.2 Splat-based Implementation, Splatting, first used in high-dimensional Gaussian filtering
2]. embeds input values in a high-dimensional space. Studies like [38] and 78] followed this
5

---

Splat-Blur Blur-Slice pipeline. Inspired by these, we implemented a splat- pasat-based zero-divergence loss: to compute divergence, we need the partial derivative the field. The irregular arrangement of particles in 3D complicates this. Thus, we propose "splatting" unstructured flow estimates onto uniform 3D grid, then applying zero-divergence regularization these grid points, as shown in the right of Fig. 3 In formal terms, the dense grid is denoted by (sj, sk, sl) with j, k, 1 E Z indicating the 3D indices the grid point. The parameter s corresponds to the grid's spacing. Then, given grid point (sj, sk, s()", we employ the inverse squared distance as interpolation weights to approximate the flow at that particular point,
f(x)
1
Σ
f,
(3)
[x)x
x +E
(xEN(x)
where f; is the estimated flow value point x;- N (xx) denotes the neighborhood among the point set of X, for grid point x. The parameter is introduced to maintain numerical stability. By employing splatting, we convert the variable particle distance into fixed grid spacing, thus achieving efficiency and effectiveness.
3.2.4.3.3 Divergence Calculation. Once Splatting has been employed, the divergence that point,
specified by (sj, sk;, sl)", can be defined as: (V (x) Σ f(x+tau) where ug is a unit vector with at the k-th entry. Finally, the zero-divergence regularization can be formulated as, Lav 1
1-1 ((VFF)((), [[[(s,ss,, sk, st)7)|l ||1 where J, K, and L represent the number of grid points along the respective dimensions.
JKL. ΣØ= E E/=
3.2.4.3.4 Zero Sero-Divergence Loss v.s. Smooth Loss Zero-Divergenn loss is similar to Smooth loss in that it computes spatial gradients and requires the norm of the gradient to be small, essentially penal- izing the case that neighboring flow vectors are totally irrelevant. However, Smooth regularization is too strict for fluid particles. While the divergence constraint only requires the total divergence to be zero, it does not necessitate that any two vectors be oriented in the same direction, thus allowing for locally complex particle dynamics. In practice, we set the neighborhood set size for calculating Smooth Loss be much larger than that for calculating Zero-Divergence Loss, because smoothness is a more coarse-scale regularization. Finally, we note that Zero-Divergence Loss is calculated along three specific axes, whereas Smooth Loss is not. Therefore, using the same method (KNN) for calculating Kero-Divergenc' Loss as for Smooth Loss is not efficient.
To summarize, our final self-supervised training loss is
Lirain Liecon + Asmooth Lsmooth + -ddii
3.3 Efficient Test-time Optimization with Dynamic Velocimetry Enhancer
As shown at the bottom of Fig. 2 with the initial flow estimate from the trained network, we introduce novel Dynamic Velocimetry Enhancer (DVE) module during the test phase for test-time optimization. This provides added flexibility to accommodate unseen situations and address potential inaccuracies arising from the limited training data context, which will be demonstrated
Sec. 4.4 In principle,
our approach seeks a residual flow vector R. such that F = F Finit + R. which can be optimized to rectify the inaccuracies. Formally, DVE is essentially an optimization process using the Lrecou objective function, with the formulation as follows:
R
argmin
1 P>
Σ Y'E)"
P. min y,Ey
lyy R; ys112
(4)
ixa
This test-time supervision (Fig. g is similar to Lyecor without the regularization Aconf Solved using Adam optimizer, it only involves parameters from n, x 3 matrix. Concerning that existing test-time optimization modules 47 are slow, DVE is very efficient, as demonstrated later in Sec. 4.1
Selection of Self-supervised Losses During the Test Phase We omit both Lumooth and Lav during the test stage. During the training phase, our objective is to embed prior knowledge about particle flow into the network. Lamoath and Laiv serve not only to foster a comprehensive understanding of fluid behaviors but also function as regularizers, mitigating overfitting caused by the unconstrained reconstruction loss. However, in the test phase, our focus shifts to specific sparse particle sets.

---

In certain scenarios, such as when flows adhere to boundary conditions, these particles may not strictly adhere to the expected norms of ideal smoothness or vero-divergence typical flow field. Additionally, given that the initial flow estimate should be sufficiently accurate, regularizers become unnecessary. Our approach to customizing the loss functions in this manner aims enhance the robustness of our model against the complex challenges encountered in real-world applications, thereby improving data efficiency and cross-domain robustness.
4 Experiments
We conduct comprehensive evaluations using different data domains on our proposed framework. First, we compare our method with SOTA fully supervised methods (Sec. 4.1 Next, we examine its performance under the constrained size of training data, reflecting real-world situations where domain-specifte data is limited (Sec.
We then assess the framework's ' performance under different domains with increasing domain shift, highlighting its cross-domain robustness (Sec. (.3). Additionally, we conduct comprchensive ablation studies on the components of our framework (Sec. 4.4) validate their effects. Following the previous SOTA method GotFlow3D [52], our datasets include FluidFlow33D [52) and its six fluid cases, DeformationFlov [98] and AVIC 41 Due to page limit, experimental settings including implementation details, datasets, and evaluation metrics can be found in Appendix A.3)
4.1 Comparison with state-of-the-art methods
Since DeepPTV [50] is not open-suurced, we enrich the comparison by including scene-flow methods [56 70; 96 92 as our baselines. We benchmark our method against established fully supervised
models, such FlowNet3D [57. FLOT 770]. PointPWC-Net [96). PV-RAFT [92], and GotFlow3D [53. all utilizing the FluidFlow3D training set. All baseline models are evaluated using the default hyperparameters. As shown in Figure 4 our purely sulf-super approach outperforms all the fully supervised baselines. Additionally, we introduce Ours (1 Juur method trained on just 1 1% of the data-whieh still demonstrates comparable performance.
Comparison Across Flow Cases: We further analyze our framework's ' performance across six distinct flow cases from the FluidFlow3D dataset, with details available in Appendix [A.3. For three representative cases- Uniform Flow, Turbulent Channel Flow, and Forced MHD Turbulence-aa we present detailed results in Fig. 4 while the remaining are documented in Appendix [A.4.] In the simple Uniform Flow case, our method shows slight improvement over the baseline. However, in more complex scenarios, such Forced MHD Turbulence, method significantly outperforms the baseline, reducing the EPE/NEPE metric by nearly half compared to the SOTA GotFlow3D.
Test-time Efficiency: In Figure 4 the Tiest column in the table illustrates the time consumption of each method during the test phase. Our method demonstrates time efficiency, incurring less inference time cost than even bascline supervised methods (without test-time optimization). Our method requires only a few epochs to converge (See Appendix A.p53.. Furthermore, our network's 's relatively small size (refer to Puaain comparison in Figure 4 facilitates rapid forward pass. Time profiling is conducted on a single RTX 3090 Ti.
4.2 Training with Limited Data
Handling rare scenarios, such as unique geometric flow fields [88 or cytoplasmic flows in disease contexts, presents challenges in assembling large datasets. To address this, we explore an evaluation setup with limited training data (still without labels) by randomly sampling from the FluidFlow3D training dataset. We established three distinct training settings: 100% sampling rate (13,621 samples), 10% sampling rate (1,300 samples), and a 1% sampling rate (130 samples). Testing was conducted on the 3l test data (see Appendix I. We compared our method with FLOT (70]. PV-RAFT 192). and the current SOTA GotFlow3D [53]. We present the results of the major metric, EPE, with further details in Appendix A.442
Fig. 5 aa) illustrates the robustness of
our approach to reductions in training data size. Our metrics remain stable even with significant decreases in training samples, while other methods show substantial performance declines. This disparity becomes more pronounced in complex cases, as discussed below.
7

---

Methods
Pasts
Twu (5)
EPE
Ace Striet
Ace Relax
Outllers
Flownet3D
1,23 million
0.478
0.0623
19.34%
38.23%
61.77%
0.3452
FLOT
0.11 million
0.030
0.0587
24.99%
45.59%
54.41%
0.3324
Pett
7.72 million
0.485
0.0172
46.75%
71.61%
28.39%
0.0874
PV-RAFT
0.19 million
1.079
0.0165
72.98%
83.69%
16.31%
0.0898
GotilowDD
1.44 million
0.758
0.0049
93.15%
96.38%
3.62%
0.0244
million
0.218
0.0046
98.77%
1.31%
Ours(1%)
0.218
0.0085
97.61%
97.76%
2.40%
0.0317
Ours(156) w/o
0.58 million
0,218
0.0090
97.45%
97.60%
2.56%
0.0419
Loss
(urs(1%)
DVE
0.58 million
0.019
0.0219
89.92%
95.56%
21.04%
0.1087
Flow
Method
EPE]
o Strictf
Ace Relax]
Outtlers]
NEPEL
Hownett3
25.41%
73223%
0.075
FLOT
0.0206
69.25%
96.43%
3.35%
0.013
PointPWC:Net Not
010214
64.31%
17.32%
2.8%
0.044
GoWFovaD
0.0021
99.85%
90
30.98%
0.00%
0.006
0
91442%
0.004
Turbulent U Channel rlow
Method
Relaxj Outllers)
HowNet3
013677
28%8
70.78%
231222%
0.086
FLOT
0.0351
20.88%
72.72%
27 25%
0.083
PointPWC-Ne Not
87.51%
0E02
P-RAFT
0.0065
93.71%
97.83%
2.17%
0.015
0.26%
Ours
0.0019
99.38%
19.41%
0.62%
0.005
0oc MU rurounecce
ethod
EPE
Aoz Striett Acc Relaxt Outllers
NEPE
HooNet3D
0.09K)
FLOT
0.066
0.04%
0,28%%
98.72%
0.576
Pei
0.0171
8.48%
33,53%
66.47%
0.185
IV-RAFT
0.005
41.44%
61.37%
38 63%
07200
GouFlowED
0.0060
83.01%
91.81%
8.19%
0.053
Cases
Isometric Vlew
End View
Figure 4: (Top) Benchmarking Against Fully Supervised Methods. Purin signifies the count of trainable parameters. Tics stands for inference time for each sample. The best results marked in bold. (Bottom) Performance Across Flow Cases. The best results are marked in bold, with the runners-up underlined. The subplots on the right visualize these three cases. The warmer color indicates higher flow speed. All models are trained on full data, except Ours (1%).
Performance Drop Across Flow Cases: We further tested our method with limited training data on different flow cases from the FluidFlow3D test data mentioned above. The performance drop associated with limited data is shown in Fig. ((b). As illustrated, complex flow cases such as Forced Isotropic Turbulence are more susceptible to limited data, while simpler flow cases like Uniform Flow and Turbulent Channel Flow maintain stable EPE as the training data decreases.
4.3 Analysis of Robustness Across Different Domains
Natural fluids exhibit a range of behaviors, including convection and laminar flow. Gathering data under all possible conditions presents significant challenges. To address this, we examine the cross- domain knowledge transfer capability of our proposed method. We explore gradual increase in domain shift: initially, investigate fluid case domain shifts (Sec. 4.3.1, where train on five

---

MethodiFlow Cases
Beltrami
Channel
Isntropic
Mhd
Transition
Uniform
0.0200
Ours
CL017660
0.00250
0,01840
0L00510
0.00200
0.00200
0.0175
Plyy
Go
0402263
0.1089
0,05570
0.03422
0.01076
0.02439
0.0150
0.0125
0.0300
Tran Size
0.12964
0.0075
PV-RAPT
0.01630
0.06298
0U10087
0.0050
GOLROWGD
0,00187
0,02032
0,02773
0.0025
Outs
0.00460
0.00660
0.00650
10 Train
100
(c)
Figure 5: (a) Leave-one-out domain EPE Comparison: "Flow Cases" stands for the flow test on with the model trained on the rest five cases. (b) Comparison EPE with Limited Training Data. (c) Performance Drop related Limited Training Data. The Y-axis shows the major matric EPE, and the X-axis indicates the percentage of the training dataset utilized.
800
Time(e)
600
400
Oxy
Oxs Patt
Oxc SThol
10
(a) Dataset
(b) Initial Estimate
(c) Time Profiling
Figure 6: (a) DeformationFlow data. (b) Initial estimation by our method. (c) Time-consumption
comparison between SerialTrack and Ours+ST. "Perlt" denotes time per PTV iteration.
certain fluid cases and test on the leave-one-one domain within the same dataset. Next, we examine the Sim2Real domain shift (Sec. 4.3.2p, where training occurs synthetic fluid dataset and testing on real-world fluid data. Lastly, we assess more extensive Sim2Real domain shift (Sec. 4.3.3) by training on synthetic physical fluid data and testing biological datasets.
4.3.1 Testing within the Same Synthetic Fluid Dataset: In this section, we employ the complete FluidFlow3D training set in a six-fold cross-vale tross-validttion setup, training on five sub-cases and testing on the remaining one. We benchmark our method solely against the state-of-the-a fluid motion learning method, GotFlow3D [53], as other baselines are not fluid-specific. The EPE metric results, shown in 5ig (with additional results in Appendix
indicate that our method outperforms GotFlow3D in various scenarios, especially in complex conditions like MHD and isotropic turbulence. Moreover, our method shows consistent performance, highlighting its robustness in unfamiliar scenarios.
Sim2Real Experimental Setting: Synthetic data with ground-truth labels often serves as a benchmark for method evaluation. However, the domain gap between synthetic and real data negatively impact performance, underscoring the importance validation on real-world datasets. In this and the following section, we validate our method using two real-world datasets. DeformationFlow (98) and Aortic Valve Interstitial Cell (AVIC) [41] (details in Appendix A...1). 1. to demonstrate its practical application potential. Two challenges real-world evaluation are: 1) the lack of ground- labels in real-world data, and 2) the requirement for complete PTV method for particle tracking application. Therefore, we use the following setting: our method, trained on FluidFlow3D, is integrated into PTV algorithms (specified below) to provide initial motion estimates. Due to the absence of ground truth, we primarily demonstrate
generalizability our method across various domains through qualitative results. Quantitatively, we emphasize efficiency in the physical domain, where multiple frames are involved. By contrast, the biological domain, we focus on validating the plausibility of estimates, especially given the significant cell deformation, where efficiency is less critical.

---

Table 1:
Comparison on AVIC data. C2E, C2N, E2N stands for settings: Cyto-D treatment to Endo-1 treatment, Cyto-D treatment to Normal and Endo-1 treatment to Normal. MNDS stands for the mean neighbor distance score.
Tracked Matches
MNDS &
Method
C2E
C2N
E2N
C2E
C2N
E2N
Fm-track
8040
7744
8097
0.358
0.283
0,399
Ours+Fm
8048
7750
8097
0.357
0.246
0.359
4.3.2 Testing from Synthetic Real-World Fluid Data: We employ SerialTrack (ST) as the PTV framework and designate integrated version Ours+ST. Quantitative results highlight the advantages of our method: it identifies 22,882 matches, exceeding the 22,001 particles tracked by vanilla SerialTrack,, and significantly reduces tracking time by providing accurate initial estimates that expedite match finding. Results Fig. showcase the initial estimations and time efficiency of Ours+ST. These outcomes affirm that our method delivers sufficiently precise estimations to improve PTV, demonstrating strong simulation-to-ree to-rllltt (Sim2Real) capabilities.
4.3.3 Testing from Synthetic Physical Fluids to Biological Data: This study analyzes dataset PEG hydrogel, using microspheres to track hydrogel movements. AVICs were subjected to three conditions: regular, Cyto-D exposure, and Endo 1 treatment. Cell deformation, challenging to observe directly, was quantified by tracking nearby particles with Fm- track [1], using our framework for initialization. Results are presented in Tab and visualization of the cell deformation we estimate is in Appendix/A4.4] We evaluate particle movement using neighbor distance score (See Appendix A.3.2.. with higher scores indicating less accurate estimations.
of AVIC images embedded
Our results Tab. [ show Ours+Fm slightly outperforming Fm-track. Notably, our method, trained exclusively on the FluidFlow3D Dataset, demonstrates strong adaptability across domains and provides insights into biological fluid dynamics.
4.4 Ablation study on different modules.
In addition to the aforementioned assessments, an ablation study regarding different proposed modules including different feature extractors, Zero-divergence Loss, and DVE is performed demonstrate their effectiveness. We show our method w/ and w/o Div Loss (Zero-Divergence Loss) and DVE module in Fig. The full results are illustrated Appendix(A.. A.5
Conclusion
In this paper, we introduce test-time self-supervised framework for learning 3D fluid motion from dual-frame unstructured particle sets. We address the challenge of improving data efficiency and ensuring cross-domain robustness, which crucial for practical applications. We demonstrate the viability of our approach through two real-world studies and suggest that our findings could inform further research into extensive real-world applications, the exploration of constraints specific to particular scenarios, and the development of novel model architectures for enhanced adaptability.
10

---

References
[1] AA Adamczyk and L Rimai. 2-dimensional particle tracking velocimetry (ptv): technique and image processing algorithms. Experiments in fluids, (6)):373-880, 1988.
Andrew Adams, Jongmin Baek, and Myers Abraham Davis. Fast high-dimensional filtering
using the permutohedral lattice. In Computer graphics forum, volume 29, pages 753-762.
Wiley Online Library, 2010.
[3] RJ Adrian. Dynamic ranges of velocity and spatial resolution of particle image velocimetry. Measurement Science and Technology, 8(12):1393, 1997.
[4] Ronald Adrian. Particle-imaging techniques for experimental fluid mechanics. Annual review of fluid mechanics, 23(1):2611334 1991.
[5] Stefan Baur, David Emmerichs, Frank Moosmann, Peter Pinggera, Bjorn Ommer, and Andreas Geiger. Slim: Self-supervised lidar scene flow and motion segmentation. In International Conference on Computer Vision (ICCV), 2021.
[6] Giuseppe CA Caridi, Elena Torta, Valentina Mazzi, Claudio Chiastra, Alberto L. Audenino, Umberto Morbiducci, and Diego Gallo. Smartphone-baser particle image velocimetry for cardiovascular flows applications: A focus on coronary arteries. Frontiers in Bioengineering and Biotechnology. 10: 121001800 1806, 2022.
[7] Vincent Casser, Soeren Pirk, Reza Mahjourian, and Anelia Angelova. Depth prediction without the sensors: Leveraging structure for unsupervised learning from monocular videos. In Proceedings of the AAAI conference on artificial intelligence, volume 33, pages 8001-8008, 2019.
[8] Xiaoxue Chen, Hao Zhao, Guyue Zhou, and Ya-Qin Zhang. Pq-transformer: Jointly parsing 3d objects and layouts from point clouds. IEEE Robotics and Automation Letters, 7(2):2519-2526, 2022.
[9] Yuhua Chen, Cordelia Schmid, and Cristian Sminchisescu. Self-super learning with geometric constraints in monocular video: Connecting flow, depth, and camera. In Proceedings of the IEEE/CVF international conference computer vision, pages 7063-7072, 2019.
[10] Yuhua Chen, Cordelia Schmid, and Cristian Sminchisescu. Self-supervised learning with geometric constraints in monocular video: Connecting flow, depth, and camera. pages 7062- 7071, 10 2019. doi: 10.1 (09/CCVV2011900777
[11] Wencan Cheng and Jong Hwan Ko. poitoint:owwne: Bidirectional learning for point cloud based scene flow estimation. In Computer Vision ECCV 2022: 17th European Conference, Tel Aviy, Israel, October 23-27, 2022, Proceedings, Part XXVIII, pages 108-124. Springer, 2022.
[12] Christian Cierpka, Benjamin Lütke, and Christian J Kähler. Higher order multi-frame particle tracking velocimetry. Experiments in Fluids, 54:1-12, 2013.
[13] Paul E Dimotakis, Francois D Debussy, and Manoochehr M Koochesfahani. Particle streak velocity field measurements in a two-dimensional mixing layer. The Physics of Fluids, 24(6): 995-999 1981.
[14] Carl Doersch, Yi Yang, Mel Vecerik, Dilara Gokay, Ankush Gupta, Yusuf Aytar, Joao Carreira, and Andrew Zisserman. Tapir: Tracking any point with per-frame initialization and temporal refinement. In Proceedings of the IEEE/CVF International Conference Computer Vision, pages 10061-10072, 2023.
[15] Alexey Dosovitskiy, Philipp Fischer, Eddy 1lg. Philip Hausser, Caner Hazirbas, Vladimir Golkov, Patrick Van Der Smagt, Daniel Cremers, and Thomas Brox. Flownet: Learning optical flow with convolutional networks. In Proceedings of the IEEE international conference on computer vision, pages 2758 2766, 2015.
[16] TD Dudderar and PG Simpkins. Laser speckle photography in a fluid medium. Nature, 270 (5632):45-477 1977.

---

[17] C Ross Ethier and DA Steinman. Exact fully 3d navier-stokes solutions for benchmarking. International Journal for Numerical Methods in Fluids, 19(5):66--375, 1994.
[18] Arthur Fage and Hubert CH Townend. An examination of turbulent flow with an ultrami- croscope. Proceedings of the Royal Society of London. Series A, Containing Papers of Muthematical and Physical Character, 135(828))656-677, 1932.
[19] Haoqiang Fan, Hao Su, and Leonidas Guibas. A point set generation network for 3d object reconstruction from single image. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 605-613, 2017.
[20] Behrooz Ferdowsi, Carlos P Ortiz, Morgane Houssais, and Douglas Jerolmack. River-bed armouring granular segregation phenomenon. Nature communications, 8(1):1363, 2017.
[21] Yossi Gandelsman, Yu Sun, Xinlei Chen, and Alexei Efros. Test-time training with masked autoencoders. Advances in Neural Information Processing Systems, 35:29374-29385, 2022.
[22] Huan-ang Gao, Beiwen Tian, Pengfei Li, Xiaoxue Chen, Hao Zhao, Guyue Zhou, Yurong Chen, and Hongbin Zha. From sumi-super omni-supervised room layout estimation using point clouds. In 2023 IEEE International Conference on Robotics and Automation (ICRA), pages 2803-2810. IEEE, 2023.
[23] Xiuye Gu, Yijie Wang, Chongruo Wu, Yong Jae Lee, and Panqu Wang. Hplftownet: Hierar- chical permutohedral lattice flownet for scene flow estimation on large-scale point clouds. In Proceedings of the IEEE/CVE conference on computer vision and pattern recognition, pages 3254-3263, 2019.
[24] Moritz Hardt and Yu Sun. Test-time training on nearest neighbors for large language models. arXiv preprint 23Xi:2335..1466 2023.
[25] Ahmed Hatem, Yiming Qian, and Yang Wang. Point-tta: Test-time adaptation for point cloud registration using multitask meta-auxiliary learning. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 16594,16044, 2023.
[26] Su Min Hoi, Ean Hin Ooi, Irene Mei Leng Chew, and Ji Jinn Foo. Sptv sheds light on flow dynamics of fractal-inducee turbulence over a plate-fin array forced convection. Scientific Reports, 12(1):76, 2022.
[27] Jiarong Hong, Mostafa Toloui, Leonardo P Chamorro, Michele Guala, Kevin Howard, Sean Ri- ley, James Tucker, and Fotis Sotiropoulos. Natural snowfall reveals large-scale flow structures in the wake of 2m5-mw wind turbine. Nature communications, 5(1):4216, 2014.
[28] Pinshane Y Huang, Simon Kurasch, Jonathan S Alden, Ashivni Shekhawat, Alexander A Alemi, Paul McEuen, James P Sethna, Ute Kaiser, and David Muller. Imaging atomic rearrangements in two-dimensional silica glass: watching silica's 's dance. science, 342(6155):
224-227, 2013.
[29] Junhwa Hur and Stefan Roth. Self-supervised monocular scene flow estimation. In CVPR, 2020.
[30] Eddy Ilg, Nikolaus Mayer, Tonmoy Saikia, Margret Keuper, Alexey Dosovitskiy, and Thomas Brox. Flownet 2.0: Evolution of optical flow estimation with deep networks. In Proceedings of the IEEE conference on computer vision and patlern recognition, pages 2462-2470, 2017.
[31] Won-Dong Jang and Chang-Su Su Kim. Interactive image segmentation via backpropagating refinement scheme. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5297-5306, 2019.
[32] Christian J Kähler, Sven Scharnowski, and Christian Cierpka. On the resolution limit of digital particle image velocimetry. Experiments in fluids, 52:1 52:1629-1639, 2012.
[33] Alex Khang, Andrea Gonzalez Rodriguez, Megan E Schroeder, Jacob Sansom, Emma Lejeune, Kristi S Anseth, and Michael Sacks. Quantifying heart valve interstitial cell contractile state using highly tunable poly (ethylene glycol) hydrogels. Acta biomaterialia, 96:354-367, 2019.
12

---

[34] Ali Rahimi Khojasteh, Sylvain Laizet, Dominique Heitz, and Yin Yang. Lagrangian and eulerian dataset of the wake downstream of a smooth cylinder at a reynolds number equal to 3900. Data in brief, 40:107725, 2022.
[35] Mirae Kim, Daniel Schanz, Matteo Novara, Philipp Godbersen, Eunseop Yeom, and Andreas Schröder. Experimental study on flow and turbulence characteristics of jet impinging on cylinder using three-dimensional lagrangian particle tracking velocimetry. Scientific Reports, 13(1))10929, 2023.
[36] Yair Kittenplon, Yonina C Eldar, and Dan Raviv. Flowstep3d: Model unrolling for self- the supervised scene flow estimation. Proceedings the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 41 4114-4123, 2021.
[37] Artur Kopitca, Kourosh Latifi, and Quan Zhou. Programmable assembly of particles on chladni plate. Science advances, 7(39) 2021.
[38] Philipp Krähenbühl and Vladlen Koltun. Efficient inference in fully connected crfs with gaussian edge potentials. Advances in neural information processing systems, 24, 2011.
[39] Itai Lang, Dror Aiger, Forrester Cole, Shai Avidan, and Michael Rubinstein. Scoop: Self-supervised correspondence and optimization-basee scene flow. In Proceedings of the IEEE/CVE Conference on Computer Vision and Pattern Recognition, pages 5281-5290, 2023.
[40] Itai Lang, Dror Aiger, Forrester Cole, Shai Avidan, and Michael Rubinstein. SCOOP: Self- Supervised Correspondence and Optimization-Bann -aased Scene Flow. In Proceedings of the IEEE/CVF Conference on Computer Vision and Puttern Recognition (CVPR), 2023.
[41] Emma Lejeune, Alex Khang, Jacob Sansom, and Michael S Sacks. Fm-track: A fiducial marker tracking software for studying cell mechanics in a three-dimensional environment. SoftwareX, 11:100417 2020.
[42] Bing Li, Cheng Zheng, Silvio Giancola, and Bernard Ghanem. Setn: Sparse convolution- transformer network for scene flow estimation. In Proceedings of the AAAI Conference on Artificial Intelligence, pages 1254-1262, 2022.
[43] Pengfei Li, Ruowen Zhao, Yongliang Shi, Hao Zhao, Jirui Yuan, Guyue Zhou, and -i-QQn Zhang. Lode: Locally conditioned eikonal implicit scene completion from sparse lidar. In 2023 IEEE International Conference on Robotics and Automation (ICRA). pages 8269-8278 IEEE, 2023.
[44] Ruibo Li, Guosheng Lin, Tong He, Fayao Liu, and Chunhua Shen. Herf-flow: Scene flow from point clouds with continuous high-order erfs and position-aware flow embedding. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pages 364-373, 2021.
[45] Ruibo Li, Guosheng Lin, and Lihua Xie. Self-point-flow Self-supervised scene flow estimation from point clouds with optimal transport and random walk. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 15577-15586, June 2021.
[46] Ruibo Li, Chi Zhang, Guosheng Lin, Zhe Wang, and Chunhua Shen. Rigidflow: Self- supervised scene flow learning point clouds by local rigidity prior. In Proceedings the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 16959- 16968, June 2022.
[47] Xueqian Li. Jhony Kaesemodel Pontes, and Simon Lucey. Neural scene flow prior. Advances in Neural Information Processing Systems, 34:7838 1457838-7851, 2021.
[48] Xueqian Li, Jhony Kaesemodel Pontes, and Simon Lucey. Neural scene flow prior. Advances in Neural Information Processing Systems, 34:7838-7511, 2021.
[49] Li, Eric Perlman, Minping Wan, Yunke Yang, Charles Meneveau, Randal Burns, Shiyi Chen, Alexander Szalay, and Gregory Eyink. A public turbulence database cluster and applications to study lagrangian evolution of velocity increments in turbulecce Journal of Turbulence, (9):
131, 2008.
13

---

[50] Jiaming Liang Shengze Cai, Chao Xu, Tehuan Chen, and Jian Chu. Deepptv: particle tracking velocimetry for complex flow motion via deep neural networks. IEEE Transactions Instrumentation and Measurement, 71:1-16, 2021.
[51] Jiaming Liang, Shengze Cai, Chao Xu, Tehuan Chen, and Jian Chu. Deepptv: Particle tracking velocimetry for complex flow motion via deep neural networks. IEEE Transactions Instrumentation and Measurement, 71:1-16 2022. doi: 10. 10.1109//0.. 1120127.
[52] Jiaming Liang, Chao Xu, and Shengze Cai. Gotflow3d: recurrent graph optimal transport for learning 3d flow motion in particle tracking. arXiv preprint arXiv:2210.17011 2022.
[53] Jiaming Liang, Chao Xu, and Shengze Cai. Recurrent graph optimal transport for learning 3d flow motion in particle tracking. Nature Machine Intelligence, pages 1-13, 2023.
[54] Jian Liang, Dapeng Hu, and Jiashi Feng. Do we really need to access the source data? source hypothesis transfer for unsupervised domain adaptation. In International conference on muchine learning, pages (028-6039 PMLR, 2020.
[55] Yancong Lin and Holger Caesar. lcp-flow: Lidar scene flow estimation with icp. In Proceedings of the IEEE/CVF Conference on Computer Vision and Puttern Recognition, pages 15501- 15511, 2024.
[56] Xingyu Liu, Charles R. Qi, and Leonidas J. Guibas. Flownet3d: Learning scene flow in 3d point clouds, 2019.
[57] Xingyu Liu, Charles R Qi, and Leonidas Guíbas. Flownet3d: Learning scene flow in 3d point clouds. In Proceedings of the IEEF/CVF conference computer vision and pattern recognition, pages 529-552925 537, 2019.
[58] Nikolaus Mayer, Eddy 1lg. Philip Hausser, Philipp Fischer, Daniel Cremers, Alexey Dosovit- skiy, and Thomas Brox. A large dataset to train convolutional networks for disparity, optical flow, and scene flow estimation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4040-4048, 2016.
[59] Moritz Menze and Andreas Geiger. Object scene flow for autonomous vehicles. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3061-3070, 2015.
[60] Roland Meynart. Instantaneous velocity field measurements in unsteady gas flow by speckle velocimetry. Applied optics, 22(4):535-540, 1983.
[61] Himangi Mittal, Brian Okorn, and David Held. Just go with the flow: Self-supervised scene flow estimation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.
[62] Kazuo Ohmi and Sanjeeb Prasad Panday. Particle tracking velocimetry using the genetic algorithm. Journal of Visualization, 12:217-232, 2009.
[63] Charles Pecora. Particle tracking velocimetry: A review. 2018.
[64] Raul Perianez. particle-tracking model for simulating pollutant dispersion in the strait of gibraltar. Marine Pollution Bulletin, 49(7-8))613-6223 2004.
[65] Christopher JD Pickering and Neil A Halliwell. Laser speckle photography and particle image velocimetry: photographic film noise. Applied optics, 23(17):2961 1984.
[66] Jhony Kaesemodel Pontes, James Hays, and Simon Lucey. Scene flow from point clouds with without learning. In 2020 international conference 3D vision (3DV), pages 261-270. IEBE, 2020.
[67] Mihir Prabhudesai, Tsung-Wer Wei Ke, Alexander Cong Li, Deepak Pathak, and Katerina Fragki- adaki. Diffusion-tta:: Test-time adaptation of discriminative models via generative feedback. In Thirty-seventh Conference on Neural Information Processing Systems, 2023.
[68] Soorya Pradeep and Thomas Zangle. Quantitative phase velocimetry measures bulk intra- collular transport of cell mass during the cell cycle. Scientific Reports, 12(1):6074, 2022.
14

---

[69] Horst Punzmann, Nicolas Francois, Hua Xia, Gregory Falkovich, and Michael Shats. Gen- eration and reversal of surface flows by propagating waves. Nature Physics, 10(9))658-663, 2014.
[70] Gilles Puy, Alexandre Boulch, and Renaud Marlet. Flot: Scene flow on point clouds guided by optimal transport. European conference on computer vision, pages 527-544. Springer, 2020.
[71] Charles R Qi, Hao Su, Kaichun Mo, and Leonidas J Guibas. Pointnet: Deep learning on point sets for 3d classification and segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 652-660, 2017.
[72] Charles Ruizhongtai Qi, Li Yi, Hao Su, and Leonidas J Guibas. Pointnet++:: Deep hierarchical feature learning on point sets in a metric space. Advances in neural information processing systems, 30, 2017.
[73] Tomas Sakinis, Fausto Milletari, Holger Roth, Panagiotis Korfiatis, Petro Kostandy, Kenneth Philbrick, Zeynettin Akkus, Ziyue Xu, Daguang Xu, and Bradley Erickson. Interactive segmentation of medical images through fully convolutional neural networks. arXiv preprint arXiv:1903.0820 2019.
[74] Torsten Seelig, Hartwig Deneke, Johannes Quaas, and Matthias Tesche. Life cycle of shallow marine cumulus clouds from geostationary satellite observations. Journal of Geophysical Research:
Amoosperes, 126(2)):20221J1035577, 2021.
[75] Yaqi Shen, Le Hui, Jin Xie, and Jian Yang. self-supervised 3d scene flow estimation guided by superpoints. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 5271 5271-5280, June 2023.
[76] PG Simpkins and TD Dudderar. Laser speckle measurements of transient benard convection. Journal of Fluid Mechanics, 89(4):665-61 671, 1978.
[77] Konstantin Sofiiuk, Ilia Petrov, Olga Barinova, and Anton Konushin. f-brs: Rethinking backpropagating refinement for interactive segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 8623-8632, 2020.
[78] Hang Su, Varun Jampani, Deqing Sun, Subhransu Maji, Evangelos Kalogerakis, Ming-Hsuan Yang, and Jan Kautz. Splatnet: Sparse lattice networks for point cloud processing. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2530- 2539, 2018.
[79] Deqing Sun, Xiaodong Yang, Ming- Yu Liu, and Jan Kautz. Pwe-net: Cnns for optical flow using pyramid, warping, and cost volume. In Proceedings the vision and pattern recognition, pages 8934-8943, 2018.
the IEEE conference on computer
[80] Yu Sun, Xiaolong Wang, Zhuang Liu, John Miller, Alexei Efros, and Moritz Hardt. Test-time training with self-supervision for generalization under distribution shifts. In International conference on machine learning, pages
PMLR, 2020.
[81] Alexandra M Tayar, Fernando Caballero, Trevor Anderberg, Omar A Saleh, M Cristina Marchetti, and Zvonimir Dogic. Controlling qquid-liquid phase behaviour with an active fluid. Nature Materials, pages I 1-8, 2023.
[82] Zachary Teed and Jia Deng. Raft: Recurrent pll-s field transforms for optical flow. In Computer Esiio-EECV 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part 11 16. pages
Springer, 2020.
[83] Zachary Teed and Jia Deng. Raft-3d: Scene flow using rigid-motion embeddings. In Pro- ceedings of the IEEE/CVE Conference on Computer Vision and Pattern Recognition, pages 2021.
[84] Beiwen Tian, Liyi Luo, Hao Zhao, and Guyue Zhou. Vibus: Data-efficient 3d scene parsing with viewpoint bottleneck and uncertainty-spectre modeling. ISPRS Journal of Photogram- metry and Remote Sensing, 194:302-318, 2022.
15

---

[85] Yun- Yun Tsai. Fu-Chen Chen, Albert YC Chen, Junfeng Yang, Che-Chun Su, Min Sun, and Cheng- Hao Kuo. Gda: Generalized diffusion for robust test-time adaptation. arXiv preprint urXiv22404000PD 2024.
[86] Hsiao-YP Yu Tung, Hsiao-Wa Tung, Ersin Yumer, and Katerina Fragkiadaki. Self-supervised learning of motion capture. In Advances in Neural Information Processing Systems, pages 5236-5246, 2017.
[87] Cédric Villani et al. Optimal transport: old and new, volume 338. Springer, 2009.
[88] BX Wang, JH Du, and XF Peng. Internal natural, forced and mixed convection in fluid- saturated porous medium. Trans Phenom Porous Media, pages 357-382, 1998.
[89] Dequan Wang, Evan Shelhamer, Shaoteng Liu, Bruno Olshausen, and Trevor Darrell. Fully test-time adaptation by entropy minimization. arXiv preprint arXiv:2006. 10726, 2020.
[90] Dequan Wang, Evan Shelhamer, Shaoteng Liu, Bruno Olshausen, and Trevor Darrell. Tent: Fully test-time adaptation by entropy minimization. arXiv preprint arXiv:2006.1072) 2020.
[91] Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E Sarma, Michael M Bronstein, and Justin M Solomon. Dynamic graph enn for learning on point clouds. ACM Transactions on Graphics (tog), 38(5):1-12 12, 2019.
[92] Yi Wei, Ziyi Wang, Yongming Rao, Jiwen Lu, and Jie Zhou. Pv-raft: Point-voxel correlation fields for scene flow estimation of point clouds. In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pages 6954-6963, 2021.
[93] Freda Werdiger. Martin Donnelley. Stephen Dubsky, Rhiannon P Murrie, Richard P Carnibell,, Chaminda R Samarage, Ying Y How, Graeme R Zosky, Andreas Fouras, David W Parsons, et al. Quantification of muco-obstructive lung disease variability in mice via laboratory x-ray velocimetry. Scientific Reports, 10(1):10859 2020.
[94] Wenxuan Wu, Zhiyuan Wang, Zhuwen Li, Wei Liu, and Li Fuxin. Pointpwe-net: A coarse- to-fine network for supervised and supe-suue scene flow estimation on 3d point clouds. arXiv preprint arXiv:1911.12400 2019.
[95] Wenxuan Wu, Zhi Yuan Wang, Zhuwen Li, Wei Liu, and Li Fuxin. Pointpwe-net: Cost volume on point clouds for (self-) supervised scene flow estimation. In European Conference on Computer Vision, pages 88-107. Springer, 2020.
[96] Wenxuan Wu, Zhi Yuan Wang, Zhuwen Li, Wei Liu, and Li Fuxin. Pointpwe-net: Cost volume on point clouds for (self-) supervised scene flow estimation. Computer Rision-ECCV 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part V 16, pages 88-107. Springer, 2020.
[97] Xin Wu, Hao Zhao, Shunkai Li, Yingdian Cao, and Hongbin Zha. Sc-wls: Towards inter- pretable feed-forward camera re-localization. In European Conference Computer Vision, pages 601. Springer, 2022.
[98] Jin Yang, Yue Yin, Alexander K. Landauer, Selda Buyuktozturk, Jing Zhang, Luke Summey, Alexander McGhee, Matt K. Fu, John O. Dabiri, and Christian Franck. TraialTrack: ScalE and Rotation Invariant Augmented Lagrangian Particle Tracking. 2022. doi: 10.48550/ARXIV, 2203. htRp::
org/ abt.abaa/02 2203. 12573.
[99] Yang Zhang, Yuan Wang. Bin Yang, and Wenbo He. A particle tracking velocimetry algorithm based on the voronoi diagram. Measurement Science and Technology, 26(7):075302, 2015.
[100] Yifci Zhang, Hao Zhao, Hongyang Li, and Siheng Chen. Fastmac: Stochastic spectral sampling of correspondence graph. Proceedings of the IEEE/CVF Conference on Computer Vision and Paitern Recognition, pages 17857-17867, 2024.
[101] Zeng Zhang, Misun Hwang, Todd J Kilbaugh, Anush Sridharan, and Joseph Katz. Cerebral microcirculation mapped by echo particle tracking velocimetry quantifies the intracranial pressure and detects ischemia. Nature communications, 13(1):666, 2022.
16

---

[102] Chengliang Zhong, Peixing You, Xiaoxue Chen, Hao Zhao, Fuchun Sun, Guyue Zhou, Xi- aodong Mu, Chuang Gan, and Wenbing Huang. Snake: Shape-aware neural 3d keypoint field. Advances in Neural Information Processing Systems, 35:7052-7064, 2022.
[103] Chengliang Zhong, Yuhang Zheng, Yupeng Zheng, Hao Zhao, Li Yi, Xiaodong Mu, Ling Wang, Pengfei Li, Guyue Zhou, Chao Yang, et al. 3d implicit transporter for temporally consistent keypoint discovery. In Proceedings of the IEEE/CVF International Conference Computer Vision, pages 3869-3880, 2023.
[104] Tao Zhong, Zhixiang Chi, Li Gu, Yang Wang, Yuanhao Yu, and Jin Tang. Meta-dmoe: Adapting domain shift by meta-distillation from mixture-of-experts. Advances in Neural Information Processing Systems, 35:22243-2223 2022.
[105] Yuliang Zou, Zelun Luo, and Jia-Bin Huang. Df-net: Unsupervised joint learning of depth and flow using cross-tase task consistency. In European Conference on Computer Vision, 2018.
17

---

A Appendix
A.1 Methods in Detail
A.1.1 Feature extractor
We draw significant inspiration from GotFlow3D's feature extractor [52] to obtain a robust represen- tation of particle features. We briefly describe the structure of the feature extractor here.
The feature extractor is based on a graph neural network. Initially, a static graph Gstatic = (V,, Es) is established in the 3D spatial space for the input point cloud. V, contains the given input points, while Eg consists of connections between the k-nearest neighbors of each point. This process yields the original point cloud feature fi for point as hexadecimal set, including the three-dimensional coordinates, the radial distance, the azimuthal angle, and the polar angle. After processing through several GeoSetConv layers [72), obtain high- nigh-dimensional geometrie local feature:
F (a (8" (Si, F F F)]
where F. denotes the feature extracted from the static graph at the n-th layer for point i, and N₂ represents its k-nearest neighbors in the static graph. 4h stands for the n-th GeoSetConv layer. F,) is initialized by the 3D coordinates of the input point cloud.
Subsequently, the dynamic graph is generated. It takes these high-dimensional features F; as inputs and, following the same steps as the construction of the static graph, seeks the nearest neighbors based on these features to establish connections, thus obtaining the local structure on the feature manifold. However, the distinction lies in the fact that the dynamic graph possesses a greater receptive range since it accepts figh-dimensional features. Moreover, it is reconstructed during every training session, whereas a static graph is constructed only once.
The dynamic feature is obtained in the same manner as F,, with the sole difference being the use of EdgeConv layers [91] instead of GeoSetConv layers. The final feature obtained by concatenating all hierarchical features from different layers of both static and dynamic graphs.
A.2 Optimal Transport for Soft Correspondence
Let xx and y represent the features extracted from two input partiele sets, A' and ), respectively. We initiate by computing the point-to-point similarity matrix, given by:
xx
S.j
ys
(5)
1
x: 2
By
In this context, 4x, pertains to the feature of the i-th point in tx, and analogously for bz.
Inspired by pioneering research, we formulate the correspondence linking problem through the framework of optimal transport [39]. Assigning a mass of 지 to each source point x,, we consider its transport to the target point y, with the cost matrix defined by C., higher cost indicates a lower similarity between two points within the feature space. Our objective to identify the optimal transport plan T that satisfies.
S.j. In this context,
T arg min
ΣTa TTJy Ci. +e
Σ
(log Ti.j - -1)
TER"
(6)
+ [LT,TI,
1 1) (T 1, | 11
X
where KI. denotes the Kullback-Leibler (KL) divergence, 1 E x is a vector filled with ones, and the terms involving and are regularizing terms with coefficients controlling their respective strengths.
The optimal transport plan T yields the soft correspondence weight of point x;; point yj as:
Ti.
Wii
EnEM Ti
18

---

where My(x;) represents the set of L points from )' corresponding to the top L values of T and a that can be chosen for specific particle tracking case. Consequently, we can estimate the new location of given Ti, the estimated position is given by,
y Σ W
(8)
Y,EV
And the confidence score Pt, which quantifies the reliability of the estimated position y; for each point x; is given by,
Σ WanS
Pj = max
(9)
(x.)((s))
Finally, the flow estimate ft is determined as,
y Xi.
(10)
A.3 Experimental Setup
A.3.1 Datasets
FluidFlow3D Dataset The FluidFlow3D [53) is large synthetic dataset designed for the study of 3D fluid flow. Specifically, offers enough data for training and serves as a benchmark to evaluate the flow estimation capabilities of supervised 3D fluid flow motion learning techniques. This dataset utilizes physically accurate simulated flow structures sourced from public database [49], ensuring that the provided ground truth flows adhere computational fluid dynamics principles. It encompasses six typical categories of flow cases, namely, uniform flow, isotropic turbulent flow, magneto-hyydoddynmii (MHD) turbulence, fully developed turbulent channel flow, transitional boundary layer flow, and the Beltrami flow [17]. We utilize this synthetic dataset to evaluate our method, comparing it with baselines (see Sec. 4.1 training with restricted data capacity (see Sec. 4.2), or testing cross-domain transferability by training on and evaluating different turbulence types (see Sec. (4.3). It should be noted that, our knowledge, this dataset the only large-scale benchmark specifically designed for dual-frame fluid motion learning. Other fluid datasets, such as CylindeFFlw (34,, focus on fluid trajectories across multiple frames, requiring the integration of the full PTV process. This approach deviates from the main problem we aim to address.
DeformationFlow The DeformationFlow dataset [98] showcases real-world physical dynamics by recording the indentation created when stainless steel sphere is placed on soft polyacrylamide hydrogel due to gravitational forces. Fluorescent fluid particles within the gel were scanned both pre and post-indentation, yielding 3D volumetric images. We employ this dataset because it diverges significantly from the synthetic training set. Notably, the zero-divergence constraint is no longer applicable, making it an ideal candidate evaluate the cross-domain generalization capability of our proposed test-time self self-supervise supervised framework.
Aortic Valve Interstitial Cell (AVIC) Dataset The AVIC dataset [1 represents real-world biolog- ical dynamics revealed through PTV analysis. In the dataset, AVICs, extracted from dissected porcine heart leaflets, were suspended in peptide-modified modified PEG hydrogel with fluorescent microbeads and subsequently incubated for 72 hours. We employ this dataset demonstrate the strong cross-domain versatility of our proposed framework with the data from the biological realm.
A.3.2 Evaluation Metrics
For thorough evaluation of our framework's performance and its comparison with previous works, we utilize five prominent metrics commonly applied in particle tracking velocimetry evaluations: EPE, NEPE, Ace Strict, Acc Relax, and Outliers.
To introduce the metrics, we first define point error e; and the relative point error eell
e, = |ff
rel
e
(11)
19

---

where e, represents the Euclidean distance (L2 norm) between the predicted flow f," and the ground- truth flow f for specific point x, The relative error etdl provides normalized measure, indicating the magnitude of the point error relation the magnitude of the ground-truth flow point X;.
Then, the metrics used for evaluating the quality of predicted flows are defined as follows. The EPE (End Point Error), calculated as EPE [ Ex,ex Cs representing the average point errors, and the NEPE (Normalized End Point Error), given by NEPE शि. Σx,xx i reflecting the normalized average of point errors. Furthermore, the Acc Striet metric denotes the percentage of points with e < 0.055(m) [m] or e!el < 5%, while Acc Relax captures the points having e < 0.10(m) or e < 10%. Lastly, Outliers indicates points where e; > 0.30 [m] or et 10%, signifying notable deviations between predictions and ground truth, potential indicator of the model's challenges in generalizing across diverse data.
Furthermore, when performing experiments on real datasets, the absence of ground truth in actual data prompts us to incorporate additional metrics for evaluating our methodology. For the SerialTruck assessment, our evaluative metrics include Matches, Tracking ratio, UpdateNorm, and Time. We use mamathh to represent the matches identified between the source and target point clouds, and Thoo_missing signifies the count of source points that are matched in the target point cloud. Consequently, Matches equivalent to Amatch and Tracking ratio is expressed as the fraction Pinash The
museng"
UpdateNorm metric captures the change in PTV parameters for each iteration, while Time measures the duration taken for a single iteration. For the Fm-track assessment, our primary metric is the neighbor distance score (NDS), delineated for a point as,
NDS;
Σ
[ - fy
(12)
N
(x,)(x,)
where (xx,) is the K-nearest-neighbor set of point x,, f, denotes the flow vector at point x,,
A.3.3 Implementation Details
In method, there are some pre-defined hyperparameters. For the sake of experimental repro- ducibility, we list them below.
MODEL STRUCTURE For the feature extractor, the K-value of our K-nearest-neighhor is chosen be 32, the embedding-dit to 128, and the dropout rate to be 0.5. The grid size for splatting 10 x 10 x 10.
LOSS TERM The selected number of neighboring points for the reconstruction loss Lrecon is 32. Likewise, the number of neighboring points for smooth flow loss Lemooth is 32, and for zero- divergence loss Laiv is 2. Acout is 0.1, Asmooth 10, and Adiv is 0.1.
During the training phase, we utilize a mini-batch training process with a batch size of 4. To achieve convergence, we train the full-data model and 10%-data model for 100 epochs, and 1%-data model for 300 epochs. A default learning rate of 0.001 is set and the training is run on a single RTX 4070TI.
During the test phase, DVE runs for 150 steps with an update rate of 0.01.
A.4 Extended Results
A.4.1 Full Comparison with the state-of-the-art methods different flow cases of FluidFlow3D dataset
We present a comprehensive comparison with state-of-thh-an methods across all flow cases in Table Notably, our model, trained on only 130 samples, outperforms the current state-of-the-as GotFlow3D. and is marked in italics. In particularly complex scenarios such as Forced MHD turbulence and Forced isotropic turbulence, our method significantly outperforms Got even when trained with only 1 % of the training set samples. In more common scenarios, our model, trained with the full dataset, surpasses GotFlow3D in most metrics, while the model trained with limited samples still demonstrates commendable performance.
A.4.2 Full Results of the Training with Limited Data Experiment
We present the full results of our limited-dat training experiment here in Tab.
20

---

Table 2:
This table illustrates the performance of our method relative to baseline methods, with the size of the training set indicated in parentheses after each technique- 1300 corresponds to 10% of the full dataset, and 130 corresponds to 1 1.. The leading results are emphasized in bold, while the second- bestdd-bestto ones are underlined.
Beltrumi Flom
Turbulent Channel Flow
Method
Method
EPE
Acc Striet
Acc Relax
Outliers
NEPF
EPE
Acc Striet
Acc Relax
Outlies
NEPF
FlowNesSDeatt
0.04435
2,30%
13.73%
86.27%
0.273
0.0367
24.89%
70.78%
20 29.224
0.086
FLOTaal)
0.02417
13.11%
42.22%
51.78%
0.169
FLOTall)
0.0354
26.89%
72.72%
27.25%
0.083
Neotall))
0.01116
24.36%
62.19%
37.81%
0.109
PoinPWCC Not(all)
0.0127
87.51%
98.41%
1.59%
0.032
(V-RAFT(al))
0.00899
69.69%
91.00%
9.00%
0.047
PVRRFTT
0.0065
93.71%
83%
2.17%
0.015
GotPlow3UKal
0.00291
95.035%
98.445
1,566
0
0.00241
99.1346
99.74%
0.26%
0:006
Ours(all)
0.0039
98.95% %
98.99%
1.6%%
0,0191
Ounscally
0.0019
99.38%
99.41%
0.62%
0.0051
(3r5()300)
0.00164
98.3918
98.45%
1,22%
0.0243
Qurs (300)
0.0024
99.3156
99.359
0.00%
0.0063
Ours(130)
0.010]
97.58%
97. 97.66%
2.44%
0,0319
Ours(130)
0,0025
29.25%
99.31%
0.75%
0,0065
Forced bottrpic Turbulence
Forced MIID Turbulence
Method
Method
EPE
Aoc Striet
Acc Relax
Outliers
NEPF
EPF
Anc Striet
Acc Relax
Outliers
NEPF
FlowNesSDeat)
0,12465
0.20%
1.52%
98.48%
0.558
FlowNNeSDDaal
0.0940
0.06%
0.45%
99.55%
0.851
FLOTaal)
0.13090
0.14%
1.15%
98.85%
0.587
FLOTalll
0,0984
0.04%
0.28%
99.7216
0.876
0,02719
18.47%
$7.06%
42.94
0.0171
8.46%
33.53%
66.47%
0.165
PV-RAFT(al))
0.04190
51.37%
62.82%
37.18%
PVRAFII(I
0.0259
41.44%
61.57%
38.63%
GotPlow SUall))
0.01153
86.62%
90.74%
9.261
D:045
0.009996
83.01%
91.81%
8.19%
0.083
Ourstall)
0.0117
96.94%
97.04%
3,07%
0.0378
Ounstally
0.0037
98.89%
98.95%
1.13%
0.0366
(Lr5(3300)
0.0156
96.11%
96.20/8
3.92%
0,0515
Qurac 3300)
0.0051
98.515
98.599
1.53%
0.0412
Ours(130)
0.0193
95.01%
95.22%
5.02%
0,0615
0,0066
97, 95.
98, 12%
2,00%
0,0505
Transitiona Boundary Flow
Unilorm Flom
Method
Method
EPF
Aoc Striet
Acc Relax
Outliers
NEPF
EPF
Anc Striee
Acc Relax
Outlies
NEPF
FlowNesSDeat)
0,02003
67.23%
91.01%
8.99%
0.048
Flow/NesDD(at)
0.0375
25.41%
79.23%
20.775
0.075
FLOTall)
0.01715
70.37%
94.63%
5,37%
0,043
FLOTCall
0,0296
68.25%
96.45%
3.55%
0,043
PoinPPW--Netlll
1.01163
89.13%
1,71%
0.0214
64.31%
2.68%
PV-RAFT(al)
0.003344
99.6558
0.063%
0.005
PVRAFTIAM
0.0033
99.87%
99.99%
0.015%
0.007
DDcall)
0.00222
99.87%
99.97%
0.03%
0
GotalewSDDall
0.00244
09.85%
99. 90.98%
0.00%
0.005
Olursjall)
0.0028
99.01%%
99.18%
0.96%
0.0067
Churstally
0.0018
99.42%
99.44%
01584%
0.084
Ours(L300)
0.004
98.60%
98.81%
1.37%
0,0093
Ourst (300)
0,002
99.40%
99.42%
0.60%
0,0013
Ours(130)
0.0066
97.55%
97.87%
2,42%
0,0149
O3rs(130)
0.0019
99.41%
99.43%
0.59%
0.09H!
A.4.3 Full Results of the Cross-Domain Robustness Experiment within the Same Synthetic Fluid Dataset
We present the full results of our cross-domain robustness experiment here in Fig. Our technique consistently outperforms the state-of-the method GotFlow3D across different fluid scenarios in terms of accuracy. Furthermore, our method demonstrates a consistent performance across these scenarios, unlike the pronounced variability observed with GotFlow3D. This suggests that, after training on certain fluid cases, our model can proficiently handle scenarios it hasn't 't directly observed. This adaptability can be attributed to the DVE module, which allows for on-the-fly adjustments to unseen examples during testing.
A.4.4 Full results of the AVIC experiment
Experimental Settings: In this work. we analyzed a dataset containing images of porcine aortic valve interstitial cells (AVICs) embedded in polyethylene glycol (PEG) hydrogel. We employed 0.5 [um] microspheres as tracking particles to monitor the movements within the PEG hydrogel. For detailed information on AVIC encapsulation within PEG hydrogels, readers are referred to previous study B3 We subjected the AVICs to three distinct conditions: initially, under regular conditions; secondly, after exposure to Cytochalasin-D, a compound that disrupts actin polymerization and cellular contraction; and thirdly, following treatment with Endo 1. Direct observation of cell deformation being challenging, we tracked the motion of particles adjacent to the cells within the gel under different treatments as a proxy measure. We utilized (4)) a specialized particle tracking velocimetry (PTV) method for cell tracking, providing it with initialization value.
21

---

Table 3: Comparative performance of different methods under varying training data sizes. The table lists End-Point Error (EPE), Accuracy Strict, Accuracy Relaxed, and Outliers percentages for each method (FLOT, PV-RAFT, Gotflow3D, and Ours) at 100%, 10%, and 1% training data utilization. This data highlights the resilience of our method against reductions training size, maintaining high accuracy and low outlier rates across all sampling levels.
Method
Train Size
EPE
Ace Striet
Ace Relax
Outliers
100%
0.05870
24.99%
45.59%
54.41%
FLOT
10%
0.08050
43.65%
74.10%
73.13%
1%
0.12954
11.55%
45.13%
92.13%
100%
0.016500
72.98%
83.69%
16.31%
PV-RAFT
10%
0.06298
45.83%
82.87%
68.03%
1%
0818097
7.11%
29.82%
95.62%
100%
0.00487
93.15%
96.38%
3.62%
Gotflow3D
10%
0.02032
46.63%
66.31%
33.69%
1%
0.02773
33.66%
57.08%
42.92%
100%
0.00460
98.69%
98.77%
1.31%
Ours
10%
0.00640
98.27%
98.37%
1.74%
1%
0.00850
97.61% %
97.76%
2.40%
Ours GotFlow3D
Figure 7: Cross-domain Robustness Analysis. Our method is compared with the state-of-the-an method GotFlow3D. Both techniques are trained on five flow cases and evaluated the rest different case, which indicated on the X-axis. The Y-axis showcases the metric of evaluation.
To illustrate the estimated cellular deformation before and after treatment with Cytochalasin-D, we provide a visualization in Figure ((a) This figure successfully captures the distinctions between untreated cells and those treated with Cytochalasin-D. The left-hand side of Figure displays the gel particle flow field, where the color indicates the angle between the cell surface's normal and the
22

---

Init Cell
Pinal Call
b
125
125
z-position μm
100
z-position jm
100
75
75
y-position
50
50
25
25
100
50
50
x-position jmm
y-position jmm
x-position jum
Figure 8: Visualization of Our Results on AVIC. (a) Enhanced Fm-track using our method illustrates cellular deformation pre and post-treatment with Cytochalasin-DD D. (b) Display of flow field profiles across the x, y, and axes.
flow vector. In the central region of this flow field, which is further magnified on the right-hand side of the subplot,, a directional flow and specific vortex patterns are evident, attributes linked cell contraction. Further analysis is provided in Figure B)), which elaborates on the flow field profiles across the æ, y. and axes, shedding light on degree cellular deformation in each dimension and aiding in the interpretation of the observed biological phenomenon.
A.4.5 Subset Selection Robustness
Different data subset selection may influence our results on limited data. We control experiment randomness with seeds, conducting multiple runs to show model performance various subset data. See results Table. 4
A.5
Ablation study on different modules
A.5.1 Feature extractor
We compared two feature extractors: our graph- feature extractor and Pointnet++ 722 generic point cloud feature extractor, to assess their efficiency in extracting features from fluid particles. Both extractors have an embedding dimension of 128 and use a K-value of 32 for the K-nearest-neighhor algorithm. The results in shoo that our feature extractor consistently outperforms Pointnet++ across all metrics, indicating that Pointnet++ has limitations in fully capturing the features of fluid particles.
23

---

Table 4: Various runs with different seeds.
Seed
Train Size
EPE
Acc Strict
Acc Relax
Outliers
100%
0.00460
98.69%
98.77%
1.31%
42
10%
0.00640
98.27%
98.37%
1.74%
1%
0.00850
97.61%
97.76%
2.40%
100%
0.00470
98. 78%
98.77%
1.28%
0
10%
0.00530
98.56%
98.63%
1.45%
1%
0,00810
97.92%
98.01 %
2.09%
100%
0.00490
98.66%
98.72%
1.35%
1
10%
0.00530
98.56%
98.63%
1.45%
1%
0.00760
98.05%
98.14%
1.97%
100%
0.00430
98.79%
98.84%
1.22%
2
10%
0.00600
98.36%
98.43%
1.66%
1%
0.00780
97. 97.94%
98.03%
2.08%
100%
0.00450
98. 98.76%
98.81 %
1.25%
3
10%
0.00550
98.53%
98.59%
1.49%
1%
0.00720
98.07%
98.16%
1.95%
100%
0.00440
98.77%
98.82%
1.24%
4
10%
0.00580
98.44%
98.51
1.57%
1%
0.00790
97.95%
98.04%
2.07%
Table 5: Comparison between our feature extractor with Pointnet+++ Both models are trained on the full Fluidflow3D dataset.
Feature Extractor
EPE
Strict
Ace Relax
Outliers
Ours
0.0046
98.69%
98.77%
1.31%
Pointnet++
0.0583
57.57%
81.98%
59.65%
A.5.2 Zero-Divergence Loss
When investigating the intrinsic properties of the velocity field, is understood that incompressible fluids, by definition, exhibit zero divergence. Consequently, we have introduced regulation specifi- cally targeting zero-divergence the estimated velocity field. However, this regulation is not entirely accurate, many fluids can be compressed. Therefore, this section scrutinizes the effectiveness of the zero-divergence regulation. We tested it using the FluidFlow3D test data(See Table, 6 and its six fluid cases(See Table 7
As shown in Table 6 the zero-divergence regulation appears ineffective for models trained on full samples when tested on the FluidFlow3D-NOO0 NORM dataset. However, this regulation becomes more effective for models trained on fewer samples. Furthermore, the lower the sample count, the greater the benefit gained from zero-divergence regulation. Our interpretation is that the zero-divergence regulation provides additional prior information, which can be learned with sufficiently large sample size but may be challenging to access when the sample size is limited. Thus, our zero-divergence regulation plays compensatory role that improves data efficiency.
We conducted further tests to determine the suitability of the zero-divergence assumption for various fluid cases. Our findings, illustrated in Figure 7 revealed that our regulation performs excellently in fluid cases that exhibit zero divergence, such as Beltrami Flow, Turbulent Channel Flow, Transitional Boundary Flow, and Uniform Flow. Specifically, our model trained with 130 samples (with zero- divergence regulation) outperformed an unregulated model trained with 1300 samples across all metrics in Uniform Flow. more complex fluid cases, such as Foreed MHD Turbulence, where zero-divergence law does not hold, employing zero-divergence may capture incorrect fluid features the model is trained with insufficient data. However, with full training dataset, it still enhances performance.
24

---

Table 6: Comparison on FluidFlow3D test data. v/x means our method with/without the zero- divergence loss term. Better results are marked in bold.
Zero-Diverge Di---ieveggnee
Train Size
EPE
Acc Strict
Ace Relax
Outliers
100%
0.0046
98.69%
98.77% %
1.31 %
10%
0.0061
98.27 %
98.37%
1.74%
1 %
0.0085
97.61 %
97.76% %
2.40%
X
100%
0.0047
98.69%
98.76%
1.32%
X
10%
0.0064
98.25%
98.37%
1.76%
X
1%
0.009
97.45%
97.60%
2.56%
Table 7: Comparison on different flow cases. V/x
means our method with/without the zero-
divergence loss term. Better results are marked in bold.
Beltrani Flaw
Turbulent Channel Flrw
Train
Zeru-Bivergace
EFE
Asc Striet
Aux Relle
Outliess
NEPE
EFE
Aue Striet
Ace Riclos
Outien
NEPE
98,989
98.99%
1.86%
0J1191
1009
99.38%
99.41%
0.62%
[UNS1
97.58%
97.66%
156
99.25%
IUNK5
100%
0.040
96.952
95.97%
1082
00195
100%
0.00
99 3855
99.4%%
0.75%
0,000
109
0.085
98.375
95.17
1.49
00253
1078
0.0122
90.95
99.65
18
0.0101
97.518
595
2,58
00319
188
0,0028
2116
99.206
0.794
0,007
Forced
Turbulence
Forced MIDD Turbulesce
Zere-Drregence
Zero-Dirergence
EFE
Aoz Strict
Aux Riclas
Qulliers
NEPE
EFE
Aux Strict
Ace Ricloo.
Outiie
NEPE
1005
0.8117
96.94%
97.04%
3,00%
1009
0,8137
98.89%
95.06%
1.13%
RJ3366
105
0.0156
96.115
W5.209
39299
000155
107
0.0051
98.515
98.599
1.535
011112
95.015
116
96.809
100%
0.0313
109
0.0146
96.33%
96.39%
3,80%
0,142
10%
0.8048
98.55%
98.645
1.4%%
0.0387
16
0.0193
95.08%
95.27%
4.96%
0.0533
1%
0.0057
97.99%
96.125
2.5%
0.0519
Houndary
Zere-Divergence Train Size
Zero-Utrergrnce
Trait Size
Strict
Outien
NEPE
100%
0.0078
99.025k
99.18%
0.95%
0.0067
100%
0,8018
99.4255
99.44%
0.58%
0.004
1093
6,0040
08.40%
98.91%
1.37%
0J0803
107
0,8020
99.40%
99.42%
0.01%
010043
11
0.8066
97.85%
97.87%
3.42%
0J1199
118
0.8019
99.41%
99.875
0.59%
IL161
99.09%
1009
10%
0.0089
96.57%
96919
3,40%
0.088
1%
0.0020
99.39%
99.41%
0.615
0.0042
Table 8: Comparison on FluidFlow3DD test data. V/x means our method with/without the DVE
module. Better results are marked in bold.
DVE
Train Size
EPE
Acc Striet
Ace Relax
Outliers
100%
0.0046
98.69 %
98.77% %
1.31 %
10%
0.0064
98.27%
98.37% %
1.74%
V
1%
0.0085
97.61% %
97.76%
2.40%
100%
0.011
95.72%
97.88%
9.29%
10%
0.016
93.38%
96.93%
14.59%
1%
0.0219
89.92%
95.56%
21.04%
A.5.3 Dynamic Velocimetry Enhancer(DVE)
We have developed the Dynamic Velocimetry Enhancer (DVE), which used the testing phase of the process optimize the initial flow and can effectively improve cross-domain robustness and data efficiency. To illustrate this, we trained models with and without the DVE module on different sizes of training sets and compared their performance on various test sets. We tested it using the FluidFlow3D test data(See Table, g and its six fluid cases(See Table
As evidenced by Tab. g the model without the DVE module is less effective than the model with the DVE module, even though the latter trained on only 1% of the samples. Furthermore, we observe that the model without the DVE module is highly sensitive to the training size, and the performance
25

---

Table 9: Comparison different flow cases. v /x means our method with/without the DVE module. Better results are marked in bold.
Beltrami Flow
Turbalent Channel Flow
DVE
Train Size
DVE
Train Size
EPC
Acc Striet
Acc Relax
Outliers
NEPE
EPE
Acc Striet
Ace Relax
Outliers
NEPE
100%
0,0039
98.95%
98.99%
1.05%
0.0191
1005
0,0019
99.38%
99.41%
0.62%
0.0051
10%
1100064
98.45%
99.31 %
99.35%
15%
0.0101
97.58%
97.66%
2.44%
0.03199
155
0.0025
99.25%
99.11%
0.75%
0.0065
100%
0.0099
46-4
98. 1896
7.11%
0.0442
100%
0.0036
99.03%
99.39%
1,14%
0.0094
10%
0.0158
96. 809S
44%
0.0709
10%
98.57%
99.22%
1.86%
0.0145
15
0.0230
385
95. 10%
22 23.63%
0.0097
0.0074
08.0399
99.125
2.65%
0.0187
Forced Inotropic Turbulence
Forced MHD Turbulence
DVE
Train Size
DVE
Train Size
Relax
Outlices
Relax
Outliess
100%
0.0117
96.94%
97.04%
3.07%
0.0378
100%
0.0037
98.89%
98.95%
1.13%
0.0306
10%
0.0156
96.11%
96.26%
3.92%
0.0515
10%
0.0051
98.51%
98.59%
1.53%
0,0412
1%
0.0193
95.01%
98.22%
5.02%
0.0645
1%
0.0066
97.98%
98.12%
2.06%
0.0505
100%
0.0244
89.05%
94, 19%
19.15%
0.093
x
100%
0.0111
95.41%
98.01%
19.35%
0.1032
10%
0,0318
85.55%
92.43%
26.19%
0.1267
x
10%
0.0169
92.66%%
96.97%
29.36%
1.5522
18
0.0396
80.94%
90. 18%
34. 4646
0.1618
18
0.0226
89.06%
95.68%
39.86%
0,2103
Tranitional Boundary Flow
Uniform Filw
DVE
Train Size
DVE
Thain Size
EPS
Apc Striet
Auc Relus
Outliers
NEPE
EPE
Aux Striet
Ace Relax
Ouiliers
NEPE
100%
0,0028
99.02%
99.18%
0.96%
0.0067
100S
0.0018
99.42%
99.44%
0.58%
0.004
10%
%
1.37%
99.40%
99.42%
1%
0.0066
97.55%
97.87%
2.42%
0.0149
15
0.0019
99.41 %
99.43%
0.59%
0.0041
100%
0.0082
97.425
99. 0.9
2,68%
0.0183
x
100%
010033
99.09%
99,43%
0.97%
0.008
10%
0.0134
94.49%
98. 30%
5.65%
0.0297
10%
0.0058
98.74%
99.35%
1.46%
0.0124
x
1%
0.0222
88.17%
96.35%
12.17%
0.0194
X
15
0.0074
98.38%
99.33%
1.92%
0.0159
Table 10: Runtime Profiling of Our Method. Different Train Size settings share the same inference time T T_test. The inference process includes the Forward of the trained network and our DVE.
Train Size
Training Epochs
T train(h)
T test(s)
100%
100
16.9
0.218
10%
100
1.241
Forward
DVE
1%
300
0.502
0.019
0.199
of each metric rapidly declines as the training size decreases. In contrast, the model with the DVE module exhibits certain degree of robustness. This demonstrates the significant impact of our DVE module on data efficiency.
Tab. 9 shows the performance of the DVE module also varies across different flow cases. In simple cases, the presence absence of the DVE module has minimal impact the metries, since the initial flow already closely matches the ground truth with a relatively small error. However, in challenging cases where the initial flow is far from the ground truth, the DVE module plays crucial role. It significantly enhances the model's accuracy through a straightforward and focused optimization process.
Time Profiling of DVE A significant challenge in test-time optimization is implementing per- sample adaptation at test time without adversely affecting inference efficiency. We demonstrate the time profiling of our method in Table 10 illustrate the efficiency of our proposed test-time optimization paradigm. Additionally, we present the convergence graph of DVE to study the influence of the number of refinement steps. As shown in Figure 9 DVE can converge within 150 epochs due to its simple structure, resulting in fast inference and high efficiency.
A.6 Limitations
Our research addresses the challenge of estimating dual-frame fluid motion. However, this field suffers from scarcity of large-scale benchmark datasets. To mitigate this issue, we incorporate real-
26

---

0,250
0.225
0,200
0,175
0.110
0,125
0.075
202
420
600
502
1930
Numbbr of Inaratios
Figure Convergence graph of EPE with respect to the number of refinement steps (iterations).
world datasets and conduct extensive studies across diverse flow cases. Nonetheless, it anticipated
that a broader variety of fluid motion data will become available in the future.
A.7 Impact Statements
Our research focuses advancing dual-frame fluid motion estimation method designed for optimal data efficiency and cross-domain robustness in turbulent flow analysis. This innovative approach is anticipated empower scientists analyze fluid dynamics with significantly reduced data requirements while enhancing overall method robustness. It is crucial to acknowledge potential limitations, particularly in real-world applications like blood analysis in medical science, where emros arising from our method could potentially be harmful. Our intent to continually refine and improve methodology minimize any such unintended consequences.
27

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: Please see Abstract and Introduction.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results
be expected to generalize other settings.
is fine to include aspirational goals as motivation as long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: Please see Appendix A.6
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification. asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low images are taken in low lighting. Or speech-to-text system might be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach address problems of privacy and faimess.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory Assumptions and Proofs
Question:
For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [NA]
28

---

Justification:
Our work does not involve theoretical provement.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental Result Reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless whether the code and data are provided or not)?
Answer: [Yes]
Justification:
We provide all needed implementation details in Appendix A.3.3
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make clear how to reproduce that algorithm. (b) the contribution is primarily a new model architecture, the paper should describe the architccture clearly and fully.
(c) If the contribution is new model (e.g., a large language model), then there should either way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reprodueibility. the case of closed-source models, may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
29

---

Answer: [Yes]
Justification:
We provide link to demo data and code in the Abstract.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips.co cc/
for more details.
While we encourage the release of code and data, we understand that this might not be possible, "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (c.g., for a new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
7/nips.com wttpcccpppbccccceee guwdes/dde/eeee Cooeeeoodeesssseeeeeee /ttp://bb///uudddd///eeeeeeeesss
ssiniiiiccy for more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible supplemental material (appended the paper) is recommended, but including URLs to data and code is permitted.
6. Experimental Setting/Details
Does the paper specify all the training and test details (c.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary understand the results?
Question:
Answer: [Yes]
Justification: We provide experimental settings both in each experiment section of the main paper and in the appendix.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, as supplemental material.
7. Experiment Statistical Significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer:
[NA]
Justification: Our method is deterministic.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, statistical significance tests, least for the experiments that support the main claims of the paper.
The factors of variability that the enror bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g., Normally distributed errors).
30

---

It should be clear whether the error bar is the standard deviation or the standard error of the mean.
is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables figures symmetric error bars that would yield results that are out range (e.g. negative error rates).
If error bars are reported in tables or plots, The authors should explain the text how they were calculated and reference the corresponding figures tables in the text.
8. Experiments Compute Resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification:
We provide time profiling of our method in Appendix I0
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
9. Code Of Ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics http:///aaaappccppppaaaaaaacaaaa
Answer: [Yes]
Justification:
We confirm this.
Guidelines:
The answer NA means that the authors have reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., there a special consid- eration due to laws or regulations in their jurisdiction).
10. Broader Impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification:
We provide impact statement in Appendix A.7
Guidelines:
The answer NA means that there no societal impact of the work performed.
If the authors answer NA or No. they should explain why their work has no socictal impact why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
31

---

The conference expects that many papers will be foundational research and not tied particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology being used intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms monitor how system learns from feedback over time, improving the efficiency and accessibility of MI..
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: Our work poses no such risks.
Guidelines:
The answer NA means that the paper poses such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Lieenses for existing assets
Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper. properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
Justification: We have cited original papers.
Guidelines:
The answer NA means that the paper does existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (c.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (c.g., website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswi thcode. com com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
32

---

If this information is not available online, the authors are encouraged to reach out to the asset's 's creators.
13. New Assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [Yes]
Justification:
We have open-sourced well-documented code.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details the dataset/cooe/mmodll as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create anonymized URL or include an anonymized zip file.
14. Crowdsourcing and Research with Human Subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
Answer: [NA]
Justification: Our work does not involve crowdsourcing research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included the main paper.
According to the NeurIPS Code Ethics, workers involved data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Question:
Answer: [NA]
Justification: Our work does not involve crowdsourcing nor research with human subjects.
Guidelines:
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent)
may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
33


Reviews:
[{"review_id": "SXsjAVBuBV", "reviewer": "Reviewer_jdpi", "summary": "In this paper, the author proposes a self-supervised learning based 3D particle tracking velocimetry (PTV) technique for dual-frame fluid motion estimation. The proposed method surpasses its supervised counterparts while utilizing only 1% of the training samples (without labels) compared to previous methods. A zero-divergence loss is utilized for turbulence flow and it is implemented in splat-based approach for efficiency and effectiveness. On the benchmarks, the proposed method shows the best performance comparing to baselines.", "questions": "See weaknesses.", "limitations": "Yes. The author discusses the limitations in the appendix.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The self-supervised learning based method can achieve better performance given only 1% of training samples compared to the previous supervised methods.\n- The proposed method shows the best performance overall on benchmarks especially on complex case such as Forced MHD Turbulence.\n- The paper is well organized and easy to follow.", "weaknesses": "- As for tests on the real-world dataset, the author mentioned the proposed method is integrated with the PTV framework. Just wondering how will the proposed method perform stand alone?\n- As mentioned in the paper, there are datasets with sparse flow field, which may influence the performance of the proposed method. It seems that there is no investigation of the particle density. Just wondering how will the method perform given trained on sparse/dense and inference on dense/sparse?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### Q1: How will the proposed method perform stand alone compared with being integrated into PTV?\n\n**Answer:**\nWe cannot perform this evaluation because the task of PTV is a superset of dual-frame motion estimation. Here, we clarify these differences:\n\nOur method targets particle motion vector estimation between two frames, whereas PTV tracks the movement of particles across the whole sequence. The relationship is similar to that between two-frame optical flow (e.g., RAFT [B]) and long-term point racking (e.g., OmniMotion [A]). The PTV framework [C] involves particle detection, particle positioning, dual-frame motion estimation and whole-sequence particle tracking. While our method provides good dual-frame motion initialization, it does not perform whole-sequence matching. Thus, it is not feasible to evaluate our method independently on PTV tasks.\n\n[A] Tracking everything everywhere all at once, ICCV 2023\n\n[B] Raft: Recurrent all-pairs field transforms for optical flow, ECCV 2020\n\n[C] 2-dimensional particle tracking velocimetry (ptv): technique and image processing algorithms. Experiments in fluids, 6(6):373–380, 1988.\n\nTo summarize, the key differences are:\n\n- PTV tracks particles across the whole sequence, while our method deals with frame pairs.\n- Dual frame motion estimation methods (like ours and GotFlow) output motion vectors while PTV methods associate particles in a long sequence.\n\n### Q2: The method performance given trained on sparse/dense and inference on dense/sparse?\n\n**Answer:**\nFirst, we note that our data originates from physical phenomena (simulation or real-world capture). Manually downsampling it might not accurately reflect the sparse distribution of fluid particles in a real-world scenario.\n\nSecond, we provide experiments where models are trained on dense and tested on sparse particle distributions. The particle density is controlled by downsampling particles in each sample. And for fair comparison, we evaluate the model trained on 1% samples of the whole dataset, which corresponds to the main experiment Ours(1%) row in Fig.4-Top.\n\nThird, we demonstrate our model trained on dense particles (100% original particles) and tested on particles with varying levels of sparsity (sample ratios).\n\n| Down-sample Ratio | EPE| \tAcc Strict|\tAcc Relax | Outliers |\n|----------|----------|----------|----------|----------|\n|100%| 0.0085|\t97.61%|\t97.76%|\t2.40%|\n| 99%    | 0.0128   | 96.64% | 96.78% | 3.39%  |\n| 95%    | 0.0347   | 91.19%|91.49%|8.84%   |\n| 90%    | 0.0669|83.99%|84.46%|16.04% |\n| 80%    | 0.1363   | 69.91%|70.59%|30.14%   |\n| 50%    | 0.2842|41.02%|\t41.86%|\t59.03%|\n\nIt is important to note that this naive downsampling can cause many particles to lose their corresponding targets, adding substantial difficulty to flow learning.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lkivAnhiRC", "reviewer": "Reviewer_wM8x", "summary": "The paper proposes a self-supervised method to learn 3D particle tracking and modelling turbulent fluid flow. They regularize their method with a zero-divergence loss function and inspired by the splat operation they propose a splat-based implementation for this loss. They also incorporate a GCNN feature extractor to learn geometric features. Their method also supports test-time optimization, with their Dynamic Velocimetry Enhancer (DVE) module.", "questions": "In Fig 4 (top table, ours (1%) w/o DEV) it is not clear whether the time reported is the time for the forward pass or includes the optimization, can you comment on that?", "limitations": "Yes.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Strengths \nThe paper writing is good.\nNovelty: the paper suggested a novel approach to solving the large data dependency problem. I really liked the idea of using EdgeConv to incorporate geometric information.\nThe paper tackles an important problem.", "weaknesses": "* The method needs test-time optimisation, which will make the result a bit questionable if you optimize the test set.\n* The test time complexity is not reported clearly. In Fig 4 (top table, ours (1%) w/o DEV) it is not clear whether the time reported is the time for the forward pass or includes the optimization.\n\n**Suggestion:**\nPlease use single tables format their different formats in the main text and appendices. Also, splitting the tables and figures will make it easier to follow the paper and understand the result.\nAnd remove the unnecessary **bold** used in the main text.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### Q1: Test-time optimisation will make the result a bit questionable\n\n**Answer:**\nWe note that the test-time optimization is self-supervised without accessing ground truth labels, making the setting realistic. This approach has garnered significant attention recently, as seen in references introduced in the related works (Section 2.1) of our main paper.\n\nAdditionally, to make comparisons with baselines fair, we incorporate the DVE test-time optimization module with other baseline methods to ensure a fair comparison.\n\nDue to computing resource constraints, we selected two baselines for comparison: the state-of-the-art method GotFlow3D and FLOT. Both methods include an optimization module in their network and learn to optimize. We integrate our DVE module with their estimated outputs, iterating for the same number of steps as our method. The results are shown below:\n\n|Methods    |T_test\t|EPE\t|Acc Strict|\tAcc Relax|\tOutliers\n|----------|----------|----------|----------|----------|----------|\n|Ours|\t0.218s|\t0.0046\t|98.69%|\t98.77%|\t1.31%|\n|GotFlow3D|\t0.758s\t|0.0049\t|93.15%|\t96.38%|\t3.62%|\n|GotFlow3D w/ DVE|\t2.260s|\t0.0024|\t99.12%|\t99.13%|\t0.86%|\n|FLOT\t|0.030s\t|0.0587\t|24.99%|\t45.59%|\t54.41%|\n|FLOT w/ DVE|\t0.520s|\t0.0300\t|90.38%\t|91.15%|\t10.00%|\n\nIt can be seen that adding the DVE module can significantly improve the performance of GotFlow3D and FLOT, demonstrating that the DVE module is generalizable to other methods. Although GotFlow3D with the DVE module performs better than our method on various metrics, its network is heavy and time-consuming.\n\n### Q2: In Fig 4 (top table, ours (1%) w/o DVE) it is not clear whether the time reported is the time for the forward pass or includes the optimization.\n\n**Answer:**\nTo clarify, The forward pass takes 0.019s and the test-time optimization module takes 0.199s, adding up to 0.218s. This has been stated in Table.9 of the supplementary material.\n\nAs for the T_{test} number in Fig.4-Top:\n\nOurs(1%) includes both forward pass and test-time optimization, so the time is 0.218s.\n\nOurs(1%) w/o Div Loss includes both forward pass and test-time optimization, so the time is 0.218s.\n\nOurs(1%) w/o DVE means the test-time optimization module is not included, so the time is 0.019s.\n\n### Q3: Suggestion: Please use single tables format their different formats in the main text and appendices. Also, splitting the tables and figures will make it easier to follow the paper and understand the result. And remove the unnecessary bold used in the main text.\n\n**Answer:**\nThanks. We will re-formatted the manuscript according to these points to improve the readability. We'll change all tables to three-line, header bolded. Please see Table 1 in the global rebuttal pdf as an example.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pqJ7uKwx99", "reviewer": "Reviewer_N6ux", "summary": "In this paper the authors propose a graph based network that combines feature extraction with test time optimization to perform two-frame particle tracing. The core problem is that given two sets of points, e.g., point cloud data, to find the correspondences between the point clouds and estimating their velocity from this correspondence. While this problem can be well analyzed in synthetic data, real world problems are often sparse in data and, accordingly, using methods that do not require labeled data (and not a lot of it) would be beneficial. The proposed methodology works by proposing an unsupervised loss formulation based on assuming smoothness and divergence-freedom of the solution space and uses parts of this loss function as a test-time target for optimization. The method is evaluated and trained on synthetic data and then evaluated on dissimilar data to show the generalization capabilities of the approach.", "questions": "Note that these questions are not in order of importance.\n1. What is the variance and statistical certainty of the training across different initializations for the proposed methods?\n2. Why is the EPE error lower for the 1% case in Table 3?\n3. What would the training results be with equal weight updates across methods, and if they do not change from the proposed hyperparameters, why were these hyperparameters chosen?\n4. Is it possible to apply DVE to other state of the art methods and, if so, what would the results be?\n5. How was the 1% subset of the data chosen? Does this influence the result, i.e., do different subsamplings (and strategies) result in consistent (within a small deviation) results?\n6. Is there any data to highlight the limitations of including the regularization terms during test time optimization?\n7. How were the lambda parameters chosen?\n8. Why is the computational time of the method at test time different based on the dataset size used during training?\n9. What was the total time to train a network from start to finish for the proposed method with and without the various terms?", "limitations": "The primary limitations of the paper relate to the evaluation of the methodology and the divergence-free loss term. In summary there are three key issues:\n1. The evaluation should include a broader evaluation that at least considers seed initialization and variance due to the 1% sampling. As the paper currently stands, it is impossible to tell, with certainty, that the results are consistent and not just due to a lucky seed. While performing such evaluations for all setups is computationally prohibitive, at least the core contributions should be evaluated\n2. There is a lot of inconsistency regarding the divergence-free loss. On one hand it is argued as very important during training and how it can work even in compressible cases. Then the term is wholly disabled during test time optimization because it isn’t needed and then towards the end in a case with divergence the term is left off anyways. There is no clear evaluation as to the influence of including the divergence-free condition during test optimization, especially for cases which are compressible (and/or not divergence-free). This makes it difficult to judge how the method truly generalizes\n3. While not a direct limitation, the authors themselves state that the use of synthetic data leads to hand-crafted biases (in the introduction), but they also demonstrate how their method is immune to this. As this does not seem to be a limitation of the synthetic data nature, this should be either clarified in the introduction or an actual limitation where the method does not generalize due to its limitation from hand-crafted data.\nRating:", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The overall idea and motivation of the paper is useful and would solve an important problem. The strengths of the paper are:\n* A method that can train with significantly smaller datasets than current state of the art methods at better performance\n* Test time optimization for particle tracking\n* A physically motivated loss term to drive a self-supervised learning setup \n* The proposed method is, fairly, straightforward and works in a variety of settings with one-out training", "weaknesses": "Overall the paper has an interesting core idea, however, several issues exist, especially regarding evaluation and the lack of statistical evaluations and ablation studies in general. There are a few core weaknesses listed below in a short list. Following these a list of all potential issues and weaknesses is included for completeness at this point. Note that many of these issues are also addressed separately and are not questions in themselves.\n* The authors perform no evaluation regarding the statistical significance of their proposed method. While the computations are deterministic (as they note in the checklist), this does not mean that network initialization does not play a role, especially when many of the evaluation metrics are relatively close and some results seem counter intuitive, e.g., the EPE improving for smaller datasets. Furthermore, there is no clear evaluation on the choice of the 1% of data, i.e., did the authors try different subsets? Were the 1% chosen consistently across class? Across different trained methods?, etc.\n* It is not clear why the proposed DVE test-time optimization cannot be performed on top of data trained on supervised data, especially as the authors change the loss functions anyways from training to testing.\n* The authors make a fairly broad claim in the introduction as to synthetic data being always limited due to hand-crafted priors but show that their method does not suffer from this problem, even though it uses the same synthetic data. This needs to be clarified.\n* The divergence-free loss term is not very well evaluated. On the one hand the assumption seems to be relatively central to the approach and the authors go out of their way to state that assuming divergence-freedom can even be assumed in compressible cases (where it does not hold), but then also state that they disable the divergence-free term during test-time optimization especially for cases where it does not hold. * Either the authors need to more clearly evaluate the impact of the divergence-free loss term during test-time optimization in _all_ cases, or consider the importance of the argument.\n* The  formatting of the paper at times is very odd with subsubsubsubsections and randomly highlighting of statements with inconsistent (and incomplete) highlighting in tables. This should be improved.\n\nGeneral:\n* In the paper checklist under 7 the authors note that their ‘method is deterministic’ and thus does not require error bars. However, choosing random subsets of data, training different architectures and more usually involves sources of randomness and seed variance can be quite significant. This needs to be evaluated properly and adjusted accordingly.\n* Table 3 seems odd as the EPE _improves_ for the proposed method when training on smaller datasets.\n* Table 9: Why is the Epoch count increased to 300 for the 1% train size and is this done consistently across all methods and what would the results be with only 100 Epochs or a normalized training with the same number of weight updates across dataset sizes?\n* It is wholly unclear as to why the proposed scheme is 6 times slower during testing just because it is trained on a larger dataset.\n* Why can other methods not utilize DVE as a test-time optimization process? This would make for much fairer comparisons\n* The authors should make it more clear in the related work as to which graph-based feature extractor they use as their architecture is heavily informed by prior work.\n\nDataset:\n* The authors claim that synthetic data is always limited as it leads to ‘hand-crafted priors’ but they utilize a synthetic dataset to train their method and show that it is not limited by the biases induced from the prior, i.e., their method can generalize even from synthetic data.\n* The authors state that in cases of boundary conditions, or other situations, smoothness and divergence-freedom is not holding up. This should be more clearly shown, i.e., a demonstration that using  these regularization terms leads to issues in such cases.\n\nReduced Data Training:\n* The 1% setup is interesting, however, the authors do not clearly show how sensitive their method (or methods in general) are to selection, i.e., there is no clear evaluation on how the choice of the 1% affects the outcome\n* There are no statistical evaluations, which would be especially important for the 1% evaluations and with the difference between methods being so narrow at times, initial seed choice might be important too\n* There is no evaluation as to what the method would do if the data is incomplete, i.e., if particles are missing from one of the frames. This would be important for real-world applications but due to their reliance on hand-crafted synthetic priors and other narrow datasets, this is not evaluated clearly.\n\nDivergence-Freedom/Loss Terms:\n* The Divergence-free loss term is interesting but not using it during testing seems odd as it should be helpful in all cases. However, even the authors note that this would be a limitation in non divergence free settings (which they evaluate one of). Considering the prominence of the divergence-free claim, this needs to be further elucidated (especially considering the statement in line 674)\n* The authors claim (178) that even for compressible fluids divergence-freedom can be assumed in practice. While sometimes done, this should be more clearly evaluated as the authors clearly do not use the divergence-free loss in datasets where the assumption would not hold.\n* The distinctions when smoothness and divergence-freedom are not to be used for optimization are not clear as there might be reason to not use the divergence-free loss, i.e., if one expects a dataset with divergence, but not requiring smoothness is odd. This should be done in an ablation study.\n* The neighborhood sizes are odd and should be more clearly written as the information in 715 is not easily understood, i.e., is the divergence-loss computed with only 2 points?\n* The formulation of the divergence as a central finite difference scheme is not necessary, other information would be more prudent.\n\nFormatting/Writing:\n* The formatting is very inconsistent at times, e.g., table layouts, subsubsubsubsection headings are sometimes italicized, sometimes underlined and sometimes bold.\n* The authors do not need to state that they are excluding information due to page limits, twice.\n* The highlighting in Figure (mostly table) 4 is inconsistent. In the top part the second best performing method is not highlighted with underlining but also boldfaced and the T_test column has no highlighting\n* Table 6 should also highlight cases where the method without divergence-freedom performs better\n\nMissing Details:\n* How were the lambda parameters chosen?\n* KNN should be properly introduced (208) and evaluated as to why it is not efficient\n\nEvaluation:\n* The method performs significantly worse in some cases, e.g., the EPE is worse by a factor of 2 for the MHD case compared to the turbulent channel flow. It would be nice if this was more clearly evaluated and highlighted, especially for the Beltrami case (Figure/Table 5) where Gotflow performs worse by a factor of 2 relative to the proposed method, which is fairly close considering this case is about 8 times worse than the best case for the proposed method and only 3 times worse for Gotflow.\n* The improvement of 22882 matches over 22001 seems fairly minor (315) and it would be prudent to add the tracking timing in this section in the main paper.\n* How does the proposed method perform on its own (323), i.e., when it is not used as an initializer?\n* The difference in EPE from 100% to 10% with and without DVE is relatively close even though the text (800) notes that the method is highly sensitive.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### Q1: 1% chosen consistently across class?\n**Answer:**\nYes, the sub-sampled dataset is chosen consistently.\n### Q2: Statistical Significance\n**Answer:**\nWe control experiment randomness with seeds, conducting multiple runs to ensure variability while maintaining consistency with the same seed. Results in Tables 2 and 3 of the rebuttal PDF affirm our method's robustness.\n### Q3: DVE performed on top of data trained on supervised data？\n**Answer:** DVE integration during training isn't feasible as gradients cannot propagate through the DVE module and it's also against our training objective.\n### Q4: Use synthetic data yet effectively mitigates its limitations.\n**Answer:** \nWe train solely with synthetic data, aiming to develop a robust feature extractor that recognizes basic correspondences consistent across synthetic and real data. Our strategy maximizes the advantages of synthetic data and addresses domain shifts at test time. Our method effectively mitigates these shifts using the DVE module during test-time optimization, as explained in Section 4.3 of our main paper.\n### Q5: Why to disabe zero-divergence during DVE.\n**Answer:** We deactivate the zero-divergence loss during test time when it may not be precise, as inaccuracies in loss terms can negatively impact optimization. Specifically, zero-divergence only holds when the fluid is incompressible—true for our evaluation scenarios—and the flow field is sufficiently dense to compute divergence accurately, which isn't always the case in our tests.\n### Q6: Evaluation of the Zero-Divergence/Smooth loss term during DVE\n**Answer:** \nWe argue against using regularizers like zero-divergence or smooth loss during test-time optimization if they might reduce accuracy. Our studies, detailed in Table 4 of our rebuttal PDF, show that smooth loss generally degrades performance, and zero-divergence loss only improves it in specific scenarios like Transition. This is because zero-divergence applies effectively only in incompressible and sufficiently dense flow fields, conditions not consistently met in our tests.\n### Q7: Formatting Issues:\n**Answer:** We'll fix the writing issues and adjust the formatting in the revised version.\n### Q8: Odd Results:\n**Answer:** \n- Figure 4 upper table: T_test for \"Ours\" should be 0.218s rather than 1.353s. \n- Table 3: Ours should be as Table 5 in rebuttal PDF.\n### Q9: Epoch increased to 300 for the 1% train size\n**Answer:** It's because the 1% training size setting requires a longer time to converge.\n### Q10: Training with the same epoch num across dataset sizes:\n**Answer:** We trained our method with 300 and 100 epochs across dataset sizes, as shown in Table 6 of the rebuttal PDF. The results indicate that 100 epochs are insufficient for convergence when training on only 1% of the data.\n### Q11: Why our method is 6 times slower when trained on a larger dataset.\n**Answer:**  Please see Q8.\n### Q12: Other methods with DVE for much fairer comparisons.\n**Answer:**  Integrating DVE into other methods wouldn't ensure a fair comparison as it is essentially comparing our basic feature extractor against more complex systems.\nResults in Table 7 of the rebuttal PDF show that adding the DVE module can significantly improve the performance. Although GotFlow3D w/ DVE performs better, its network is time-consuming.\n### Q13: Lack of graph-based feature extractor in the related works:\n**Answer:**   We have introduced its related works in Section 3.2.2 of the main paper. We will revise the related works.\n### Q14: Missing particles from frames:\n**Answer:**  \nTo simulate missing particle scenarios, we downsample particles in either the source or target frame. The results are detailed in Table 8 of the rebuttal PDF.\n### Q15: Not using zero-divergence during testing.\n**Answer:**  Assuming it works for all cases is not the best practice. Please refer to Q5 and Q6 for details.\n### Q16: Neighborhood sizes\n**Answer:**  It is not sensitive. See Table 9 of the rebuttal PDF.\n### Q17: Use smooth loss for DVE\n**Answer:**  We argue that smooth loss is only a regularizer and we should not assume it holds for all datasets. Please refer to Q5 and Q6 for more details.\n### Q18: Need other information about divergence calculation\n**Answer:**  We do not understand what \"other information\" refers to here. For our implementation, we use torch.gradient.\n### Q19: Lambda chosen\n**Answer:** By cross validation.\n### Q20: Evaluation on KNN.\n**Answer:** Direct KNN application is ineffective in sparse fields. Our method as a grid-based gradient computation is more precise due to the well-established techniques in gradient calculations on grids. A  comparison is shown in Table 10 of the rebuttal PDF.\n### Q21: More clearly evaluated why the method performs significantly worse in some cases.\n**Answer:** \nWe provide visualizations of different flow cases in rebuttal PDF. Our method excels in simpler flow cases like \"Channel,\" \"Transition,\" and \"Uniform,\" but performs less better in complex scenarios due to inherent limitations.\n### Q22: Prudent to add the tracking timing (Line 315)\n**Answer:** It is already included in Figure 6-c. We will revise it further.\n### Q23: Method performance on its own when not used as an initializer\n**Answer:** Refer to Q1 of our reply to jdpi.\n### Q24: Performance too close in DVE ablation study\n**Answer:**  We argue that the difference is significant. EPE has a notable scale difference (e.g., from 0.0046 to 0.011 at 100% and from 0.0064 to 0.016 at 10%), and other metrics, such as the outlier rate, are also significant.\n### Q25: Hyperparamter chosen for baselines\n**Answer:**  Default hyperparameters ensure optimal performance, with further experimentation shown in Table 11 of the rebuttal PDF where GotFlow3D overfits at 100 epochs. \n### Q26: Apply DVE to other methods\n**Answer:**  Please refer to Q12.\n### Q27: Training time\n**Answer:**  See Table 9 of main paper. It should be roughly the same without loss terms.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We present all tables and figures mentioned in the rebuttal.

Meta-review: All review scores are positive: WA, WA, and BA. While no A and SA are given, AC confirmed that most reviewers' comments are positive. AC also verified that their final evaluations lean toward accepting this paper. Reviewers jdpi and N6ux recognize that the proposed method can train with significantly smaller datasets than current state-of-the-art methods at better performance. Reviewers jdpi and wM8x appreciate that the paper is well organized. Reviewers wM8x and N6ux agree that this work tackles an important and interesting research problem. Finally, reviewers jdpi and N6ux recognize that the experimental results demonstrate the effectiveness of the proposed method. While reviewer N6ux was initially a bit underwhelmed by this paper, the authors' rebuttal did a good job explaining the issues to convince this reviewer. For the reasons stated above, AC recommends this paper for presentation in NeurIPS. Since several issues remain, we expect the authors to revise the manuscript based on the reviewers' comments.
Decision: Accept (poster)