Flow Field Reconstruction with Sensor Placement Policy Learning
Authors: Ruoyan Li
Guancheng Wan
Zijie Huang
Zixiao Liu
Haixin Wang
Xiao Luo
Wei Wang
Yizhou Sun
Venue: NeurIPS 2025 poster ()
Keywords: Sparse Sensor Measurement
Optimal Sensor Placement
Graph Neural Network
Proximal Policy Optimization
Primary area: machine_learning_for_sciences

Abstract: Flow‐field reconstruction from sparse sensor measurements remains a central challenge in modern fluid dynamics, as the need for high‐fidelity data often conflicts with practical limits on sensor deployment. Existing deep learning–based methods have demonstrated promising results, but they typically depend on simplifying assumptions such as two‐dimensional domains, predefined governing equations, synthetic datasets derived from idealized flow physics, and unconstrained sensor placement. In this work, we address these limitations by studying flow reconstruction under realistic conditions and introducing a \emph{directional transport‐aware Graph Neural Network (GNN)} that explicitly encodes both flow directionality and information transport. We further show that conventional sensor placement strategies frequently yield suboptimal configurations. To overcome this, we propose a novel \emph{Two‐Step Constrained PPO} procedure for Proximal Policy Optimization (PPO), which jointly optimizes sensor layouts by incorporating flow variability and accounts for reconstruction model's performance disparity with respect to sensor placement. We conduct comprehensive experiments under realistic assumptions to benchmark the performance of our reconstruction model and sensor placement policy. Together, they achieve significant improvements over existing methods.

Flow Field Reconstruction with Sensor Placement Policy Learning
Ruoyan Li', Guancheng Wan', Zijie Huang', Zixiao Liu', Haixin Wang', Xiao Luo', Wei Wang', Yizhou Sun! University of California, Los Angeles
Abstract
Flow-field reconstruction from sparse sensor measurements remains a central chal- lenge in modern fluid dynamics, as the need for high-fidelity data often conflicts with practical limits on sensor deployment. On one hand, existing deep learn- ing based methods have demonstrated promising results, but they typically rely on overly simplified assumptions such as 2D domains, predefined governing equations, synthetic datasets derived from idealized flow physics, and unconstrained sensor placement. In this work, we address these limitations by studying flow recon- struction under realistic conditions and introducing directional transport-aware Graph Neural Network (GNN) that explicitly encodes both flow directionality and information transport. On the other hand, conventional sensor placement strate- gies frequently yield suboptimal configurations. To overcome this, propose a novel Two-Step Constrained PPO procedure for Proximal Policy Optimization (PPO), which jointly optimizes sensor layouts by incorporating flow variability and accounts for reconstruction model's performance disparity with respect to sensor placement. We conduct comprehensive experiments under realistic assumptions to benchmark the performance of our reconstruction model and sensor placement policy. Together, they achieve significant improvements over existing methods.
Introduction
Flow field reconstruction from sparse sensor data (Berkooz et al. 1993, Schmidd (010)) has emerged as pivotal challenge modern fluid dynamics, particularly as the demand for high-fideltt measure- ments clashes with the practical constraints of sensor deployment. Such reconstruction techniques underpin real-world applications such as aerodynamic shape optimization and active flow control in aerospace and turbomachinery (Luo et al. 2017). With the rapid advancement of deep learning, leveraging deep learning models to transform limited experimental data into detailed, reliable repre- sentations of complex flow phenomena has been a promising solution (Zhong et al.) 2023 Yadav
ct al. 2025 Xu et al. 2023 Jing et al. 2024 ci al. 2025).
On one hand, we note that many existing studies rely assumptions that may not hold in realistic scenarios. Specifically, these works commonly assume that: (1) Domain: Experiments are predomi- nantly conducted in awo-dimensional (2D) domains. However, real-world applications take place in three-dimensional settings. (2) Physics: The governing physical PDEs, such as the Navier-Stokes equations, are known a priori, which is integrated to inform and constrain the models. Yet, empirical fluid dynamic data rarely conform precisely to the Navier-Stokes equations (Maadjionnstaninou 2006 Stubbe 2020). (3) Datasets: Datasets usually generated through pseudo-spectral solvers (Orszag 1969) or reynolds-
Navier-Stokes (RANS) (Tennekes & Lumley| [1992), but these numerical solvers rely simplified assumptions about fluid behavior, yielding datasets that diverge from real-world fluid dynamics. (4) Sensor Placement: Sensors are assumed arbitrarily placed within the flow field without influencing the fluid dynamics. In reality, measurement sensors should either
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Table Comparison of related works on problem assumptions.
Paper
3D-Domain Unknown Physies
Complex Data
Placement Optimization
Zhong al.
2023
X
Mo & Magni
2024
Yaday et al 2125
X
Jing et al.
2024
X
He et al. 2022
X
Zhang et al. (2022
Hosseini & Shiri 2024
x
X
Zhang et al. 2025
X
Shan ot a.. 2024
x
Xu et al. (2023
x
Ours
remain fixed at the domain boundaries or be advected with the fluid flow. A comprehensive review of these assumptions is provided in Table
To address this, we generate four three-dimensional (3D) turbulent flow datasets using Direct Numer- ical Simulation (DNS) in COMSOL (COM) (020) with varying geometries. The simulations initiate with randomly generated velocity field, and the inlet velocity is modeled both time-dependent and stochastic. We argue that training on these datasets enables improved transferability to real-world tasks and yields more reliable evaluation results. Although large-scale, real-world sensor datasets remain unavailable, we propose that the combination of high-fidelity simulations in COMSOL and the incorporation of dimeddepen,n stochastic inlet conditions provide more faithful representation of actual fluid phenomena. Further, we restrict sensor placement to the domain boundaries, since permitting sensors to be advected with the fluid flow leads to their accumulation in vortical regions and makes reconstruction of other areas infeasible. Thus, we aim to develop reconstruction model that can adapt arbitrary mesh-based geometries without assuming any underlying PDEs, while confining sensor placement solely to the boundaries.
We propose directional transport-aware GNN that explicitly encodes directionality and information transport in the message-passing stage. The explicit parameterization of directional weightings and transported quantities not only mimics the continuous advection operator a discrete, mesh-based ting but also corresponds to learnable interpolation algorithm. This encourages learning meaningful representations and yields robust imputation capability across various sensor configurations.
One the other hand, we reveal that traditional methods for sensor placements, such as singular value decomposition (SVD) QR pivoting (Chmielewski al. (002), often perform poorly when integrated with our reconstruction models. We attribute the performance degradation to reconstruction model's performance disparity with respect sensor placement. To address this, we train PPO policy that determines optimal sensor configurations and introduce a novel Two Step Constrained PPO training procedure enforce sensor constraints, Our policy not only captures variability in the fluid field but also accounts for model's performance disparity. Experimental results demonstrate substantial improvements reconstruction accuracy when using the learned sensor placements.
Our contributions are as follows: (i) Problem Identification: We introduce a realistic problem formulation for fluid-field reconstruction and generate extensive datasets that closely mimic real- world scenarios; (fi) Practical Solution: To tackle fluid field reconstruction on arbitrary mesh-based geometries, develop directional transport-aware GNN that explicitly encodes both directionality and information transport; (iii) Further Scientific Discoveries: We find that conventional sensor placement algorithms fail identify effective sensor locations, and thus we propose novel Two-Step Constrained PPO training strategy to learn a policy that identifies the optimal placement of sensors; (iv) Experimental Validation: We conduct comprehensive experiments under realistic assumptions to validate the superiority of our reconstruction model and sensor placement policy.
2 Related Work
AI for Computational Fluid Dynamies (CFD) Recent advances machine learning have led to various Baaring--bseed surrogate models for accelerating scientific discoveries et al. 2025 Huang
al. 2024b) Wang et al.] 2024). In the study flow field reconstruction, /Zhong et al. (2023) leverages a combination of MLP with CNN to reconstruct unsteady vortical flow fields near airfoils.

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Reconstruction Model Ma
Accurate Parameter Efficiant
Information Aggregation
Sensor No Sensor
Enode
Directional Alignment
Eedge
D
Transported Information
Emask
x L
Optimal Sensor Placement
Penalized Training
Constrained Training
(
R R AIVV o|
Value,(G))
log I'a = k|V
Saddlepoint Approx
R
Canolraine Ratiefaction
Gumble Top-k
af((a)
Ty(a
I
Figure 1: Overall framework of the proposed method. The directional transport- aware reconstruction GNN Ma takes boundary sensor inputs and infers missing field values through message passing with explicit directional alignment and information transport. A two-stage PPO training procedure identifies optimal sensor placements via a penalized stage that softly enforces sensor-count constraints through rewards, followed by a constrained stage where sampling from a constrained probability distribution ensures compliance with sensor limits.
He et al. 2022 introduces the Flow Completion Network, which employs GNNs to reconstruct both structured and unstructured data. Hosseini & Shiri (2024); [Shan et al. ); Yadav et al. (2025); ): Xu
al. (2023); Jing et al. (2024) leverages underlying physics PDE develop physics-informed
neural networks for enhancing reconstruction quality, Mo & Magri (202)) injects artificial noise into sensor data and develops a physics-constraine CNN for reconstruction. For deep learning-based
optimal sensor placement, Marcato et al. (2023) employ differentiable programming to integrate sensor placement into the training of a neural network model. Nonetheless, this method is constrained by fixed number of sensors and makes the assumption that sensors can be positioned arbitrarily without interfering with the fluid dynamics.
3 The Fluid Field Reconstruction Model
Problem Statement Let EV be graph representing a discretized mesh of the domain boundary, where each vertex V; V corresponds to node in the mesh and each edge ei, E E captures the local connectivity among these nodes. Each node is characterized by a feature vector [u;, Ps, a.). where u, E 3 denotes the velocity, P; E R denotes the pressure, and a; {0, 1) a binary mask indicating the availability sensor. a; = implics that the velocity and pressure at node i are known (i.e., sensor is present), while a, 0 indicates that these values must be imputed and u, and p, are randomly generated. Our objective is to develop a reconstruction model Ma that takes the graph G' input and outputs estimated values [ú, P.] for all nodes where a, = 0.
Directional Transport-Aware GNN This problem is substantially more complex due the absence of known governing equations, the use of irregular 3D geometries, and the restriction of sensors to boundaries. Our proposed model based on an Imcoder-Processor--- is designed to handle these challenges by learning flexible, geometry- representations that generalize across diverse domains without relying on explicit physical priors. In the encoding stage, we employ three distinct MLPs denoted by Emasks Enode. Eedge to embed the node and edge features into latent space. Formally,
framework.
ā; <- Emaas((a)) Tf Enode (u:llpi|lã..) Eig EEdgge(ess))
where || denotes concatenation. Next, we introduce the directional transport-awan processor that integrates the notion of directionality and information transport into the -aassgng framework. In our approach, the directional information, d, is computed as the inner product between a node's latent representation and its corresponding edge features. This inner product quantifies the degree of alignment of a node with respect to the direction of information transfer, thereby acting as a proxy for the node's 's contribution feature reconstruction. The computed directional score is then
3

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used to weigh the differences between the latent states of adjacent nodes, effectively capturing the information transported from node node j. The resulting directional transport function, T. is parameterized by MI.ss, and, subsequently, a node aggregation function, S, synthesizes the weighted edge messages to update the latent state of each node.
d (0) E()"', El. + T
of of S
Σ
(2)
We apply L layers of this processor with residual connections. Finally, the decoder D uses one MLP to map each node embedding vf to the desired output space: [ú,, P.] =(wf).
Remark 1 (Connection to advection operator) The design of the processor is closely connected to the advection operator. Advection describes the transport of properties, such as heat or pollutants, via the bulk motion of fluid. Mathematically, this process is typically characterized by the operator v V where v denotes the velocity and yp is the field being transported.
Our d encodes the direction of information flow, acting in the role of v. The difference between neighboring latent states, o oi(-1, serves as a discrete analogue to the spatial gradient. By com- bining these two quantities, we recreate the behavior of the advection operator in high-dimensional latent spaces and thereby enable efficient information propagation within the processor.
Remark 2 (Connection to interpolation algorithm) Our processor can be viewed as learnable inter- polation operator. In generic interpolation scheme, one writes U, + EjeN(e) 6(U,, 1J,, et ei,)((v,, vy), where 0; is the interpolated value, b(v,, Uy, fi.y) is the weight assigned to neighbor j. and g("), VU) is the contribution of node j. In our formulation, the weight function is the directional information, d. while the neighboring contribution is the difference between the latent states. We also include self- contribution by defining b(U;, U/. eg,;) (v;; 212) = de Rather than performing a fixed weighted sum, our processor replaces these terms with learnable MLP-based embeddings for both the transported information and the aggregation step.
Remark 3 (Connection to conventional message passing GNNs) Compared to conventional message passing GNNs, which typically concatenate node and edge information and process with MLPs, our method explicitly incorporates the directionality of information flow. This explicit encoding facilitates the learning of more meaningful representations, as it encourages the propagation of information along physically motivated pathways.
Sensor Placement Optimization
Building upon the directional transport- reconstruction model described above, explore methodologics for optimal sensor placement aimed at further enhancing reconstruction accuracy.
Traditional Approach We evaluate sensor placements by applying two greedy column-selection strategies: QR pivoting and D D-optimality (Manohar et al., 2018). Details on implementation and experimental results are in Appendix G These methods incur substantially higher reconstruction errors than uniformly distributed sensors. Deep learning models often exhibit performance disparity. Although a sensor placement strategy may be ideal for capturing flow-field variability, our reconstruc- tions model may reconstruct this configuration with lower accuracy than they do for other placements. An effective sensor placement strategy must address both the variability of the fluid field and the reconstruction model's differential subgroup performance. Thus, we aim to train a policy to account for these two factors in sensor placement.
Given reconstruction model MM, we aim to learn a policy (a( that takes a mesh G containing complete fluid field information as input and outputs probability parameter p, E [0, of a Bernoull distribution for each node for its sensor placement. We interpret a random variable a, ~ Bernoulli(p) p:) such that a; = 1 if node i has a sensor and a; = 0 otherwise. The policy aims to maximize the reward under such actions, which is the negation of the Mean Squared Error (MSE) of the reconstructed velocity and pressure. The objective function is defined as R( = -SSE [(;; 6 = Mo a; = 0), (vsla, = 0}}- We impose the constraint Σ 11 a; = V where [V] denotes the cardinality of the set and k E [0, 1] denotes the proportion of the mesh nodes that contains a sensor. Empirical results show that increasing the number of sensors consistently improves reconstruction accuracy. Therefore, instead of imposing upper bound on sensor count, we enforce the exact number of sensors via this equality constraint. The placement strategy is constrained to assign sensors exclusively to predefined nodes, rather than
Problem Statement

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Algorithm Two Step Constrained PPO
Require: Initial policy parameters do. initial value function parameters 1/b
Penalized Training
2: for w = .,.... Ti do
3: Collect v= = {(Gu, : aw))) by rumning T&.. (a G).
4: Compute penalized reward R = R X - ||al3)² and advantage estimates A
5: Compute log Th. (a G)) and compute ratio r(Ø).
6:
Update policy: Φw+1 arg maxş [D_] ΣG,a min A")) clip(r(0). 1 -, 1+ A
7: Fit value function 1 arg ming
[Do] ΣG,a (Value,, R
8: end for
9: # Constrained Training
for w Ti.... . do
11:
Collect D., = {[G.., an }} by running constrained Tdw (a G, a kV)).
12:
Compute reward R. and advantage estimates A
13:
Estimate log T.. (a G, 1 k)l)) and compute ratio r(Ø).
14:
Update policy: Φu+1 = arg max pal ΣG.a
A", clip (r(()( 1 - 1 1+e) AW)
15:
Fit value function M arg ming
D... EG.a (Value, RT)²²
end for
allowing arbitrary locations, thereby reflecting practical deployment considerations real-world scenarios. The objective is formally defined as
M Σ a. =
Maximize Eg (a G)] subject
We utilize the Proximal Policy Optimization (PPO) algorithm (Schulman al. 2017). and decompose this problem into two parts: (1) How to sample from T while respecting the constraints and (2) How compute the log probability log (a
Σ a, klV)).
4.1 Constrained Sampling
We first address the problem of sampling from the constrained distribution. Sampling from discrete distributions has been investigated in the literature. In particular, the (Jang al.) 2017: Maddison al.] 2017) leverages continuous relaxations reparameterize the categorical distribution by perturbing the class logits with Gumbel noise and passing them through temperature- scaled softmax. Extending this idea, reparameterizable subset sampling (Xie & Ermon 2019) generalizes the Gumbel-softmax trick to k-subset sampling, thereby rendering it amenable to backpropagation. Moreover, recent work by /Ahmed et al. (2023) employs dynamic programming to sample exactly from the constrained distribution.
approach
The sampling strategy from Ahmed al. (202k necessitates constructing dynamic programming
table of T [EM1] a; = k| V|V)) with varying |V| and k/[V], which runs in (k]V complexity. Then, the sampling algorithm runs in O(VV)) complexity. We refer the readers to Ahmed et al. (2023) for additional details. However, this method becomes computationally infeasible when applied to meshes adopt the Gumbel approach with top-k| V| selection (Kool et al. 2020). For each node vz, we sample independent Gumbel noise and compute the perturbed scores:
with many nodes. Consequently,
$ log P. + with 9: Gumble(0, 1).
Then, we select the indices corresponding to the top-k|V| largest values:
Top-k|V| indices of {s:)]//.
(5)
This procedure runs in [[VV log k VI) with min-heap streaming. Since PPO does not require differentiation through the sample, we avoid the (Ok (V2) complexity incurred by differentiable
Gumble -substt sampling by Xie & Ermon 22019)
Assuming perfect parallelization, the vectorized complexity of computing the dynamic programming table in Ahmed et al. (202)) reach (log k/V| |og V)), while sampling achieves O(log [V]. In
5

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contrast, the Gumbel approach with top-k|V] selection attains (log V]) vectorized complexity. offering significantly faster alternative.
4.2 Constrained Log Probability
We now address the problem of computing the log probability log T ΣM a; = k
By Bayes' rule, it can be expressed as log Ta(a E 0; k|V]) log Te
Σ i=1 a;
k|V[] - log & D
k)V)), where |
IV a,
] denotes an indicator function. The term
log & E ==1 a; xVI) appears intractable (Ahmed al. 2023). since a brute force computation would entail ((( complexity. An efficient method proposed by Ahmed et al. 2023) leverages
dynamic programming to compute this log probability exactly with ([+[V])). However, even this improvement remains computationally prohibitive within the context of our problem.
To address this issue, we employ the saddle point approximation (Daniels 1954) to estimate the log probability log Te &h (Ei1 a; = k|V)). The saddle point approximation provides a highly accurate method for approximating any probability distribution function and is particularly effective for the distribution of the sum of independent random variables. In the Bernoulli setting, where there are [V] independent Bernoulli variables with parameters pi, we define the cumulant generating function
[V]
(t)
log (1 + p.ct).
(6)
Consequently, the probability can be approximated by
Σ a = xa= ~
/2T (t*) exp (e((t) -k)))
(7)
where t* is the saddle point found by solving (tt)) klV In practice, it is determined using a differentiable numerical root-finding algorithm, such as Newton-Raphson, over finite number of iterations. We present the approximation complexity below. The linear complexity and vectorized log complexity are very favorable in our problem setting, where the number of mesh points could be extremely large. We refer the readers to Appendix [or for detailed proof.
Proposition 1 (Saddle Point Approximation Complexity). Suppose a, Bernouli(p.).. When the probability म (E a, k|V)) is approximated using the saddle point method, the algorithmic complexity of computing ( MM a, k\V) is O(VV). Assuming perfect parallelization, the vectorized complexity can achieve (
Additionally, we establish an error bound for the approximation, with the proof presented in Ap- pendix E This bound indicates that the relative error diminishes as the number of nodes grows, which is especially beneficial for our problem.
Proposition 2 (Saddle Point Approximation Error Bound). Let a; Bernoull((p.) Then, the asymptotic relative error aT(a I a k(Vl) when approximating no T
[V] a = k\V)) using the saddle point approximation is given by TT(a E. k(V)) a MMM a
x|VI) (1 11-00 0 -01 (I²) where Ta(a Σ. a; k\V)) denotes the ground truth probability and
(a Σ 1 a; kVV) denotes our approximated probability.
4.3
Algorithm
With (I) sampling from the constrained distribution and (2) estimating the log probability addressed, the intuitive solution is to integrate them into the standard PPO algorithm. However, such a high dimensionality. we observed that initiating the training with constrained policy for a randomly initialized model is overly restrictive, leading a failure to learn. Consequently, we propose Two-Step Constrained PPO training procedure.
In the first stage, the PPO is trained in an unconstrained manner, meaning that both the sampling and log probability computations are performed without enforcing any constraints. To softly enforce the constraints, we adopt penalized objective function as presented in Proposition
Proposition (Penalized Reward Function (from Liu et (2024). Lei E (0, 10, and assume
that constraint is 1" a k. Assume the reward function R is Lipschitz with respect a. Then,

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if a" optimizes the reward R. it also optimizes R = R - -0-(-V V - 11al11) all X > 0, and the optimal reward of R is equivalent to R.
Although converting a constrained optimization problem into its penalized form has been widely studied (Boyd& & Vandenberghe] 2004, [eetsekas 1999), these approaches typically require the convexity of the original objective function to ensure equivalence in the global optimizer. In light of recent work on non-convex optimization theory (Liu et al. 2024), we show that as long as the objective function, in this case, R. is Lipschitz, the penalization formulation ensures the global maximizer of R is also the global maximizer for R. which provides a theoretical guarantee for our training procedure. Since R represents MSE of the reconstructed data from our reconstruction network Me. the Lipschitz assumption merely requires that the gradient of the MSE is differentiable with respect to the input of Ma. This is a reasonable assumption given that training Ma demands non-exploding gradients, and the Lipschitz condition holds in experimental domain.
After training under the penalized reward setting for Ti iterations, we switch to constrained train- ing, where we sample from the constrained distribution using the Gumble Top-A/VV| discussed in Section 4.1 and compute the constrained log probability using saddle point approximation in Section 4.2 The complete algorithm is provided in Algorithm | During inference, we sample from the constrained distribution using Gumble Top-k|V].
5 Experiment
Dataset Four three-dimente -icce-iimesionall turbulent flow datasets were generated using DNS in COMSOL
(COM 2020). The datasets comprise variations in four distinct geometries: (1) Sphere with variable radius. (2) Ellipsoid with varying semi-axis lengths, (3) Cylinder with varying height and radius,
and (4) NACA 4-digit airfoil with varying chord length and thickness. The initial velocity field was randomly generated, while the inlet velocity was modeled as time-dependent and stochastic, expressed as u u(x f) = f(x f, θ + (xx θ)e,, where f(x, i, θ) specifies the prescribed inlet velocity at each spatial location and h(2, θ) controls the magnitude of the random noise term, O contains the parameters of the geometries. Although these datasets were generated by numerical simulation, they effectively mimic the dynamic behavior observed in actual turbulent flows. Code and datasets are available at Github
Task Setup and Baselines We evaluate our proposed directional transport-aware GNN (DTA- GNN) on all datasets under varying sensor placement configurations. Specifically, two sensor distribution strategies are considered: (i) Uniform, in which sensors are evenly distributed across the computational domain, and (ii) Random, in which sensor locations are selected randomly. For each strategy, sensor densities of 5%, 10%, 20%, and 30% of the total mesh points are used.
To benchmark our approach, we compare it against several baselines. First, two naive interpolation methods, Mean and k-Nearest Neighbors (KNN), are included. Additionally, we compare with DiffusionPDE (Huang et al. 2024a) and OFormer (Li et al., 2023). both of which have demonstrated strong performance in PDE forward modeling with cartial-observatti
Considering that our data are mesh-based, we also include MeshGraphNets (Pfaff et al.) (02T]) and Graph Kernel Operator (GKO)
(Li et al. 2020), which are recognized for their excellent performance in mesh-based PDE forward simulation. Moreover, Flow Completion Network (FCN) (He et al., (022), designed specifically for sparse sensor measurement interpolation, is also considered in our experiments. The details of these baseline implementations are provided in the Appendix C Performance is quantified using the MSE
of the normalized velocity and pressure fields MSE( (normalizedd Pnommulized] [Unommaizedd Pnommlizea)).
5.1 Main Results
We present the results in Table 2 Across all four geometries and under both uniform and random sensor placement strategies, our model consistently achieves the lowest reconstruction error, often by wide margin. We also observe that several baselines, including FCN, DiffusionPDE, and MeshGraphNets, achieve strong performance under particular sensor placement strategies on certain datasets. However, our reconstruction model is able to achieve consistent improvements across different shapes, sensor densities, and distributions, which demonstrates that our approach more effectively captures underlying flow field and yields robust predictions even in highly under- regimes. In the most challenging NACA 4-digit scenario, our model able to achieve approximately 10% improvements in many placement strategies.
7

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Mean
KNN
OFormer
GKO
FCN
MeshGraphNets
DiffusionPDE
DTA-GNN
5% Uniform
2 10
221.413
9.571
10.819
8.367
5.612
9.923
5.534
10% Uniform
2 10
121.119
4,400
6.745
6.153
3,537
3.755
3,092
20% Uniform
2 104
72.916
7.285
4.877
4,856
4,280
2.283
1.724
Sphere
30% Uniforn
2 10
53.369
2.532
4,346
4.754
3,537
1,775
1.204
5% Random
2 10
419.095
18.614
15.879
9.654
8,741
8,978
7,180
10% Random
2 10*
220.894
10.494
12,410
7.655
4,541
5.872
4.110
20% Random
2 10*
120.537
6,460
7.305
6.998
2.778
2.730
2.155
30% Random
2 10"
88.131
3.681
5.121
4.927
1,563
1,687
1.446
5% Uniform
2 10a
248.818
28.395
63.162
69.162
53.003
68.809
21.306
10% Uniforn
10a
136.951
20.380
49.931
41.602
18.105
43.315
9.676
20% Uniform
2 10
85,819
17.231
45,390
28.203
9.612
14.055
7.083
Ellipanid
30% Uniform
2 10
67.111
16.384
42.654
15.146
9.112
12.128
6,310
5% Random
2 10
492.094
60.837
140.946
41.612
53.010
99.697
39.033
10% Random
② 10
238.163
25.683
73.284
29.975
27.494
$7.804
14.175
20% Random
> 10
132.324
16.882
53.019
10.513
15.684
11,450
10.116
30% Random
2 10
98.768
15.197
48.623
10.267
10.705
7.728
5,930
5% Uniform
> 10*
195.235
24.665
32.694
25.983
16.684
24.340
15.901
10% Uniform
2 10a
124.619
21.659
27.479
21,511
8.353
16.612
7.932
20% Uniform
10
53.977
14.011
24.583
16.765
2.622
5.310
2,299
Cylinder
30% Uniform
2 10a
28.973
6,368
23.610
11,656
2.331
3.311
1.913
5% Random
2 10
355.989
51.832
48.321
53.574
46,980
43,895
43.723
10% Random
2 10
195.992
15.717
21.46
28.640
14.091
14.193
12.797
20% Random
2 10
108.420
8.732
12.358
26.532
6,123
6.346
5.765
30% Random
> 10
73.905
7,091
10,955
18.663
2.895
3.134
2.152
5% Uniform
10a
1735.205
65.524
87.756
102.914
78.421
72.974
59.308
10% Uniform
10
1110.690
44.384
72.311
88.425
74.549
66.277
43.647
NACA 4-digit
20% Uniform
10
769.226
31.714
54.879
52.877
32.596
50.627
28.173
30% Uniform
10
656.289
30.634
46.273
29.817
44.358
24.883
5% Random
2 10a
2623.504
148.324
156.791
113.718
103.126
97.781
97.076
10% Random
> 10
1438.99 933
57.024
92.245
71.875
62.337
60.306
56.224
20% Random
2 10
1028.617
46.368
77.994
55.630
48.519
49.497
44,748
30% Random
2 10
816.640
29.923
55.475
44.426
31.806
29,912
29.803
Table 2: Comparison MSE scaled by 10 across multiple datasets and sensor placement schemes.
Parameter Efficiency Besides the performance gain, we also show that our model is parameter efficient. We present the number parameters for various models Table 3
5.2 Sea Surface Temperature
We evaluate our reconstruction model on NOAA OISST V2 weekly mean sea surface temperature dataset recorded from De- cember 31, 1989 through January 29, 2023 (Reynolds et al.) 2008) at 1 x 1 spatial resolution and with sensors 10% of the grid locations. We train on 80% of the data, use 10% for validation and the remaining 10% for testing. We compare with three strong baselines from Section 5.1 FCN, DiffusionPDE, and MeshGraphNets.. As shown in Tabld2 approach achieves a relative improvement of 14.01% in MSE over the strongest competing method DiffusionPDE.
Model
MSE
FCN
15.761
DiffusionPDE
7.969
MeshGraphNets
8.176
DTA-GNN
6,853
Figure 2: Comparison of different
models on the global ocean surface temperature dataset. All numbers are scaled by 10-2
Figure illustrates representative reconstructions for both summer and winter seasons. addition to accurately recovering the large-scale seasonal cycle and major ocean gyre structures, model

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also succeeds at resolving the much smaller-scale temperature anomalies that arise from equatorial upwelling, coastal current meanders, and tropical instability waves. These fine-scale features play an outsized role in modulating air-sea heat fluxes and driving interannual phenomena such as El Miio-Southern Oscillation. Our model accurately reconstructs both the global trend and these small, dynamically driven variations with high fidelity. We present additional visualizations in Appendix N
Figure 3: Visualization of ground truth sea surface temperature data and reconstructed sea surface temperature data. The first row corresponds to ground truth data and the second row corresponds reconstructed data from DTA-GNN.
5.3 Optimal Sensor Placement
further enhance the reconstruction accuracy of our model by identifying optimal sensor locations. After training the reconstruction model M& described in Sectior 5.1 we incorporate it into the reward function of the Two Step Constrained PPO. We refer the readers to Appendix [ for training and model hyperparameters. We consider two standard baselines. uniform placement and random placement, and assume that 10% of the mesh points have sensors.
Figure4 reports the MSE of the reconstructed data for each method. Our sensor placement policy achieves approximately a 15% reduction MSE relative to uniformly placed sensors. The perfor- mance gain can be attributed to concentrating sensors in regions of fluid high variability, such as high velocity and pressure gradient. Additionally, our policy accounts for performance disparity in reconstruction models. By explicitly optimizing for both criteria, our policy achieves significant improvements in reconstruction accuracy.
1190
-Us83m
+190
4Urdum
+Rarsee
Sphere
(b) Ellipsoid
(c) Cylinder
(d) NACA 4 digits
Figure 4: Comparison of sensor placement schemes across datasets. We assume that 10% of the mesh points have sensors. All numbers are scaled by 10-4
Conclusion
We introduce realistic problem formulation for fluid-field reconstruction and design a directional transport-ane aware GNN that achieves superior reconstruction accuracy across multiple datasets and mansor-placement configurations. We observe that conventional sensor placement algorithms often fail to identify optimal sensor locations, and we propose a Two Stage Constrained PPO training procedure train sensor placement policy that yields additional improvements.

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Jang, E., Gu, S., and Poole, B. Categorical reparameterization with gumbel-softmax. International http://///ppeeeeeeeeeeefforeeeee 7 Acknowledgments
This work was partially supported by NSF Center for Computer Assisted Synthesis (2202693), National Artificial Intelligence Research Resource (NAIRR) Pilot (240280, 240443), National Science Foundation (2106859, 221 2211557, 2119643, 2200274, 2303037, 2312501, 2531008), National Institutes of Health (U54HG012511) U24DK097771, U54OD036472), NEC, Optum AI, SRC JUMP 2.0 Center, Amazon Research Awards, and Snapchat Gifts.
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A Limitations
Our work uses datasets simulated with DNS Comsol and adopts time-dependent and stochastic inlet velocity to better reflect real-world scenarios. Since there are large-scale CFD datasets collected from the real world, we believe that our simulated datasets better represent real-world scenarios. However, we acknowledge that models trained using such simulated datasets could have performance degradation when inferring on real-world datasets. The performance degradation could be attributed to the inaccuracies from real-world sensors. Future work could focus conducting extensive real-world experiments to collect real-world datasets and enhance the real-world applicability of related works.
B Broader Impacts
Our reconstruction framework and optimal sensor placement strategy offer transformative potential across aerospace, automotive, and environmental engineering. In the aerospace sector, for example, it can dramatically reduce wind tunnel testing time by inferring complete flow fields from just a handful of strategically positioned probes. In automotive design, it paves the way for rapid, cost-effective acrodynamic optimization by filling in the gaps between sparse on-vehicle measurements.
By accurately reconstructing full CFD fields from partial observations, our approach lowers both the logistical and financial barriers to advanced flow analysis. enables engineers and researchers to iterate designs more quickly, run fewer physical experiments, and integrate real-time monitoring into digital-twin platforms. Ultimately, this work democratizes access to high-fidelit, flow data, accelerates research and development cycles, and fosters deeper scientific insight across multiple disciplines.
C Baseline and Model Implementation Detail
OFormer We adopt the model from Li et al. (2023). The hidden dimension is set to 64. We add separate MLP in the encoder to encode the binary mask indicating sensor placement. The mask encoder latent dimension is set to 16. The number of attention blocks is set to 2. The models are trained with batch size 16 for 300 epochs. We use a cosine annealing learning rate with a starting learning rate at 1 x 10-4 to an end learning rate at 1 x 10-5
GKO We adopt the model from al. (2020). We add separate MLP in the encoder to encode the binary mask indicating sensor placement. The mask encoder latent dimension is set to 16. We choose width=256 and depth=6. The models are trained with batch size 16 for 300 epochs. We use cosine annealing learning rate with a starting learning rate of 1 x 10-4 to end learning rate at 1 x 10-5
FCN We adopt the model from He et al. (2022). The model consists of three graph convolution layers and two spatial gradient attention layers. The latent dimension is set to 64 with a separate MLP to encode the binary mask indicating sensor placement. The mask encoder latent dimension is set to 16. The models are trained with batch size 16 for 300 epochs. We use a cosine annealing learning rate with starting learning rate at x 10 to end learning rate 1 x 10-5
DiffusionPDE We adopt the model from Huang et al. (2024a). The score network in DiffusionPDE has been modified by incorporating bessage-passing graph neural networks, addressing the limitation that the original UNet architecture in DiffusionPDE: is not directly compatible with mesh data. We include 6 message passing layers with latent size 64. The encoder is the same as our methods. We use layers of MLPs with relu activation functions for each encoder block. The latent dimension for the encoding of the binary mask is 16 and the other encoder's 's latent dimensions are 64. For diffusion parameters, 3min = 0.0001 and 8max = 0.02. The total number of diffusion steps is 1000. The models are trained with batch size 16 for 300 epochs. We use a cosine annealing learning rate with a starting learning rate at x 10 to an end learning rate at 1 x 10-5
MeshGraphNets We adopt the model from Pfaff et al. (2021). We include 6 message passing layers with latent size 64. The encoder is the same as our methods. We use layers of MLPs with relu activation functions for each encoder block. The latent dimension for the encoding of the binary mask is 16 and the other encoder's latent dimensions are 64. The decoder consists of single 4-layer MLP with relu activation functions. The models are trained with batch size 16 for 300 epochs. We
20

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44 to end learning rate at use cosine annealing learning rate with starting learning rate of 1 x 10-4 1 10-5
Ours We include 6 message passing layers with latent size 64. We use 4 layers of MLPs with relu activation functions for each encoder block. The latent dimension for the encoding of the binary mask is 16 and the other encoder's 's latent dimensions are 64, The decoder consists of single 4-layer MLP with relu activation functions. The models are trained with batch size 16 for 300 epochs. We use a cosine annealing learning rate with starting learning rate at x 10 to an end learning rate 10-5.
Model Parameters We present the number of parameters for different models in the following table:
Model
parameters
OFormer
727,316
GKO
336,228
FCN
388,836
DiffusionPDE
317,236
MeshGraphNets
315,412
Ours
266,260
Table 3: Comparison of parameter counts for various models.
D Proof of Proposition i
Suppose a; ~ (p.oo The cumulative generating function and its derivatives have closed
form:
V
(tt)
Σ
log (1 P: + pree)
(8)
M
Pict
1 (1)
(9)
+ Dset
pic² (1 -Pi)
(0())
Σ
(10)
(1 pi + +2,et)2
(11)
Their evaluations all cost O(V)). Newton's method converges quadratically, so to reach an error tolerance e, we need Olog ()). Thus, the overall runtime O(V log log({)). Since specify fixed precision and a maximum number of iterations (this is chosen to be a small number, since Newton Raphson generally converges fast), the runtime becomes ([VV).. For vectorized computation, computing (t), vp (t), and 10 (t) take (log [V) complexity and the overall runtime becomes (log [V]).
E
Proof of Proposition 2
Let a; ~ Bernoulli(p.). Then,
V
k|Vl) p(a) Vl
= A|VI
p(a
p( M( a, k/VI)
i=1
21

---

In the standard asymptotic regime, the relative enror in the likelihood estimated using the saddle point approximation is of order
[V (Goodman
2022). Then,
p(a) p(a) !N K|VI]
p(a
Σ 01
k(V))
~
pl 1 a; k[V1(1 0(+))
IV
R p(a
Σ
a; x|Vl) 1+0
i=1
By Taylor Expansion,
p(a
a [V p(a
Σ a. -V
F Hyperparameters for Optimal Sensor Placement
The actor and critic network consists of layers of message-passing GNNs with latent dimension 128. The value predicted by the critic is taken the mean of the decoded graph. À from the penalized reward function is taken as 0.00015. The number of gradient descent steps is 5, and we adopt a clip value of 0.2. We train under the penalized scheme for 1500000 steps (Tī 1500000) and under the constrained scheme for 500000 steps (72 = 2000000). We employ a cosine learning rate for both actor and critic with starting learning rate 10-5 and end learning rate 10-7
G Additional Experiment Results on Optimal Sensor Placement
Techniques that determine sensor placement through QR pivoting and SVD assume that high- dimensional states be effectively represented by latent low-dimensional structures, inherent compressibility that enables sparse sensing. particular, a high-dimensional state æ E R" is often assumed to have compact representation in suitable transform basis & Raxr, so that
I 4s where s E RR" is sparse and and r < n. The objective to design a measurement matrix C p with small number p of optimized measurements, such that the measurement vector y C E R enables accurate reconstruction of 8, and consequently æ.
We provide the reconstructed MSE in Table tested in the Sphere dataset. We notice that the reconstruction MSE from sensor locations determined by QR Pivoting and d-optimal is significantly higher than uniform randomly placed sensors. Note that for randomly placed sensors, randomly sample sensor locations for every frame of the data and then feed it into the reconstruction network, whereas the sensor locations for QR Pivoting and d-optimal are fixed for the entire trajectory.
Method
MSE
Uniform
3.092
Random 4.110
QR Pivoting
6.988
d-optimal
6.309
Table 4: Comparison of sensor locations Sphere datasets with sensor density of 10%. We report the MSE scaled by 10-4
H Independence Assumption of Random Variables
We assume that the random variables are independent. This assumption is reasonable because each sensor provides measurements only at its specific location, and the removal of one sensor does not directly influence the others. However, when the total number of sensors constrained, statistical correlation is introduced through the equality constraint.
22

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I Experiments on Turbulence Data
We compare our proposed DTA-GNN against three strong baselines, FlowCompletionNetworr (FCN), DiffusionPDE,, and MeshGraphNets, on two benchmark datasets: Kolmogorov Flow and Taylor-Green Vortex. Both datasets are generated via high-resolution numerical simulations using pseudo-spectral solver governed by the incompressible Navier-Stoker equations. The Kolmogorov Flow dataset features a time-dependent sinusoidal external forcing and is simulated at Reynolds number of 2000. The Taylor- Green Vortex dataset is initialized from its analytical solution and perturbed with Gaussian noise to produce a variety of flow trajectories. Simulations are carried out at Reynolds number of 1 1500. The performance of each method these datasets is summarized in the table below:
Table 5: Performance comparison on Kolmogorov Flow.
Method
5% Random 10% Random 20% Random
30% Random
Ours FCN
9.484
8.537
4,510
3.443
10.547
9.1557
5.109
4.690
DiffusionPDE:
11.214
9.458
6.699
4.703
MeshGraphNets
10.180
9.283
5.271
4.137
Table 6: Performance comparison on Taylor Green Vortex.
Method
5% Random
10% Random
20% Random
30% Random
Ours
6.749
4.856
2.643
1.267
FCN
8.430
6.354
3.590
2.825
DiffusionPDE
8.898
5.348
3.142
1.974
MeshGraphNets
9.801
5.677
3.119
2.831
J Generalisability Experiments
We evaluate the generalization capability our proposed DTA-GNN by training it on the Ellipsoid dataset and testing it the Sphere dataset. Notably, the Ellipsoid dataset contains sphere geometries, ensuring that the test domain represents a previously unseen configuration. Furthermore, the two datasets differ in their inlet velocity profiles, resulting in entirely distinct flow fields. Despite these differences, as shown in the table below, DTT-GNN demonstrates strong generalization performance under these shifted conditions. all test scenarios, except for the 30% Random setting (x% Random refers to sensors randomly distributed at x % of the mesh points), DTTA-GN consistently outperforms all baseline models, maintaining superior accuracy the unseen flow distributions.
Dataset
MSE (trained on this dataset)
Generalization MSE
% Drop
5% Random
7.180
7.650
6.54%
10% Random
4.110
4.468
8.71 %
20% Random
2.155
2.328
8.03%
30% Random
1.446
1.574
8.86%
K
Ablation Studies on DTA-GNN
We conduct ablation studics in the following table. ABL d means DTA-GNN without directional information in the message passing stage. ABL_diff means DTA-GNN without the difference between neighboring latent states. We simply concatenate the neighboring latent states and multiply with the direction information. MeshGraphNets corresponds to removing both the directional information and difference between neighboring latent states.
Based on these results, we draw three key conclusions: (1) Directional information plays critical role when sensor density is low or when sensors are placed randomly, as some regions may lack sufficient sensor coverage. (2) Relying solely the difference between neighboring latent states is suboptimal, as this approach does not capture edge attributes or directional cues essential for accurate information transfer. (3) Simply concatenating neighboring latent states leads to a moderate drop in performance,
23

---

Dataset
MeshGraphNets
ABL
ABL diff
DTA-GNN
5% Uniform
78.421
75.124
63.162
59.308
10% Uniform
74.549
59.064
52.294
43.647
20%
Uniform
32.596
30.368
30.660
28.173
30% Uniform
29.817
26.923
27.497
24.883
5% Random
103.126
155.466
101.878
97.076
10% Random
62.337
73.355
60.536
56,224
20% Random
48.519
49.210
48.622
44.748
30% Random
31.806
32.389
30.943
29.803
indicating that explicitly computing transported information is beneficial. Nevertheless, since message passing networks can still approximate difference operators, the performance degradation is less severe than when directional information is entirely removed.
L Ablation Studies on Two-Step Constrained PPO
We report the ablation studies on Two-Step Constrained PPO in the following table. Penalized PPO refers to the first stage, where we use a penalized objective function to guide the model toward the constraints. During inference, Constrained PPO refers to the second stage, where we use Gumble Top-k for sampling and saddle point approximation for computing the log probability. Two-Step PPO without Saddlepoint Approx refers to the original Two-Step Constrained PPO, but removing saddle point approximation in the second stage. In the second stage,
Gumble Top-k to strictly enforce the equality constraints.
unconstrained log probability.
Shape
Penalized PPO
Constrained PPO
without Saddlepoint Approx
Sphere
3.270
19.280
3.178
Ellipsoid
11,002
31.647
10.629
Cylinder 9.490
55.014
9.575
Airfoil
46.431
327.966
44.103
From these results, we draw several important conclusions: (1) As discussed in the main paper, initiating constrained training from randomly initialized policy overly restrictive and substantially limits the model's 's performance. (2) Although the penalized training in the first stage helps guide the model toward reasonable sensor placement strategy, does not strictly enforce the equality constraint. Consequently, when combined with a constraint-copplian sampling strategy during inference, its performance degrades. (3) Accurately computing the log probability the second-stage constrained training is essential. Without the saddle point approximation, in the two-Step PPO without Saddle Point Approximation, the model fails outperform even the baseline of uniformly placed sensors.
M Hardware Specification
We implement all models in PyTorch. All experiments are run on servers/worrkstation with the following configuration:
80 CPUs, 503G Mem, 8 NVIDIA V100 GPUs.
48 CPUs, 220G Mem, 8 x NVIDIA TITAN Xp GPUs.
96 CPUs, 1.OT Mem, 8 x NVIDIA A100 GPUs.
64 CPUs, 1.OT Mem, 8 NVIDIA RTX A6000 GPUs.
224 CPUs, 1.5T Mem, 8 NVIDIA L40S GPUs.
24

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N Additional Experiment Results on Sea Surface Temperature
187
470
1291
47A
DYA4
679
601
27A
1724
0/4
1178
07A
17/
Figure 5: Visualization of ground truth sea surface temperature data and reconstructed sea surface temperature data. The first column corresponds to ground truth data and the second column corre- sponds to reconstructed data from our reconstruction model.
25


Reviews:
[{"review_id": "12V5rUaCRQ", "reviewer": "Reviewer_5qAz", "summary": "The authors proposed a directional transport-awareGraph Neural Network (GNN) that explicitly encodes both flow directionality and\ninformation transport. The authors also studies the problem of optimal placement of sensors for enhanced reconstruction accuracy. They proposed a two-step constrained proximal policy optimisation based algorithm for that purpose.", "questions": "1. See Weaknesses 1-4.\n2. Do you assume the random variables $a_i,~i \\in \\{1,2,\\cdots,|V|\\}$ to be mutually independent? If yes, does this assumption make sense when a_i's are observables at nodes in the graph, and usually they are correlated?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The proposed method offers significant performance improvements over the related baselines across various datasets of differing geometries.\n2. The proposed algorithm offers improvement over existing sensor placement strategies.\n\nWeaknesses:\n\n1. **Novelty of direction information:** Direction information has been incorporated in [2] for GNNs on fluid reconstruction problem, so what are the differences with [2] and the authors directional transport aware GNN? \n2. **Lacking experiments on turbulence data:** The paper is lacking experiments on turbulent flow. Can the authors provide experimental results on datasets similar to the Forced or Decayed turbulence as in [6,8]?\n3. **Lacking Generalisability experiments:** The paper is lacking study on how the trained model generalises to unknown physical conditions as in [9]?\n4. **Insufficient analysis:** It is important to assess if the reconstructed flow field accurately predicts Force coefficients e.g. lift and drag coefficients for practical applications as done in existing works [1,2, Zhong et al. 2023 in the paper]. Have the authors analysed the boundary layer profiles of the reconstructed flow filed on the datasets?\n\n5. Related works are quite light, missing many important recent papers: [1-9]\n\n[1] AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier–Stokes Solutions, NeurIPS 2022. \n\n[2] Finite Volume Features, Global Geometry Representations, and Residual Training for Deep Learning-based CFD Simulation, ICML 2024. \n\n[3] Topology-aware Neural Flux Prediction Guided by Physics, ICML 2025.\n\n[4] PDE-Transformer: Efficient and Versatile Transformers for Physics Simulations, ICML 2025.\n\n[5] PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems, ICLR 2025.\n\n[6] Multi-Order Loss Functions For Accelerating Unsteady Flow Simulations with Physics-Based AI, CAI 2024. \n\n[7] Learning to simulate complex physics with graph networks, ICML 2020\n\n[8] Machine learning–accelerated computational fluid dynamics, PNAS 2021\n\n[9] Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction, ICML 2020.", "was_revised": true, "final_justification": "Most of my concerns are resolved.", "rebuttal": "We thank you for your insightful comments on improving our paper. We hope the following could address your concerns.\n\n## Novelty of Direction Information\n\nThe Directional Integrated Distance (DID) proposed in [1] is fundamentally different from the approach presented in our work. Below, we highlight the key distinctions across several dimensions:\n\n1. **Problem Definition**\nThe work in [1] addresses the task of predicting velocity and pressure fields for steady-state simulations. Their model takes mesh coordinates as input and directly outputs steady-state velocity and pressure. In contrast, our work focuses on flow field reconstruction, where the input includes both the mesh and partially observed velocity and pressure data. The goal is to reconstruct the velocity and pressure at unobserved mesh points. Thus, our model must learn not only the underlying physical dynamics but also how to effectively integrate available observations for accurate reconstruction.\nMoreover, the simulations in [1] assume a constant inlet velocity, leading the solution to converge to a steady state. In our case, the inlet velocity is time-dependent and stochastic, making it infeasible to infer the full flow field from mesh geometry alone.\n\n2. **Semantic Meaning of DID vs. Our Directional Information** \nIn [1], the authors note that mesh coordinates provide limited information about immersed objects and their surroundings. To address this, they propose DID, which augments mesh coordinates by encoding the average distance to domain boundaries over various angular sectors.\nIn contrast, our directional information encodes the alignment of a node with the local direction of information transfer. This metric is influenced not only by mesh geometry but also by observed sensor data.\n\n3. **Role Within the Model**  \nThe DID in [1] is a precomputed geometric feature derived from mesh coordinates and is used as an additional input to their neural network. Our directional information, by comparison, is computed dynamically during message passing using a node's latent representation and edge features. It plays a functional role in guiding the aggregation of information.\n\n\n----------\n\n\n## Experiments on Turbulence Data\nWe compare our proposed DTA-GNN against three strong baselines, FlowCompletionNetwork (FCN), DiffusionPDE, and MeshGraphNets, on two benchmark datasets: Kolmogorov Flow and Taylor-Green Vortex. Both datasets are generated via high-resolution numerical simulations using a pseudo-spectral solver governed by the incompressible Navier–Stokes equations.\nThe Kolmogorov Flow dataset features a time-dependent sinusoidal external forcing and is simulated at a Reynolds number of 2000. The Taylor-Green Vortex dataset is initialized from its analytical solution and perturbed with Gaussian noise to produce a variety of flow trajectories. Simulations are carried out at a Reynolds number of 1500. The performance of each method on these datasets is summarized in the table below:\n\n**Kolmogorov Flow**\n| Method| 5% Random | 10% Random | 20% Random | 30% Random |\n|-|-|-|-|-|\n| Ours| 9.484    | 8.537| 4.510| 3.443|\n| FCN | 10.547   | 9.1557| 5.109| 4.690|\n| DiffusionPDE | 11.214 | 9.458| 6.699| 4.703|\n| MeshGraphNets | 10.180 | 9.283| 5.271| 4.137 |\n\n**Taylor Green Vortex**\n| Method | 5% Random | 10% Random | 20% Random | 30% Random |\n|-|-|-|-|-|\n| Ours | 6.749    | 4.856  | 2.643 | 1.267 |\n| FCN | 8.430    | 6.354  | 3.590 | 2.825  |\n| DiffusionPDE | 8.898 | 5.348 | 3.142   | 1.974 |\n| MeshGraphNets| 9.801 | 5.677 | 3.119  | 2.831|\n\n\n----------\n\n\n## Generalisability Experiments\nWe evaluate the generalization capability of our proposed DTA-GNN by training it on the Ellipsoid dataset and testing it on the Sphere dataset. Notably, the Ellipsoid dataset contains no sphere geometries, ensuring that the test domain represents a previously unseen configuration. Furthermore, the two datasets differ in their inlet velocity profiles, resulting in entirely distinct flow fields.\nDespite these differences, as shown in the table below, DTA-GNN demonstrates strong generalization performance under these shifted conditions. In all test scenarios, except for the 30% Random setting (x% Random refers to sensors randomly distributed at x% of the mesh points), DTA-GNN consistently outperforms all baseline models, maintaining superior accuracy on the unseen flow distributions.\n\n|      | MSE for model trained on this dataset | Generalization MSE | % Drop in Performance |\n|-------------|-|----------|----------------|\n| 5% Random |7.180 | 7.650 | 6.54%  |\n| 10% Random | 4.110 | 4.468 | 8.71%  |\n| 20% Random | 2.155 | 2.328 | 8.03% |\n| 30% Random| 1.446 | 1.574| 8.86% |\n\n\n\n----------\n\n\n## Analysis on Lift and Drag Coefficients\nWe report the MSE on lift and drag coefficients. The experiments are conducted on the airfoil dataset with 10% uniformly placed sensors.\n\n| Method | lift mse| drag mse |\n|-|-|-|\n| Ours| 2.54e-06 | 3.42e-07 |\n| MeshGraphNets | 7.43e-06  | 1.15e-06 |\n| DiffusionPDE| 4.47e-05 | 2.17e-06  |\n| FlowCompletionNetwork | 8.27e-06| 5.37e-06 |\n\n\nAs shown in the results, our proposed model achieves the lowest error, indicating superior performance compared to baseline methods. This suggests that the model not only captures local flow patterns but also effectively reconstructs the global structure of the flow field. Moreover, the accuracy on lift and drag coefficients highlights the practical utility of our approach for downstream tasks such as aerodynamic analysis and design optimization.\n\n\n----------\n\n\n## Related Work\nWe thank the reviewer for highlighting these related works. When preparing the manuscript, our primary focus was on studies concerning flow field reconstruction, as this is the central theme of our work. However, we agree that some papers on physical simulation and neural PDE solvers are also relevant to our approach. As we are unable to revise the manuscript at this stage, we will ensure that these works are included and properly discussed in future revisions.\n\n\n----------\n\n\n## Independence Assumption of Random Variables\nWe assume that the random variables are independent. This assumption is reasonable because each sensor provides measurements only at its specific location, and the removal of one sensor does not directly influence the others. However, when the total number of sensors is constrained, statistical correlation is introduced through the equality constraint.\n\n\n----------\n\n\nWe hope this clears up any confusion and concerns, and if so, we would greatly appreciate your consideration in raising the score.\n\n[1] Finite Volume Features, Global Geometry Representations, and Residual Training for Deep Learning-based CFD Simulation, ICML 2024."}, {"review_id": "MZ2tlmGJ9Y", "reviewer": "Reviewer_dAAG", "summary": "This paper proposed a realistic flow field reconstruction method based on GNN and a two-step constrained PPO for policy learning. A directional transport-aware GNN is designed to explicitly encode both flow directionality and information transport. This method can also allow for arbitary sensor placement and 3D domains. Extensive experiments prove that the proposed method can achieve state-of-the-art performance.", "questions": "Actually, I am not familiar with this research field, can authors clarify more about the Proposition 3 part? And what additional techniques are proposed based on existing method. Even though authors have supplemented with an algorithm illustration in the main maniscript about Two-Step Constrained PPO, it is still little confusing to researchers who have limited knowledge in this domain.", "limitations": "Please refer to the weakness and question parts.", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The method releases lots of problem assumptions like arbitary sensor placement, which is more close to realistic flow scenes.\n\n2. The proposed novelty of transport-aware GNN and PPO seems to be resonable.\n\n3. The method is extensively evaluated and proved effectively in the experiments, outperforming existing methods.\n\nWeakness:\n\n1. The paper lacks adequate ablation studies for each designed novelty. Incorporating these experiments can further strengthen the paper's solid contributions. For example, for the Directional Transport-Aware GNN, Two Step Constrained PPO, authors can give the experimental results without these designs.\n\n2. The runtime of the proposed method can be tested and compared with other learning-based methods, like DiffusionPDE.\n\n3. Since the latest compared baselines in this work is DiffusionPDE (NeurIPS'24), which is a little out-of-date, could authors compare the model with more recent methods in 2025?", "was_revised": true, "final_justification": "Most of my concerns are tackled, but I would follow other reviewers because of low overlap with my expertise.", "rebuttal": "We thank you for your insightful comments on improving our paper. We hope the following could address your concerns.\n\n## Ablation Studies on DTA-GNN\n\nWe conduct ablation studies in the following table. ABL_d means DTA-GNN without directional information in the message passing stage. Formally, the edge equation in Equation 2 becomes\n$$\n\\\\tilde{e}_{i,j}^{\\\\ell} \\\\leftarrow \\\\mathcal{T} \\\\bigl(\\\\tilde{v}_j^{\\\\ell-1} - \\\\tilde{v}_i^{\\\\ell-1}\\\\bigr)\n$$\nABL_diff means DTA-GNN without the difference between neighboring latent states. We simply concatenate the neighboring latent states and multiply with the direction information. MeshGraphNets corresponds to removing both the directional information and difference between neighboring latent states.\n\n|   | MeshGraphNets | ABL_d   | ABL_diff | DTA-GNN |\n|-|-|-|-|---|\n| 5% Uniform  \t| 78.421 | 75.124| 63.162  | 59.308 |\n| 10% Uniform| 74.549 | 59.064| 52.294  | 43.647 |\n| 20% Uniform | 32.596 | 30.368 | 30.660  | 28.173 |\n| 30% Uniform | 29.817 | 26.923 | 27.497  | 24.883 |\n| 5% Random | 103.126 | 155.466 | 101.878 | 97.076 |\n| 10% Random | 62.337 | 73.355 | 60.536  | 56.224 |\n| 20% Random | 48.519 | 49.210 | 48.622  | 44.748 |\n| 30% Random | 31.806 | 32.389 | 30.943  | 29.803 |\n\nBased on these results, we draw three key conclusions:\n\n(1) Directional information plays a critical role when sensor density is low or when sensors are placed randomly, as some regions may lack sufficient sensor coverage.\n\n(2) Relying solely on the difference between neighboring latent states is suboptimal, as this approach does not capture edge attributes or directional cues essential for accurate information transfer.\n\n(3) Simply concatenating neighboring latent states leads to a moderate drop in performance, indicating that explicitly computing transported information is beneficial. Nevertheless, since message passing networks can still approximate difference operators, the performance degradation is less severe than when directional information is entirely removed.\n\n----------\n\n## Ablation Studies on Two-Step Constrained PPO\nWe report the ablation studies on Two-Step Constrained PPO in the following table. Penalized PPO refers to the first stage, where we use a penalized objective function to guide the model toward the constraints. During inference, we use Gumble Top-k to strictly enforce the equality constraints. Constrained PPO refers to the second stage, where we use Gumble Top-k for sampling and saddle point approximation for computing the log probability. Two-Step PPO without Saddlepoint Approx refers to the original Two-Step Constrained PPO, but removing saddle point approximation in the second stage. In the second stage, we use unconstrained log probability.\n\n| Shape| Penalized PPO| Constrained PPO| Two-Step PPO without Saddlepoint Approx |\n|-|-|-|-|\n| Sphere | 3.270 | 19.280| 3.178|\n| Ellipsoid | 11.002| 31.647| 10.629 |\n| Cylinder | 9.490| 55.014| 9.575|\n| Airfoil| 46.431| 327.966  | 44.103 |\n\nFrom these results, we draw several important conclusions:\n\n(1) As discussed in the main paper, initiating constrained training from a randomly initialized policy is overly restrictive and substantially limits the model’s performance.\n\n(2) Although the penalized training in the first stage helps guide the model toward a reasonable sensor placement strategy, it does not strictly enforce the equality constraint. Consequently, when combined with a constraint-compliant sampling strategy during inference, its performance degrades.\n\n(3) Accurately computing the log probability in the second-stage constrained training is essential. Without the saddle point approximation, as in the Two-Step PPO without Saddle Point Approximation, the model fails to outperform even the baseline of uniformly placed sensors.\n\n----------\n\n## Runtime and Memory Comparison\nWe report the runtime and memory consumption of various methods in the table below. The results demonstrate that our proposed model is both memory-efficient and computationally fast. This is consistent with its parameter-efficient design, as it uses the fewest parameters among all compared models.\n\n| Method                | Memory (MB) | Runtime (ms) |\n|-----------------------|-------------|--------------|\n| Ours                  | 434.16      | 66.31        |\n| MeshGraphNets         | 482.40      | 70.50        |\n| Diffusion PDE         | 491.98      | 715.65       |\n| FlowCompletionNetwork | 584.65      | 64.16        |\n| OFormer               | 1584.08     | 81.14        |\n| GKO                   | 909.80      | 79.13        |\n\n----------\n\n## Additional Baseline Comparison\nWe include [1] as our baseline and conduct experiments on the airfoil dataset. Results show that our DTA-GNN remains the best method.\n\n|        | MSE    |\n|---------------|----------|\n| 5% Uniform  | 117.122  |\n| 10% Uniform   | 75.871   |\n| 20% Uniform   | 49.478   |\n| 30% Uniform  | 30.119   |\n| 5% Random   | 130.775  |\n| 10% Random    | 64.194   |\n| 20% Random   | 51.404   |\n| 30% Random   | 34.317   |\n\n----------\n\n## Additional Clarification on Proposition 3 and Two-Step Constrained PPO\nGiven the constrained problem defined in Equation 3, a natural approach is to directly apply constrained sampling(Section 4.1) and constrained log probability computation (Section 4.2), forming the basis of the Constrained Training Stage. However, as demonstrated in the ablation studies of the Two-Step Constrained PPO, enforcing the equality constraint from the outset is overly restrictive for a randomly initialized sensor placement policy, often leading to suboptimal solutions.\nThis limitation motivates Proposition 3, which reformulates the constrained objective into a penalized objective function. Under this formulation, the policy is not required to satisfy the constraint exactly; instead, the objective function encourages the policy to produce outputs that approximately respect the constraint. Nevertheless, the ablation results indicate that this penalized formulation alone is insufficient for producing an effective placement policy for inference, as it lacks architectural mechanisms to enforce the constraint precisely.\n\nTo address this, we propose the Two-Step Constrained PPO. In the first stage, we use Penalized Training to guide the policy toward constraint-compliant behavior, helping it to learn a meaningful initialization. In the second stage, we transition to Constrained Training, incorporating architectural components that strictly enforce the equality constraint. This two-stage procedure enables the model to achieve both feasibility and optimality in sensor placement.\n\n\n----------\n\nWe hope this clears up any confusion and concerns, and if so, we would greatly appreciate your consideration in raising the score.\n\n\n[1] Qitian Wu and Chenxiao Yang and Kaipeng Zeng and Michael Bronstein, Supercharging Graph Transformers with Advective Diffusion, ICML 2025"}, {"review_id": "6Wp2ZnaCXf", "reviewer": "Reviewer_Qp1z", "summary": "This paper focuses on CFD data reconstruction from partial data. Considering an underlying mesh-based discretization with some sensors on the mesh boundary capable\nof capturing some properties (pressure, velocity), the goal is to inferre the density and velocity field on the full mesh. The paper comes with two distinct contributions. \nThe first one proposes an extended  GNN architecture with transport capabilities. Experiments comparing with several other SOAT architectures show that the proposed one achieves\nbest performance   in this specific context of mesh-based reconstruction from partial data. Then a second contribution focused on optimizing sensor placement given a certain sensor budget. \nThe solution proposed is based on an adaptation of the classical PPO RL algorithm. Results comparing to basic baselines show some improvements.", "questions": "- Motivate your choice for the PPO algorithm and why the QR-pivoting and SVD approach fail in your experiments. Also comment on the recent paper (likely published after the submission) Diff-SPORT. \n\n- Can you comment on comparing a DTA-GNN and the same GNN without the DTA part (MeshGraphNet is probably close)? \n\n- Appendix G: why changing the sensor location at each frame ? Would expect the sensor to be fixed.", "limitations": "Limitations could be more thoroughly discussed and included in the core paper (currently in the appendix)", "rating": 3, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Clarity: The paper is generally clear and well written. The exposition of the contributions is somehow not well balanced, the PPO approach with the associated sampling issue  taking a large amount of space while experimental validation is very slim. \n\nQuality: \n- Dataset: mesh sizes, topology. Some more elements and some visuals would be welcome. It's difficult\n  to figure out how difficult the problem tested are. \n- Not totally clear how the data set is split into a validation and training set. \n- Experiments for the GNN:\n  - Several of the NNs used for comparison have been modified to work with mesh. This is explained in the appendix. Should be mentioned in the core paper as this can make a significant divergent architecture from the one proposed in the original paper.\n  - The central proposed adaptation to GNN architecture (DTA-GNN) is a inner product to compute d, but proved effective as demonstrated by the experiments. I miss in the experiments a comparison with a GNN without the transport aspect and one with it, but no other change, to measure more directly the gain that adding d brings.\n- PPO:\n  - RL algorithms like PPO are effective to learn action policies to curb a trajectory toward a high cumulated reward. Optimal sensor placement does not naturally fit into that category.  Approaches like simulation based inference sound more directly related to this inverse problem of sensor placement. Could you develop in the paper some rational about the path taken and the choice of PPO beyond a validation by experiment ? \n   - Experimental results show some gains compared to uniform and random sampling, but there is no discussion or number giving the cost of the PPO based approach. Should at least be discussed in the weaknesses.\n   - QR pivoting and SVD are giving worst result than the base uniform/random strategies (in appendix). Why ? It eventually affects the experimental results as the approach proposed is only compared against very basic schemes.\n\n\n\nSignificance and Originality: the paper shows several weaknesses first on motivating the approach chosen, then in the experiments. This affects the quality and potential impact of the contribution.", "was_revised": true, "final_justification": "Many concerns raised in my review have been adressed during the rebuttal.", "rebuttal": "We thank you for your insightful comments on improving our paper. We hope the following could address your concerns.\n\n## Dataset Description\n\nThe sphere dataset contains 100 trajectories, each with a trajectory length of 500. We vary the sphere radius to create different trajectories. The minimum mesh size is 1794, and the maximum mesh size is 74148.\nThe ellipsoid dataset contains 200 trajectories, each with a trajectory length of 250. We vary the three semi-axes to create different trajectories. The minimum mesh size is 3314, and the maximum mesh size is 94544.\nThe cylinder dataset contains 100 trajectories, each with a trajectory length of 500. We vary the height and radius to create different trajectories. The minimum mesh size is 3450, and the maximum mesh size is 76304.\nThe airfoil dataset contains 50 trajectories, each with a trajectory length of 1000. We vary the maximum camber and maximum thickness to create different trajectories. The minimum mesh size is 2519, and the maximum mesh size is 53786.\n\n----------\n\n## Train, Valid, Test Split\n\nWe use 80% of the trajectories for training, 10% for validation, and 10% for testing. This means that the testing data are always out of distribution in terms of geometry configurations and inlet velocity.\n\n\n----------\n\n## Description on Baseline Modification\nWe thank the reviewer for the suggestion. As we are unable to modify the paper during the rebuttal stage, we will implement changes in our future versions.\n\n----------\n\n## Ablation Studies on DTA-GNN\n\nWe conduct ablation studies in the following table. ABL_d means DTA-GNN without directional information in the message passing stage. Formally, the edge equation in Equation 2 becomes\n$$\n\\\\tilde{e}_{i,j}^{\\\\ell} \\\\leftarrow \\\\mathcal{T} \\\\bigl(\\\\tilde{v}_j^{\\\\ell-1} - \\\\tilde{v}_i^{\\\\ell-1}\\\\bigr)\n$$\nABL_diff means DTA-GNN without the difference between neighboring latent states. We simply concatenate the neighboring latent states and multiply with the direction information. MeshGraphNets corresponds to removing both the directional information and difference between neighboring latent states.\n\n|   | MeshGraphNets | ABL_d   | ABL_diff | DTA-GNN |\n|-|-|-|-|---|\n| 5% Uniform  \t| 78.421 | 75.124| 63.162  | 59.308 |\n| 10% Uniform| 74.549 | 59.064| 52.294  | 43.647 |\n| 20% Uniform | 32.596 | 30.368 | 30.660  | 28.173 |\n| 30% Uniform | 29.817 | 26.923 | 27.497  | 24.883 |\n| 5% Random | 103.126 | 155.466 | 101.878 | 97.076 |\n| 10% Random | 62.337 | 73.355 | 60.536  | 56.224 |\n| 20% Random | 48.519 | 49.210 | 48.622  | 44.748 |\n| 30% Random | 31.806 | 32.389 | 30.943  | 29.803 |\n\nBased on these results, we draw three key conclusions:\n\n(1) Directional information plays a critical role when sensor density is low or when sensors are placed randomly, as some regions may lack sufficient sensor coverage.\n\n(2) Relying solely on the difference between neighboring latent states is suboptimal, as this approach does not capture edge attributes or directional cues essential for accurate information transfer.\n\n(3) Simply concatenating neighboring latent states leads to a moderate drop in performance, indicating that explicitly computing transported information is beneficial. Nevertheless, since message passing networks can still approximate difference operators, the performance degradation is less severe than when directional information is entirely removed.\n\n----------\n\n## Appendix G Changing Sensor Location for Randomly Placed Sensors\nWe change the sensor location for randomly placed sensors to estimate the expected MSE. We report the results for sensors fixed for the entire trajectory. For each trajectory, we randomly sample the sensor locations, and this sensor placement is fixed for the entire trajectory.\n\n| Method        | MSE   |\n|---------------|-------|\n| Uniform       | 3.092 |\n| Random (change at every frame)        | 4.110 |\n| Random (fixed for trajectory)        | 7.531 |\n| QR Pivoting   | 6.988 |\n| d-optimal     | 6.309 |\n\n\n----------\n\n\n## Rationale for Choosing PPO\nThe PPO framework used in this work is not based on the Markov decision process. As described in Section 4 (Optimal Sensor Placement), we instead cast the problem as a policy learning and optimization task, where the objective is to learn a generalizable policy that maps an input mesh geometry to a sensor placement configuration. This configuration should minimize reconstruction error while satisfying a strict equality constraint. The problem is inherently single-step decision-making.\n\nWe choose PPO over supervised regression for several reasons. Supervised regression would require differentiating through the binary placement mask, which is constrained to satisfy an equality condition. This makes direct optimization intractable, as the best-known gradient estimator [1] has a computational complexity of $O(k |V|^2)$. As discussed in the paper, this is prohibitive for large meshes. Moreover, supervised regression models may extrapolate poorly at test time when encountering geometries outside the training distribution. In contrast, PPO learns to act based on reward feedback and can generalize more robustly, provided sufficient exploration during training. Finally, in realistic scenarios where sensor data may contain noise, the policy gradient in PPO naturally averages over such noise, contributing to greater training stability.\n\n[3] also applies a **single step PPO for optimization tasks**. For example, one of its experiments employs a single step PPO to determine the angle of attack that maximizes mean lift coefficients.\n\n\n----------\n\n\n## Explanation for why QR-pivoting and SVD Fail\nBoth QR pivoting and D-optimal assume that reconstruction relies on linear independence of measurements. For example, QR Pivoting maximizes the numerical rank of the selected columns in the feature matrix. However, Deep learning models are highly nonlinear, so the important sensors determined by QR pivoting and D-optimal may be uninformative for a neural network.\nThese methods assume a basis matrix, assuming linear combinations of basis vectors reconstruct the full state. However, deep learning models do not operate in the same space. Its latent features or inductive biases may not align with the modes captured by the basis.\n\n\n----------\n\n\n## Computational Cost of PPO\nThe PPO formulation in our approach involves a single-step decision, where the sensor placement configuration is determined once and used throughout the entire trajectory. As a result, the computational overhead of executing the PPO policy is minimal, particularly when the trajectory length is large.\nMoreover, given the substantial improvements in reconstruction accuracy achieved by the learned sensor placement policy, we believe that the use of PPO is well justified and offers a favorable trade-off between computational cost and performance.\n\n\n----------\n\n\n## Comment of Recent Papers\n[2] violates several key assumptions outlined in our introduction. The experiments are limited to 2D domains, and the datasets are generated from Navier–Stokes simulations. Additionally, the method assumes that sensors can be arbitrarily placed within the flow field without affecting the underlying fluid dynamics, a simplification that is physically unrealistic in many practical scenarios.\n\n\n----------\n\nWe hope this clears up any confusion and concerns, and if so, we would greatly appreciate your consideration in raising the score.\n\n\n\n[1] SIMPLE: A Gradient Estimator for k-Subset Sampling, ICLR 2023.\n\n[2] Diff-SPORT: Diffusion-based Sensor Placement Optimization and Reconstruction of Turbulent flows in urban environments\n\n[3] Single-step deep reinforcement learning for open-loop control of laminar and turbulent flows, Phys. Rev. Fluids 6, 053902 – Published 12 May, 2021"}, {"review_id": "uo9O2Z0oRg", "reviewer": "Reviewer_4m3m", "summary": "Flow field reconstruction refers to the process of constructing complete flow field information from sparse sensor measurements. This is a crucial challenge in modern fluid dynamics because practical limits on sensor deployment often make it difficult to obtain high-fidelity, complete data. This paper develops a directional transport-aware Graph Neural Network (GNN) that explicitly encodes both directionality and information transport to address flow field reconstruction on arbitrary mesh-based geometries. Secondly, it finds that conventional sensor placement algorithms are insufficient in identifying effective sensor locations and thus proposes a novel Two-Step Constrained Proximal Policy Optimization (PPO) training strategy to learn a policy that identifies optimal sensor placement. Comprehensive experiments were conducted under realistic assumptions to validate the superiority of its reconstruction model and sensor placement policy, and it was tested on real-world meteorological data (sea surface temperature), outperforming existing methods.", "questions": "1. Scope of Real-World Validation: Address concerns about performance generalization based on the current extent of real-world dataset testing (e.g., primarily one dataset).\n2. Ablation Studies: Clarify the impact of individual components in the proposed GNN (e.g., directional transport awareness) and the Two-Step Constrained PPO (e.g., the necessity of the two stages, saddlepoint approximation).\n3. Comparison with Navier-Stokes Based Benchmarks: Explain the positioning of the work relative to methods tested on datasets derived from idealized physics or known governing equations like Navier-Stokes.\n4. Robustness to Sensor Noise and Sim-to-Real Gap: Discuss the model's expected performance with noisy or imperfect real-world sensor data, and strategies to mitigate the sim-to-real gap.\n5. Computational Cost and Scalability of PPO: Provide further details on the training time, resource requirements, and scalability of the Two-Step Constrained PPO for sensor placement, especially for larger systems.\n6. Interpretability: Comment on the interpretability of the learned GNN representations and the sensor placement strategies derived from the PPO policy.\n7. Sensor Placement Constraints: Clarify if sensor placement by the PPO is restricted (e.g., to boundaries only, as with the DNS dataset generation) or applicable to any predefined node in the flow field.", "limitations": "1. Sensitivity to Real-World Sensor Imperfections: The model's robustness to real-world sensor noise, calibration errors, or other data imperfections is not extensively evaluated, as training primarily utilized high-fidelity simulated data or clean sea surface temperature data.\n2. Computational Cost of PPO Training: Training the GNN-based Proximal Policy Optimization (PPO) policy for sensor placement can be computationally intensive and time-consuming, particularly for very large and complex flow fields, given the extensive training steps involved.\n3. Fixed Sensor Proportion in PPO: The current PPO framework optimizes sensor placement for a predefined proportion or number of sensors, rather than determining the optimal number of sensors itself.\n4. Interpretability of Learned Models: Gaining deep physical insights into the decision-making processes of the directional transport-aware GNN and the sensor placement policy learned by PPO remains challenging.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. General Model for Flow Field Reconstruction: The paper introduces a commendable general model for flow field reconstruction that distinguishes itself by not relying on many simplifying assumptions prevalent in existing methods. For instance, it doesn't require predefined governing equations (like Navier-Stokes) or restrict itself to two-dimensional domains. The core of this is a directional transport-aware Graph Neural Network (GNN) specifically designed to operate on arbitrary mesh-based geometries and infer flow fields without prior knowledge of the underlying Partial Differential Equations (PDEs).\n2. Novel Two-Step Constrained PPO for Sensor Placement: A significant contribution is the novel Two-Step Constrained Proximal Policy Optimization (PPO) procedure developed for optimizing sensor placement. This approach addresses the observation that conventional sensor placement strategies can be suboptimal when integrated with deep learning reconstruction models. It also tackles the inherent difficulty of training reinforcement learning policies under strict constraints (e.g., an exact number of sensors) by first employing a penalized objective function before transitioning to a hard-constrained optimization phase.\n3. State-of-the-Art Performance: The proposed methods, combining the GNN reconstruction model and the PPO-based sensor placement, demonstrate state-of-the-art (SOTA) performance. This is evidenced by consistently achieving the lowest reconstruction errors across four different synthetic turbulent flow datasets generated via Direct Numerical Simulation (DNS) with varying geometries. Furthermore, the model and placement strategy were validated on a real-world sea surface temperature dataset (NOAA OISST V2), where they achieved a notable 14.01% improvement in Mean Squared Error (MSE) over the strongest competing baseline.\n4. Clarity and Readability: The paper is well-written and structured logically, making the methodologies, including the intricacies of the GNN architecture and the PPO algorithm, as well as the extensive experimental results, clear and relatively easy for readers to follow.\n\n\nWeaknesses\n1. Limited Real-World Dataset Validation: While the performance on the single real-world sea surface temperature dataset is promising, the claim of general superiority on real-world data might be perceived as strong based on this sole example. Evaluating the model and placement strategy on additional real-world fluid dynamics datasets, potentially from different domains (e.g., aerodynamics, industrial flows) with varying complexities and sensor modalities, would more robustly confirm its broader applicability and effectiveness.\n2. Absence of Ablation Studies: The paper could be enhanced by including detailed ablation studies for the key design choices within both the GNN and the PPO algorithm. For the directional transport-aware GNN, this could involve analyzing the impact of the explicit encoding of directionality and information transport, or the contribution of different components in its encoder-processor-decoder framework. For the Two-Step Constrained PPO, an ablation could quantify the benefits of the two-stage training procedure versus a single-stage approach, or the sensitivity to the choice of constrained sampling (Gumbel top-k ) and log probability estimation (saddle point approximation ). Such studies would offer deeper insights into why specific components are crucial for the observed performance.\n3. No Comparison on Navier-Stokes-Based Datasets: The paper's focus is on realistic conditions, moving away from datasets based on idealized physics or known governing equations. However, many existing deep learning methods for flow reconstruction are benchmarked on datasets derived from, for example, solutions to the Navier-Stokes equations. Including a comparison on some of these established (even if idealized) benchmarks could provide a broader context for the proposed model's performance and highlight its advantages or trade-offs relative to a wider spectrum of existing approaches that do rely on such physical priors.", "was_revised": true, "final_justification": "Thanks for the responses from the authors. Most of my concerns have been resolved. However, the computational cost remains unclear. Therefore, I keep my rating as Borderline accept.", "rebuttal": "We thank you for your insightful comments on improving our paper. We hope the following could address your concerns.\n\n## Ablation Studies on DTA-GNN\n\nWe conduct ablation studies in the following table. ABL_d means DTA-GNN without directional information in the message passing stage. Formally, the edge equation in Equation 2 becomes\n$$\n\\\\tilde{e}_{i,j}^{\\\\ell} \\\\leftarrow \\\\mathcal{T} \\\\bigl(\\\\tilde{v}_j^{\\\\ell-1} - \\\\tilde{v}_i^{\\\\ell-1}\\\\bigr)\n$$\nABL_diff means DTA-GNN without the difference between neighboring latent states. We simply concatenate the neighboring latent states and multiply with the direction information. MeshGraphNets corresponds to removing both the directional information and difference between neighboring latent states.\n\n| | MeshGraphNets|ABL_d|ABL_diff|DTA-GNN|\n|-|-|-|-|---|\n|5% Uniform| 78.421 | 75.124| 63.162  | 59.308|\n|10% Uniform|74.549 | 59.064| 52.294  | 43.647|\n|20% Uniform | 32.596 | 30.368 | 30.660  | 28.173|\n|30% Uniform | 29.817 | 26.923 | 27.497  | 24.883|\n|5% Random | 103.126 | 155.466 | 101.878 | 97.076 |\n|10% Random | 62.337 | 73.355 | 60.536  | 56.224 |\n|20% Random | 48.519 | 49.210 | 48.622  | 44.748 |\n|30% Random | 31.806 | 32.389 | 30.943  | 29.803 |\n\nBased on these results, we draw three key conclusions:\n\n(1) Directional information plays a critical role when sensor density is low or when sensors are placed randomly, as some regions may lack sufficient sensor coverage.\n\n(2) Relying solely on the difference between neighboring latent states is suboptimal, as this approach does not capture edge attributes or directional cues essential for accurate information transfer.\n\n(3) Simply concatenating neighboring latent states leads to a moderate drop in performance, indicating that explicitly computing transported information is beneficial. Nevertheless, since message passing networks can still approximate difference operators, the performance degradation is less severe than when directional information is entirely removed.\n\n---\n\n## Ablation Studies on Two-Step Constrained PPO\nWe report the ablation studies on Two-Step Constrained PPO in the following table. Penalized PPO refers to the first stage, where we use a penalized objective function to guide the model toward the constraints. During inference, we use Gumble Top-k to strictly enforce the equality constraints. Constrained PPO refers to the second stage, where we use Gumble Top-k for sampling and saddle point approximation for computing the log probability. Two-Step PPO without Saddlepoint Approx refers to the original Two-Step Constrained PPO, but removing saddle point approximation in the second stage. In the second stage, we use unconstrained log probability.\n\n|Shape| Penalized PPO| Constrained PPO| Two-Step PPO without Saddlepoint Approx |\n|-|-|-|-|\n|Sphere|3.270|19.280|3.178|\n|Ellipsoid|11.002|31.647|10.629|\n|Cylinder|9.490|55.014|9.575|\n|Airfoil|46.431|327.966|44.103|\n\nFrom these results, we draw several important conclusions:\n\n(1) As discussed in the main paper, initiating constrained training from a randomly initialized policy is overly restrictive and substantially limits the model’s performance.\n\n(2) Although the penalized training in the first stage helps guide the model toward a reasonable sensor placement strategy, it does not strictly enforce the equality constraint. Consequently, when combined with a constraint-compliant sampling strategy during inference, its performance degrades.\n\n(3) Accurately computing the log probability in the second-stage constrained training is essential. Without the saddle point approximation, as in the Two-Step PPO without Saddle Point Approximation, the model fails to outperform even the baseline of uniformly placed sensors.\n\n---\n\n## Experiments on Turbulence Data\nWe compare our proposed DTA-GNN against three strong baselines on Kolmogorov Flow and Taylor-Green Vortex. Both datasets are generated via high-resolution numerical simulations using a pseudo-spectral solver governed by the incompressible Navier–Stokes equations. Kolmogorov Flow is driven by time‑dependent sinusoidal forcing at Re = 2000, while Taylor–Green Vortex starts from its analytic solution with added Gaussian noise and is simulated at Re = 1500.\n\n**Kolmogorov Flow**\n|Method|5% Random|10% Random|20% Random|30% Random|\n|-|-|-|-|-|\n|Ours|9.484| 8.537|4.510|3.443|\n|FCN |10.547|9.1557|5.109|4.690|\n|DiffusionPDE|11.214|9.458| 6.699|4.703|\n|MeshGraphNets|10.180|9.283| 5.271|4.137 |\n\n**Taylor Green Vortex**\n|Method|5% Random|10% Random|20% Random|30% Random|\n|-|-|-|-|-|\n|Ours|6.749|4.856|2.643|1.267|\n|FCN|8.430|6.354|3.590|2.825 |\n|DiffusionPDE|8.898|5.348|3.142|1.974|\n|MeshGraphNets|9.801|5.677|3.119|2.831|\n\n---\n\n## Additional Real World Validation\nUnfortunately, most existing CFD datasets and benchmark studies rely on numerical simulations with simplified assumptions and lack real-world observational data. For instance, [1, 3] are based on Reynolds-Averaged Navier–Stokes (RANS) simulations, [2] employs Large-Eddy Simulations (LES), and [4] utilizes a hybrid RANS–LES turbulence modeling approach.\nTo further evaluate the practical applicability of our proposed method, we benchmark it on a real-world air quality dataset from the United States. Following the setup used in the sea surface temperature experiment, we compare our model against three strong baselines: FlowCompletionNetwork (FCN), DiffusionPDE, and MeshGraphNets. Results are reported in the table below. In this experiment, we assume that sensors are uniformly distributed over 10% of the available spatial locations.\n\n| Method | MSE |\n|-|-|\n|Ours | 1.488 |\n|FCN | 2.257 |\n|DiffusionPDE | 1.607 |\n|MeshGraphNets | 1.628 |\n\n\nAdditionally, our Navier–Stokes-based benchmarks are derived from high-fidelity simulations, offering a more realistic approximation of real-world fluid dynamics compared to simplified models. Given that our model consistently achieves strong performance across both synthetic and real-world datasets, we are optimistic about its potential for real-world applications.\n\n---\n\n## Robustness to Noise\nTo evaluate robustness to sensor noise, we inject Gaussian noise N(0, 0.025) into the observed sensor data in the Airfoil dataset. Note that the noise is injected into normalized data that follow a standard normal distribution N(0,1). We compare the performance of three strong baselines, FlowCompletionNetwork (FCN), DiffusionPDE, and MeshGraphNets, with our proposed DTA-GNN.\n| Method| 5% Random | 10% Random| 20% Random| 30% Random|\n|-|-|-|-|-|\n|Ours| 102.098|60.083| 48.394| 30.732|\n|FCN| 134.292|81.654| 68.733| 52.278|\n|DiffusionPDE|115.646| 73.280| 58.348| 32.270|\n|MeshGraphNets|114.209| 70.263| 56.293| 34.174|\n\n---\n\n## Computation Cost and Scalability of PPO\nThe PPO framework operates as a single-step decision-making process, where the sensor placement configuration is determined once and remains fixed throughout the entire trajectory. As a result, the computational cost of executing the PPO policy is minimal, particularly for long trajectories. Given the substantial improvements in reconstruction accuracy achieved by the learned placement strategy, we consider the use of PPO to be a worthwhile and effective choice.\nThe underlying model architecture is based on a message-passing graph neural network (GNN), which scales linearly with the number of nodes, making it suitable for large-scale spatial domains.\n\n---\n\n## Sensor Placement Constraints\nThe sensor placements by PPO are restricted to the boundaries to reflect real world scenarios.\n\n---\n\n## Interpretability of DTA-GNN\nThe interpretability of our proposed model is grounded in its principled design, which draws from established physical and algorithmic analogies. First, by mimicking the structure of the advection operator, we endow our processor with a clear directional semantics. Second, the processor can be seen as a learnable generalization of classical interpolation schemes. Here, directional information modulates neighbor contributions, and the update rule mirrors interpolation by aggregating weighted differences, augmented with self-contributions. Finally, unlike conventional message passing GNNs that rely on generic concatenation and MLPs, our model introduces an explicit notion of directionality, enhancing interpretability by aligning message flow with physically meaningful pathways.\n\n---\n\n## Interpretability of PPO\nOur sensor placement policy is guided by both the variability of the underlying fluid field and the performance of the reconstruction model. It prioritizes regions with high variability while also selecting sensor configurations that the model can accurately reconstruct. However, since the reconstruction model functions as a black box, the resulting placement decisions are not readily interpretable.\nWe argue that the lack of interpretability is not a limitation of our approach, but rather a reflection of the inherently opaque nature of the problem itself.\n\n---\n\n## Fixed Sensor Proportion in PPO\nEmpirically, increasing the sensor budget consistently improves the performance. This reflects an inherent trade-off in the problem formulation.  At present, the sensor budget is specified in advance, and the model is tasked with determining optimal sensor locations. An interesting futuer direction is to reverse this setup and ask the policy to determine the number of sensors required with a desired level of reconstruction accuracy.\n\n---\n\nWe hope this clears up any confusion and concerns, and if so, we would greatly appreciate your consideration in raising the score.\n\n[1] AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier–Stokes Solutions\n\n[2] WindsorML: High-Fidelity Computational Fluid Dynamics Dataset For Automotive Aerodynamics\n\n[3] DrivAerNet: A Parametric Car Dataset for Data-Driven Aerodynamic Design and Prediction\n\n[4] AhmedML: High-Fidelity Computational Fluid Dynamics Dataset for Incompressible, Low-Speed Bluff Body Aerodynamics"}]

Author rebuttal:

Meta-review: This paper introduces a Directional Transport-Aware GNN for flow reconstruction and a Two-Step Constrained PPO for sensor placement under realistic constraints. While sensor placement is a long-standing optimization problem, the contribution here is in adapting PPO with equality constraints and coupling it with a reconstruction model under a realistic CFD setting.

Reviewers initially raised concerns about novelty (directional encoding overlap), limited turbulence experiments, and missing ablations. The rebuttal provided careful clarifications and additional results, largely addressing these points.

The AC finds the work a solid contribution: the originality is somewhat incremental, but the problem formulation is well-motivated, the empirical validation is comprehensive, and the paper illustrates the broader value of applying ML methods to challenging scientific domains.
Decision: Accept (poster)