Flow Field Reconstruction with Sensor Placement Policy Learning

We thank you for your insightful comments on improving our paper. We hope the following could address your concerns.

Ablation Studies on DTA-GNN

We 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

$$\\tilde{e}_{i,j}^{\\ell} \\leftarrow \\mathcal{T} \\bigl(\\tilde{v}_j^{\\ell-1} - \\tilde{v}_i^{\\ell-1}\\bigr)$$

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 a critical role when sensor density is low or when sensors are placed randomly, as some regions may lack sufficient sensor coverage.

(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.

(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.

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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, 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.

From these results, we draw several important conclusions:

(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.

(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.

(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.

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Experiments on Turbulence Data

We 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.

Kolmogorov Flow

Taylor Green Vortex

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Additional Real World Validation

Unfortunately, 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. To 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.

Additionally, 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.

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Robustness to Noise

To 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.

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Computation Cost and Scalability of PPO

The 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. The 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.

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Sensor Placement Constraints

The sensor placements by PPO are restricted to the boundaries to reflect real world scenarios.

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Interpretability of DTA-GNN

The 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.

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Interpretability of PPO

Our 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. We 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.

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Fixed Sensor Proportion in PPO

Empirically, 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.

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We hope this clears up any confusion and concerns, and if so, we would greatly appreciate your consideration in raising the score.

[1] AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier–Stokes Solutions

[2] WindsorML: High-Fidelity Computational Fluid Dynamics Dataset For Automotive Aerodynamics

[3] DrivAerNet: A Parametric Car Dataset for Data-Driven Aerodynamic Design and Prediction

[4] AhmedML: High-Fidelity Computational Fluid Dynamics Dataset for Incompressible, Low-Speed Bluff Body Aerodynamics

We thank you for your insightful comments on improving our paper. We hope the following could address your concerns.

Dataset Description

The 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. The 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. The 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. The 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.

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Train, Valid, Test Split

We 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.

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Description on Baseline Modification

We 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.

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Ablation Studies on DTA-GNN

We 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

$$\\tilde{e}_{i,j}^{\\ell} \\leftarrow \\mathcal{T} \\bigl(\\tilde{v}_j^{\\ell-1} - \\tilde{v}_i^{\\ell-1}\\bigr)$$

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 a critical role when sensor density is low or when sensors are placed randomly, as some regions may lack sufficient sensor coverage.

(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.

(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.

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Appendix G Changing Sensor Location for Randomly Placed Sensors

We 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.

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Rationale for Choosing PPO

The 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.

We 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.

[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.

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Explanation for why QR-pivoting and SVD Fail

Both 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. These 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.

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Computational Cost of PPO

The 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. Moreover, 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.

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Comment of Recent Papers

[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.

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We hope this clears up any confusion and concerns, and if so, we would greatly appreciate your consideration in raising the score.

[1] SIMPLE: A Gradient Estimator for k-Subset Sampling, ICLR 2023.

[2] Diff-SPORT: Diffusion-based Sensor Placement Optimization and Reconstruction of Turbulent flows in urban environments

[3] Single-step deep reinforcement learning for open-loop control of laminar and turbulent flows, Phys. Rev. Fluids 6, 053902 – Published 12 May, 2021

We thank you for your insightful comments on improving our paper. We hope the following could address your concerns.

Ablation Studies on DTA-GNN

We 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

$$\\tilde{e}_{i,j}^{\\ell} \\leftarrow \\mathcal{T} \\bigl(\\tilde{v}_j^{\\ell-1} - \\tilde{v}_i^{\\ell-1}\\bigr)$$

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 a critical role when sensor density is low or when sensors are placed randomly, as some regions may lack sufficient sensor coverage.

(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.

(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.

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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, 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.

From these results, we draw several important conclusions:

(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.

(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.

(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.

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Runtime and Memory Comparison

We 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.

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Additional Baseline Comparison

We include [1] as our baseline and conduct experiments on the airfoil dataset. Results show that our DTA-GNN remains the best method.

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Additional Clarification on Proposition 3 and Two-Step Constrained PPO

Given 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. This 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.

To 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.

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We hope this clears up any confusion and concerns, and if so, we would greatly appreciate your consideration in raising the score.

[1] Qitian Wu and Chenxiao Yang and Kaipeng Zeng and Michael Bronstein, Supercharging Graph Transformers with Advective Diffusion, ICML 2025

We thank you for your insightful comments on improving our paper. We hope the following could address your concerns.

Novelty of Direction Information

The 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:

1. Problem Definition The 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. Moreover, 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.

2. Semantic Meaning of DID vs. Our Directional Information In [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. In 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.

3. Role Within the Model
   The 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.

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Experiments on Turbulence Data

We 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. The 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:

Kolmogorov Flow

Taylor Green Vortex

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Generalisability Experiments

We 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. Despite 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.

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Analysis on Lift and Drag Coefficients

We report the MSE on lift and drag coefficients. The experiments are conducted on the airfoil dataset with 10% uniformly placed sensors.

As 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.

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Related Work

We 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.

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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 is constrained, statistical correlation is introduced through the equality constraint.

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We hope this clears up any confusion and concerns, and if so, we would greatly appreciate your consideration in raising the score.

[1] Finite Volume Features, Global Geometry Representations, and Residual Training for Deep Learning-based CFD Simulation, ICML 2024.