Graph Neural Networks for Aerodynamic Flow Reconstruction from Sparse Sensing
Authors: Gregory Duthé
Imad Abdallah
Sarah Barber
Eleni Chatzi
Venue: Submitted to ICLR 2023 (2023)
Keywords: CFD
flow reconstruction
GNNs

Abstract:   Sensing the fluid flow around an arbitrary geometry entails extrapolating from the physical quantities perceived at its surface in order to reconstruct the features of the surrounding fluid. This is a challenging inverse problem, yet one that if solved could have a significant impact on many engineering applications. The exploitation of such an inverse logic has gained interest in recent years with the advent of widely available cheap but capable MEMS-based sensors. When combined with novel data-driven methods, these sensors may allow for flow reconstruction around immersed structures, benefiting applications such as unmanned airborne/underwater vehicle path planning or control and structural health monitoring of wind turbine blades. In this work, we train deep reversible Graph Neural Networks (GNNs) to perform flow sensing (flow reconstruction) around two-dimensional aerodynamic shapes: airfoils. Motivated by recent work, which has shown that GNNs can be powerful alternatives to mesh-based forward physics simulators, we implement a Message-Passing Neural Network to simultaneously reconstruct both the pressure and velocity fields surrounding simulated airfoils based on their surface pressure distributions, whilst additionally gathering useful farfield properties in the form of context vectors. We generate a unique dataset of Computational Fluid Dynamics simulations by simulating random, yet meaningful combinations of input boundary conditions and airfoil shapes. We show that despite the challenges associated with reconstructing the flow around arbitrary airfoil geometries in high Reynolds turbulent inflow conditions, our framework is able to generalize well to unseen cases.

Under review as a conference paper at ICLR 2023
GRAPH NEURAL NETWORKS FOR AERODYNAMIC FLOW RECONSTRUCTION FROM SPARSE SENSING
Anonymous authors
Paper under double-blind review
ABSTRACT
Sensing the fluid flow around an arbitrary geometry entails extrapolating from the physical quantities perceived at its surface in order to reconstruct the features of the surrounding fluid. This is a challenging inverse problem, yet one that if solved could have significant impact many engineering applications. The exploitation such an inverse logic has gained interest in recent years with the advent widely available cheap but capable MEMS- based sensors. When combined with novel data-driven methods, these sensors may allow for flow reconstruction around immersed structures, benefiting applications such unmanned airborree/nddrwatte vehicle path planning or control and structural health monitoring of wind turbine blades. this work, we train deep reversible Graph Neural Networks (GNNs) perform flow sensing (flow reconstruction) around two-dimensional aerodynamic shapes: airfoils. Motivated by recent work, which has shown that GNNs can be powerful alternatives to mesh-based forward physics simulators, we implement a Message-Passinnn Neural Network to simultaneously reconstruct both the pressure and velocity fields surrounding simulated airfoils based on their surface pressure distributions, whilst additionally gathering useful farfield properties the form of context vectors. We generate unique dataset of Computational Fluid Dynamics simulations by simulating random, yet meaningful combinations of input boundary conditions and airfoil shapes. We show that despite the challenges associated with reconstructing the flow around arbitrary airfoil geometries in high Reynolds turbulent inflow conditions, our framework is able to generalize well unseen cases.
INTRODUCTION
Many engineering applications stand to benefit from the ability to sense and reconstruct fluid flow features from sparse measurements originating at structure's surface. Flow sensing could be crucial for improvements in the accuracy and resilience of wind turbine and unmanned aircraft controllers. Another possible application is monitoring of wind loaded structures (Barber et al. 2022 where the use of cheap micro-electromechaniica systems (MEMS) in combination with novel methods for flow sensing could lead robust structural health monitoring solutions. In this work, focus on common aerodynamic structures: we aim to reconstruct the flow around 2-D airfoils. Traditionally, computing the flow around an airfoil requires approaches from Computational Fluid Dynamies (CFD), which are forward-physics simulators. In CFD, the inflow, outflow and wall boundary conditions are set, and over many iterations solution for the discretized Navier- Stokes PDEs is reached, which then yields pressure distribution at the airfoil surface. We aim to solve the inverse problem: given only the pressure distribution at the airfoil surface, a solution for the flow field and farfield boundary conditions is to be found. Moreover, our aim is to do so for any airfoil geometry subject to wide variety of turbulent inflows.
Adopting the notation of Erichson et al. (2020), the problem can be described in the following manner. An airfoil equipped with distributed barometric sensors placed in a steady flow of air, providing surface pressure measurements IR multiple locations around its perimeter. The sensors sample from the surrounding flow field x e R" through mcasurement operator H:
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The goal is construct estimate of the flow field e surrounding the airfoil, by learning from training data a function F that approximates the highly nonlinear inverse measurement operator G such that:
F(s) = 8 G(s)
(2)
Meshes an extremely useful tool, indispensable in many engineering domains and especially in CFD, Contrary to Cartesian grid representations, mesh representations offer high flexibility for irregular geometries and allow for variable spatial density. This makes them ideal for discretizing complex physical problems, where one can balance the trade-off between numerical accuracy and computational efficiency in certain regions interest. Furthermore, meshes can also be described in terms of nodes and edges, i.c. as a graph. In this context, the flow reconstruction problem can be described as follows. graph 9 = (V, () is constructed from an airfoil CFD) mesh, with m fluid nodes y and p airfoil boundary nodes Va. The flow features of all V are unknown, whilst the features of Va are known. Our aim is then to learn graph operator f that estimates the information at the fluid nodes using the information contained at the airfoil boundary nodes, the input graph-level attributes Unn and the edges E:
1 = (Va,
An ancillary goal is to estimate the global context U of the graph, this contains relevant information for applications. Figure I provides an description the flow reconstruction problem in terms of graph learning.
VV uu
G' (22 E, à)
U, Estimated input context
ú Predicted context
Learned operator
o 1/2 Known sirfoit nodes
Di- Unknown fluid nodea
1/ Known nirfoil nodes
0) Reconstructed fluid nodes
Figure I: Problem setup. We aim to learn the reconstruction operator & which estimates properties of the fluid nodes as well as the graph context. This amounts to reconstructing a solution to the Navier-Stokes equations which satisfies the boundary conditions perceived the airfoil surface.
From geometric learning perspective, flow reconstruction is challenging problem for several reasons. The first significant hurdle to overcome is the size of our graphs. We use meshes with high densities close to the airfoil in order achieve good spatial resolution in these critical regions. Thus, our dataset contains graphs with mean of around 55'000 nodes. which is an order of magnitude higher than previous mesh-graph learned simulation methods (Pfaff al. 2020). Moreover, the input information is concentrated in a very localized domain of the graph: the airoii nodes. It is difficult to propagate the necessary information to reconstruct nodes far from the airfoil with a shallow Graph Neural Network (GNN), meaning that deep GNN architectures with a large number of message- passing steps are required to push this 'information barrier' away from the airfoil nodes. However, deep GNNs go hand in hand with other issues such large memory requirements, over-smoothing and over-squashing.
In this work, combine number of existing graph-learning methods to tackle the aforementioned challenges. Our contributions may be summarized as follows:
We combine unknown Feature Propagation (Rosii et al. 2021) with very deep Grouped Reversible GNNs (Li al. 2021) to reconstruct flow features at the fluid nodes, whilst additionally gathering contextual farfield information.
Generalization of 2D aerodynamic flow field learning with GNNs, including (1) arbitrary airfoil geometries, (2) arbitrary turbulent inflow conditions (flow velocity, turbulence intensity, and angle of attack), and (3) simultancous flow field reconstruction and inference of contextual farfield flow information based only on sparse pressure data the surface of the airfoil.
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Generation of unique training dataset of OpenFOAM airfoil CFD simulations with many different geometries and inflow conditions which are parsed to graph structure and made publicly available
We gather qualitative and quantitative results on unseen airfoil and flow configurations, and perform number of experiments to understand the limitations of this framework. In particular, we test and compare three different GNN layer architectures.
RELATED WORK
Machine learning methods have recently garnered interest in the fluid mechamics community (Brunton et al. 2020). Fluid-related problems are typically nonlinear, complex and generate large amounts of data, all of which are conditions under which deep learning approaches thrive. Specifically in the context of flow reconstruction from sparse measurements, several neural network approaches can be found in the literature. In an article by Erichson et al. (2020), a "Shallow Neural Network", Le. a fully-com fully-connected network with only two hidden layers, was applied to estimate transient flows from sparse measurements. The authors trained the networks on a single specific geometrical flow configuration, for example the flow behind a cylinder, and then tested at different time steps. We aim to avoid this limitation as our goal is to estimate the flow around any airfoil geometry. The authors compare their findings against typically used proper orthogonal decomposition (POD) methods and note significant improvements in terms of the reconstruction error. This work was then extended to turbulent flow reconstruction around airfoils based on experimental data in Carter et al.) (2021), where the results were compared to Particle Image Velocimetry (PIV) measurements. The viability of neural networks over other approaches was further confirmed in the work of (Fukami et al. (2020). where multiple methods were pitted against each other to estimate the flow behind a cylinder and an airfoil. Again, here the models were trained and tested on a single flow configuration, while also being dependent Cartesian geometrical imputs. In Özbay & Laizet -hriiss-hrittffflll mappings to
the same configuration
(2022). researchers attempt to avoid this limitation by utilizing sample the points at which the flow is reconstructed. thus rendering the method geometry invariant. The authors train multiple neural network architectures on a collection of transient flow simulations around randomly generated 2-D Bezier shapes at predefined inflow Reynolds number. As inputs for the flow reconstruction, they use multiple pressure sensors on the shapes' surface as well as velocity probes in the wake. Their results indicate that, when compared to Cartesian sampling strategy, significant performance boost is achieved for all neural network types, especially in the vicinity of the immersed shape. While this work demonstrates robustness to various geometric configurations, requires additional velocity sensors and is trained on singular farfield boundary condition, both of which we aim to avoid and improve upon. Another method which avoids geometrical dependency is reported in Chen et al. (202T). In this work, to which our approach most closely relates to, the authors utilize a Graph Convolutional Network (GCN) (Kipf & Welling 2016) on graphs constructed from CFD meshes of randomly generated Bezier shapes. The GCN is used to predict the flow around the shapes at a fixed laminar (Reynolds number of 10) inflow condition without using surface measurements. In approach, we aim to reconstruct wide variety of turbulent flows given only surface readings, a significantly less constrained problem. We also aim to characterize the global properties of the flow, similarly Zhou et al. (202T). To our best knowledge, we are the first to attempt to simultancously reconstruct the flow while estimating turbulent inflow parameters at large Reynolds numbers for arbitrary airfoil geometries.
The dataset that we generate to train our GNN model is similar in terms of the geometries, meshing and CFD pipeline to the work of Thuerey et al. (020), the main differences being the chosen RANS model and the post-processing (graph parsing). Other datasets found in the literature focus only on the NACA family of airfoils (Schillaci et al. 2021).
Graph networks are based on the message- -aasiigg framework GGimerr et al. 2017). where nodes features are updated by aggregating messages emanating from its neighbors. Many different types of message- message-passing schemes can be constructed, with some using attention mechanisms (Veličkovid ct al. 2017) and others relying on strong theoretical backgrounds (Xu et al. 2018). Graph learning methods are increasingly being applied to a wide variety of physics problems "Sanchez-Gonzalee et al. 2018 2020). In Pfaff et al. (2020), the authors successfully demonstrate how GNNs can learn to wwwpeepciiee replicate forward mesh-based physics simulators and are able to predict the evolution of a transient solution. Motivated by these results, our approach is constructed upon the the same basic Encode- Process-Decode network structure. However, a key difference note is that, contrary to the next-step
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Geometry selection
Mcshing
CFD simulation
Gruph
Gmsh
OpenFOAM k-omega SST
Finite-volume
parsing
Figure 2: Illustration of the dataset generation pipeline. Airfoil shapes are selected random from database, then meshed and simulated OpenFOAM with random feasible boundary conditions. In the last step, the finite-volume scheme is used to parse the simulation mesh into graph.
prediction problem, flow reconstruction has to overcome high amounts of missing information, with known features being extremely localized. To address this hurdle, we turn to graph-base based feature propagation methods (Rossi et al. (02T), which is closely related to matrix completion approaches (Monti et al., 2017). Feature propagation an effective yet computationally inexpensive method for initializing graphs with missing features. We use this method as pre-processing step, through which graphs are passed before being fed into the rest of the GNN model.
Training GNNs for very large graphs is challenging, with typical approaches tending toward minimiz- ing the number of learnable parameters so that the problem becomes tractable (Chen et al. 20200.. This often results in relatively shallow GNNs, which could adversely influence the propagation of the information contained at the airfoil nodes through sufficient extents of the graph. Making use of subgraph sampling strategies (Hamilton al. 2017) is another possible approach, one which also allows for larger/deeper GNNs. However, these methods are not applicable in our case, as we need to feed entire graphs in pass due the heterogeneity in information localization. Moreover, subgraph sampling would yield additional difficulties for our ancillary goal of predicting global graph properties for farfield estimation. Recent work by Li al. (202T) has shown that possible train very deep GNNs on large graphs by making use of Grouped Reversible layers, which reduces memory requirements at the cost of extra computation. This method forms the core of the processing block of the proposed flow reconstruction GNN. Another issue which traditionally characterizes training deep GNNs on large graphs is over-squashing (Alon & Yahav 2020). Over-squashing a by-product of graph's structure, where bottlenecks and tree-like structures (Topping et al. (021) can cause the latent representation of certain nodes to be overwhelmed by the amount of information needed to be stored. We make use of this information while parsing the simulation meshes into graphs.
DATASET GENERATION
In this section introduce the different elements of our data generation pipeline. In total we generate 1120 converged simulations, which are separated into train, validation and test datasets in 80/10/10 split. Figure depicts an illustration of this pipeline.
Geometry selection and meshing In dataset generation pipeline. airfoil shapes drawn at random from the UUIC database of airfoils (Selig 1996). Before passing the shape to the meshing algorithm, carry out some additional interpolation and processing make sure that the selected airfoil has a sufficient amount of points at the ledging-edge as well as a properly defined trailing-edge. Then, we use Gmsh (Geuzaine & Remacle, 2020) construct unstructured O-grid type mesh around the selected airfoil. A sizing field is set close to the airfoil in order to make sure that meshes with appropriate y+ values for the CFD call-functions are generated. An overall sizing parameter is also set for sufficient farfield density, but is adjusted to ensure that an acceptable amount of cells are created (< 150'000).
CFD simulations Each mesh is associated with different inflow configuration. Three parameters control the farfield conditions: angle attack, inflow velocity and turbulence intensity. These parameters are drawn from probability distributions reflecting realistic atmospheric flows Reynolds
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OUTPUT
arrfoil
INPUT
Lalant
HLP
cnooder
Locoder
Saje encoder eoong
v Ke-w (Miuuii (mm)
Feature propagation
Encoding
Processing
Decoding
Figure 3: Overview of the Flow Reconstruction GNN achitecture. Feature Propagation is used to
initialize the unknown fluid nodes. The graph then passed through the pipeline, to obtain the reconstructed flow. During the Process step, node features are updated via message-passing within deep Grouped Reversible GNN.
numbers ranging from 2- 10s 6.5 10 with Reynolds number of around 3 10 This mean value is well beyond the typical taminar-to-turullee transition threshold of around 5 10 Incropera al. 1996). which greatly increases the difficulty of the flow reconstruction problem. The farfield conditions form the global context vectors of our graphs and are estimated at inference time. We simulate the flow around the airfoils using a steady 2-D Reynolds-Averaged Navier-Stoker (RANS) CFD solver with the OpenFOAM software package (Jasak et al. (007). For turbulence modelling, we select the K-Omega SST model (Menter & Esch] (201]) along with the standard OpenFOAM wall-functions for boundary layer treatment. Only sufficiently converged CFD simulations with pressure, velocity and turbulence residuals below 10 are kept.
Graph parsing Contrary many other mesh-based physics simulators, CFD solvers such as OpenFOAM are based on finite volume methods. It is significant difference that should be reflected in the manner which mesh converted into a graph. To do so, we use the cells themselves as the nodes, with bidirectional edges being formed between adjacent cells. This allows to gather an additional edge feature which relevant to the underlying physics. Specifically, the length (or surface for 3-D meshes) of the boundary between two cells is used of this twofold: a form of sizing fed to the network and a quantity relevant flux computation is set the edges. For the nodes, we gather 4 types of features: pressure, x-velocity component, y-velocity component and node category (fluid, farfield, wall), the latter which is inputted as one-hot vector. The global context features the farfield conditions (turbulence intensity, inflow velocity and angle of attack). To avoid unnecessary computational overhead, we do not parse the entire CFD domain, which has radius of 100 airfoil chord lengths, into graph. Instead opt only keep cells within a I chord circle centered on the airfoil. Furthermore, we set the airfoil nodes to be located the meshed airfoil boundary and add bidirectional edges between adjacent airfoil nodes. These additional edges are created with the aim of avoiding tree-like structures in our graphs, as these could potentially cause bottlenecks for the learning process (Topping et al. (02TT) The graphs of dataset have average around 55k nodes and around 85k individual edges.
edge feature. The benefit
4 GRAPH NEURAL NETWORK FRAMEWORK
4.1 ARCHITECTURE
For our GNN architecture, we adopt the Proiee-Proees--eeee logic that is now popular for learning graph-basee based physics problems SaachezzGonzaleee et al. 2020; Pfaff et al. 2020 Godwin
et al. 2022). albeit with some notable modifications. Figure shows an overview of the Flow Reconstruction GNN.
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Input features In the flow reconstruction problem, assume that the global context of the graph is unknown, however some useful physical parameters describing the flow can be estimated. Using Bernoulli's ' principle, and given that all airfoils are simulated with zero farfield static pressure, the farfield inflow velocity magnitude can be initially approximated as:
2 Pa
0.
(4)
where o the density of air (constant throughout simulations) and po the total pressure measured at the stagnation point, which can be estimated by taking the maximum pressure at the airfoil nodes Po max(p..). ). While Bernoulli's principle is not valid for turbulent flows such as the ones we try to reconstruct, it serves as good starting point for farfield velocity estimation. Another useful graph property that can be extracted the normal force coefficient acting on the airfoil. While the lift coefficient is usually used to characterize airfoils, it cannot be calculated as the angle of attack is unknown (a quantity to be inferred from the learned model). Nevertheless, the normal coefficient is directly related to the lift coefficient and brings additional physical information which may aid the network to reconstruct the flow. It can be estimated via the following equation:
Σ Pee lu 7y
Cn = ..EV
(5)
Po
where is the boundary length and ny the y component of the normal boundary vector, both of which are known properties for each mesh cell boundary. We therefore use Uin = (Ü.c, Cn) as the two-dimensional input context vector.
For the nodes, we only have access to the pressure distribution at the surface of the airfoil, while it is set to 'NaN' values the fluid nodes. The type of each node is known and is encoded as a one-hot vector, bringing the total number of input node features to four. To account for mesh geometry, the following four edge features are used inputs: the x and y components of the relative edge direction vector, the edge length, and the cell boundary length value I (see Section
Pre-processing Both the input and target node features are normalized. To avoid biasing the normalization, all pressure features normalized by the mean and standard deviation of the known airfoil surface pressure distribution, while both components of the velocity target features are normalized by the initial estimated farfield velocity Üx. We use Feature Propagation (Rossi al., 2021) as a preliminary step before feeding a graph our GNN. This step an important part of our framework as it conditions the input graph into a plausible initial state. Essentially, the feature propagator radiates surface pressure information outwards. In most cases, we found 20 feature propagation iterations to be sufficient.
Encoding In the encoding layer, ReLU activated MLPs with two hidden layers and LayerNorm are used to project the input features of the graph into latent vectors of size N. This encoding layer differs to the standard GraphNet encoder Janchez-Gonzalle et al. 2020 in that the node encoder MIP takes as input the input node features well as the latent global vector. We make this modification so that graph-level attributes are taken into account in the construction of the node latent variables, as this is not the case in the processing steps.
Processing For the processing step, we opt to use a deep Grouped Reversible GNN (Li et al. 202T) with L message-passing layers. This architecture makes modifications the typical GNN architecture by first splitting the input node feature matrix V across the feature dimension into C' groups (V1, V2,... Vc). which are then processed into grouped outputs (V1, V Vi,,., V() with Grouped Reversible GNN layer. These outputs are computed as follows:
V =
V2
(6)
k=2
V/ = (VE)
k (I,
with A the adjacency matrix and E the edge feature matrix. The Grouped Reversible framework allows for any type of message-passing architecture to be chosen

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for the GNN layer fukk We choose to test three popular types of GNN layers: the Graph Attention Network (GAT) (Veličkoiić et al., 2017). the modified Graph Isomorphic Network (GIN) (Xu et al. 2018) which accounts for edge features (Hu et al., 2019), and the Generalized Aggregation Networks (GEN) Li et al. (020) which modifies the standard GCN with different aggregation schemes while also utilizing edge features.
Decoding Only the nodes and global context are decoded back into feature space, the edges not updated. Both decoding neural networks are MLPs with two hidden layers and ReLU activations without any output normalization. At the output of the decoder, gather for each node the estimated pressure and velocity fields. The output context vector is composed of an updated version of the farfield velocity, well estimation of the inflow angle (angle of attack) and of the turbulence intensity.
4.2 TRAINING
General aspects Our models are trained on a dataset composed of 896 graphs. Models are trained with the Adam optimizer on single Nvidia GPU with 10GB of VRAM. Due the size and nature of the graphs, we can only use a batch size of one, albeit with random order shuffling occurring at each epoch. The learning rate is initially set at 5 10 and is exponentially decayed by a factor of 0.97.
Loss function We use multi-component loss function, in order to minimize both the node feature reconstruction error and the context vector prediction error, with L2 losses for both components. An additional loss component based on the velocity divergence also tested but yielded too many artefacts and was therefore discarded. The overall loss is:
L La -22
where X is hyperparameter used to balance the different components. In practice, usually set to 1.
5 RESULTS
Our trained models are tested on dataset comprising of 1 12 unseen airfoil simulations, each with a different combination turbulent inflow parameters. We show here qualitative and quantitative results for models and perform experiments aiming investigate limitations and improvements of the proposed framework.
Comparison of GNN layers Table [ we gather and compare results for the three different types of GNN layers used in the Processor: GAT, GIN and GEN. The architecture of the Encoder and Decoder networks were kept constant, with a latent size for the node, edge and global features of N 128. In the Grouped Reversible Processor, the number of Layers the number of groups was chosen as C = 4. To obtain a consistent number of learnable parameters, the hyperparameters for each GNN layer type different configurations can be found the appendix. We also study the impact of the depth and width on the performance of each model in Appendix D
set to L 30, while
carefully selected, more information about the
Velocity reconstruction One of the more challenging prediction tasks is the estimation of the velocity field away from the airfoil. Accurately capturing velocity shear and recirculation regions is non-trivial even for CFD simulators and is highly dependant on the airfoil shape and the inflow angle. To complete this task sucessfully, the GNN needs to be expressive enough to propagate relevant information throughout the graph. Table 2 multiple concentric regions around the airfoil for the three models. We observe that for all three models, the error on the x-velocity decreases further away from the airfoil, but this is not the case for the y-velocity.
the prediction errors of the velocity field in
Qualitative results Figure 4 displays some qualitative results for our two best performing models (revGAT and revGIN), compared to the CFD simulation ground truth. These results highlight the fact
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Table 1: Root mean squared prediction errors averaged over the test dataset for both the flow reconstruction task and the global parameter estimation task. Results are averaged over 3 runs with different initializations. All models have latent size of N 128 and L 30 layers. While the revGAT model performs the best in terms of pressure and y-velocity reconstruction, it is outperformed by the revGIN model when comes to x-velocity reconstruction, and graph-level attribute prediction.
Node reconstruction RMSE
Global parameter prediction RMSE
pressure
x-velocity [m/s]
y-velocity [m/]]
farficld velocity
angle of attack
turbulence intensity
[Pa]
[m/s]]
[]]
[-]
revGAT
77.98 17.99
7.96 +: 1.19
2.69 + 0.53
0.92 0.17
4.16 +: 0.10
0.04 0.003
revGIN
158.37 = 5.16
6.23 + 0.32
4,43 + 0.22
0.45 0.06
4.12 = 0.09
0.03 + 0.003
revGEN
137.51 + 17.31
10.81 + 0.25
5,47 1: 0.05
0.46 0.05
4.12 0.01
0.04 0.002
Table 2: Root mean squared prediction errors of the velocity components for different concentric regions of the flow, averaged over the test dataset. Each region is defined as the interior of an ellipse with the length of semi-major axis and the length of semi-minor (in chord lengths). Results averaged over 3 runs with different initializations.
x-velocity [m/s]
y-velocity [m/s]
revGAT
revGIN
reyGEN
revGAT
revGIN
revGEN
region (a=0.6, 1=
8.26 1.34
6,46 + 0.39
11.44 + 0.25
2.48 + 0.60
4.15 + 0.18
0.18
5.40 + 0.03
region (a=0.7. b=0.15)
7.98
1.25
6.30 + 0.35
11.00 = 0.26
2.49 0,43
5.52 = 0.05
region (a=0.8, b=0.2)
7.95
1.24
6.27 0.33
10.92 + 0.26
2.61 0.55
4.44 0.223
5.53 + 0.05
that the learned model is able to reconstruct flow features well, albeit with some artefacts. As the distance to the airfoil increases, these defects become more apparent. Moreover, some parts of the flow are not well captured. This is the case for flows exhibiting long wakes. On the other hand, we notice that flow features near the leading edge of the airfoil are in general well captured. We provide additional examples of reconstructed flows in Appendix
Farfield estimation F displays the graph-level context prediction results evaluated on the test set for the revGIN model. The GNN is able to accurately predict farfield inflow velocity, owing to the good initial farfield estimation provided as an input. For the angle of attack estimation, we observe good results small angles but less so for larger positive and negative angles. Prediction of the turbulence intensity is however relatively poor, which can be attributed to this variable having lesser impact on the airfoil pressure distribution. Moreover, this variable is not directly set in the CFD simulations as it is used to calculate turbulent boundary conditions (kinetic energy & and specific rate dissipation w the k: w turbulence model), which makes it more difficult to retrieve in this inverse context.
DISCUSSION
Our results indicate that the type of GNN architecture chosen in the Grouped Reversible Processor has clear impact on the flow reconstruction quality. From our comparison, we find that, overall, using Graph Attention Network layers usually yields the best reconstructed solutions. However, we also observe that the Graph Isomorphic Network layer is better able capture detached flows (see Appendix D.. Perhaps a combination of the two could lead to better reconstructed flows, which could be for instance implemented by alternating GIN and GAT layers. Lastly, we find that the performance of the GEN layer to be somewhat underwhelming.
Something to note that the first few layers nodes surrounding the airfoil carry a disproportionate amount of relevant information which needs to be propagated outwards to increasing number of nodes, thus creating artificial tree-like paths within the graph. While the Grouped Reversible framework provides good way to train deep networks in order to circumvent this, other methods may also feasible. One possible solution might be to use hicrarchies such as those implemented in Martinkus et al. (202T). Another option could be to apply a message- passing GNN in iterative
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(a) Airfoi I, pressure
(b) Airfoil 2, velocity magnitude
U. 58.52 mes, 3,9
Ug 31,08 més, 10.8"
Gironnd trath
Ground truth
-2.
10
Pressure [Pa]
Sanne Ims
revGATT
revGAT
10
20
10
rovGIN
TeyGIN
11
Figure 4: Reconstructed flow around two unseen arbitrary airfoils geometries at different inflow configurations for the revGAT and revGIN models. Figure (a) plots the reconstructed pressure field while the reconstructed velocity field is shown Figure (b). For each case the ground truth is shown for comparison. Qualitatively, the revGAT model displays fewer artefacts in the solution.
(b)
(c)
100
CNN Prediction
20
Bemoulli Estimation
Pred d1, [mis]
60
pred AoA (*)
0,273
60
0.721
40
20
0.175
1-
40
80
80
-20
10
0.15
025
0.30
True U. [m//]]
True ApA [*]
True [-)
Figure 5: Comparison of the global properties predicted by our revGIN model against the ground truth values, evaluated on the test dataset. The model able to accurately predict farfield flow velocity thanks to decent initial estimation (a) and to lesser extent the angle of attack (b), but it falls short for farfield turbulence intensity prediction (c).
manner, with each step reconstructing increasingly large concentric bands around the airfoil. Physics- driven learning methods could also lead to potential improvements. For instance, the message-passing framework may well be suited incorporate elements from the
method (Chen &
Doolen 1998) as it also functions in similar two step algorithm (collision and streaming). Another possible option would be minimize the gradient of the pressure, described in Taha & Gonzalez (2021).
7 CONCLUSION
In this work we applied deep graph-based learning techniques to reconstruct pressure and velocity fields around arbitrary airfoil geometries subject to high-Reynolds turbulent flows. We show that, despite the challenges posed by this problem, such the large graphs and the very localized input information, our Flow Reconstruction GNN framework able to provide good reconstructed solutions, and infer contextual farfield flow information. We compared several message-passing architectures within the Grouped Reversible Processor GNN, and found that Graph Attention Network layers yielded the best reconstrueted solutions. This work provides flexible framework which may easily applied to other mesh-based inverse physics problems, and which may be significant interest to number of engineering applications.

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Sarah Barber, Julien Deparday, Yuriy Marykovskiy, Eleni Chatzi, Imad Abdallah, Gregory Duthé, Michele Magno, Tommaso Polonelli. Raphael Fischer, and Hanna Müller. Development of wireless, non-intrusive mems-baser pressure and acoustic measurement system for large-scale operating wind turbine blades. Wind Energy Science, ((4):1383-11) 1398, 2022.
Steven L Brumton, Bernd R Noack, and Petros Koumoutsakos, Machine learning for fluid mechanics. Annual Review of Fluid Mechanics, 52:477-508, 2020.
Douglas W Carter, Francis De Voogt, Renan Soares, and Bharathram Ganapathisubramani Data- driven sparse reconstruction of flow over a stalled acrofoil using experimental data. Data-Centric Engineering, 2, 2021.
Junfeng Chen, Elie Hachem, and Jonathan Viquerat. Graph neural networks for laminar flow prediction around random 2d shapes. arXiv preprint arXiv:2107. 11529, 2021.
Ming Chen, Zhewei Wei, Zengfeng Huang, Bolin Ding, and Yaliang Li. Simple and deep graph convolutional networks. In International Conference on Machine Learning, pp. 1725-1735. PMLR, 2020.
Shiyi Chen and Gary D Doolen. Lattice boltzmann method for fluid flows Annual review of fluid mechanics, 30(1):329-364, 1998.
N Benjamin Erichson, Lionel Mathelin, Zhewei Yao, Steven L Brunton, Michael W Mahoney, and J Nathan Kutz. Shallow neural networks for fluid flow reconstruction with limited sensors. Proceedings of the Royal Society A, 476(2238) 2020.
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Andrea Schillaci, Maurizio Quadrio, Carlotta Pipolo. Marcello Restelli, and Giacomo Boracchi. Inferring functional properties from fluid dynamics features. 2020 25th International Conference on Pattern Recognition (ICPR), pp. 4091-4098. IEEE, 2021.
Michael S Selig. Uiue airfoil data site. 1996. URL https://heelligaaalliiio.. ads. html
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Nils Thuerey, Konstantin Weißenow, Lukas Prantl, and Xiangyu Hu. Deep learning methods for reynolds-averagel navier-stokes simulations of airfoil flows. AIAA Journal, 58(1):25-36, 2020.
Jake Topping, Francesco Di Giovanni, Benjamin Paul Chamberlain, Xiaowen Dong, and Michael M Bronstein. Understanding over-squashing and bottlenecks on graphs via curvature. arXiv preprint 11. 2021.
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Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural 2018
networks? arXiv preprint
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AIAA Journal, 5959)3300-33 2021.
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APPENDIX
A BENCHMARKING OF THE CFD MODEL
To ensure that the simulations which constitute our dataset are of sufficient quality, we benchmark our CFD pipeline against results reported in the literature for the NACA 0012 airfoil (Krist, 1998 Gregory & O'reilly] 1970p. In Figure we plot the pressure coefficient for this airfoil two angles of attack. We see that our CFD pipeline matches well with both the previous experimental and numerical results.
(
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Figure 6: Comparison of our CFD data pipeline against experimental (Gregory & O O'reilly 1970)
and other CFD results (Krist 1998). We plot the pressure coefficient distribution on the surface of NACA 0012 airfoil at Re = 6 10 at an angle of attack of (a) 10 and (b) 15
B ADDITIONAL HYPERPARAMETE DETAILS
For all models, 30 layers were used in conjunction with a latent space size of 128. The GAT model is implemented with 2 attention heads. Moreover LayerNorm layer is applied to the output of each GAT layer, as it was found that this greatly aided training stability. Both the GIN and GEN implementation makes use of Rel.u activated MLPs with two hidden layers and LayerNorm. These choice of parameters ensure that each individual layer has approximately 4.5k to 5k learnable parameters. In total for the baseline models with layers and latent space size of N 128, we obtain models with around 700k trainable parameters.
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C ABLATION OF ESTIMATED GLOBALS
We plot Figure the result of removing the estimated farfield conditions (Üxc and C.) from the input of the model. We note a drastic penalty in the quality of the prediction farfield quantities.
revGAT (L = 30 N = 128)
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Figure 7: Relative performance of the model when no estimated farfield parameters are used in the input compared the baseline model.
D DEPTH AND WIDTH STUDY
We display in Figure the effect model depth on the reconstruction performance each reversible model. In Figure plot the relative performance of different latent space sizes (the width) for the different models.
E ADDITIONAL RESULTS
We plot in Figure 10 and Figure 11 some additional results, showcasing the reconstructive abilities of models, well some configurations which are not well captured. Notably, detached flows are not always properly reconstructed, an issue which may stem from either the dataset not holding enough detached flows, or from challenges arising from the GNN architecture.
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revGAT = 128
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Figure 8: Comparison of the impact of the depth reconstruction performance for the different layers, relative to the baseline architectures with 30 layers (in green). We also show the relative training time for each depth.
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(a)
revGAT 30
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Figure 9: Comparison of the impact the size the latent space on reconstruction performance for the different layers, relative to the baseline architectures with a latent space size of N 128 layers (in orange). We also show the relative training time for each width.
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(a) Airfoil velocity magnitude
(b) Airfoil 2, velocity magnitude
mis,
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Figure 10: Reconstructed velocity field magnitude around two unseen arbitrary airfoils geometries different large wake inflow configurations for the revGAT, revGIN and revGEN models. For each case the ground truth shown for comparison. Qualitatively, the revGIN model is able to better capture the detached flows in the solution.
(a) Airfoil I, pressure
(b) Airfoil 2, pressure
57.15 =4.
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Pressure [Pa]
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Figure 11: Reconstructed pressure field around two unseen arbitrary airfoils geometries at different inflow configurations for the revGAT, revGIN and revGEN models. For each case the ground truth shown for comparison. Qualitatively, the revGAT model displays fewer artefacts.
16


Reviews:
[{"review_id": "w4rgL5Bm5j_", "reviewer": "Reviewer_TYBR", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper introduces a new dataset simulating the steady airflow on an airfoil. The dataset contains a large number of different geometries and initial conditions. The authors are then interested in estimating the velocity field and the global context of the simulation from the sole knowledge of sparse measurements of the pressure field on the surface of the airfoil. The proposed method rely on feature propagation and grouped reversible GNN.", "summary_of_the_review": "Despite the good quality of the work presented, my first impressions lead me to believe that the novelty is limited. Beyond that, some design choices seem questionable to me, and I hope the authors can provide us with convincing justifications. In that case, I may increase my grade.", "strength_and_weaknesses": "The paper focuses on a difficult but very interesting problem, with direct applications in aviation. Reconstruction of flows from sparse measurements has already been studied in Erichson et al. but in a very simple scenario. The task introduced by the authors is much more challenging, and the proposed solution seems reasonable. Following the structure of the paper, I organized my questions and remarks in two sections dealing with the dataset first, then the model. These are sorted in ascending order of importance.\n\nAbout the dataset\n\n- I would have appreciated more statistics about the dataset, including its total weight, average number of nodes and edges, etc.\n\n- It seems that the simulations were carried out under the assumption of incompressibility. It seems to me that this assumption does not hold for flow on an airfoil (Sanchez-Gonzales et al. 2020). Can the authors discuss the relevance of this assumption?\n\n- However, I am quite skeptical about the novelty of the dataset. The authors simulate the evolution of the velocity and pressure field over a large number of different wing profiles and under a large set of initial conditions. However,  this type of dataset already exists in other works. In particular, [1] seems to me to be the closest, but other variants exist [2], [3] (non-exhaustive list). Would it be possible to complete the related works section of the article by comparing the new dataset with the existing works?\n\nAbout the model\n\n- In appendix B, the models are limited to 4.5k parameters. It seems very low. In comparison, MeshGraphNet (Sanchez-Gonzalez et al. 2020) contains 1.5M parameters for L=15 layers of GNN (but the model is designed for dynamic forecasting). Is this a typo? Are GNN weights shared across layers?\n\n- Section 4.2 presents an auxiliary loss enforcing the divergence of the velocity field to be close to zero. This is perfectly justified in the case of an incompressible simulation. Nevertheless, it is not always respected by the simulator. Have the authors verified that the null divergence hypothesis is verified on their dataset? Also, what is the effect of ablating this term on model performances?\n\n- In general, I deplore the absence of more baselines to compare with the proposed model, which novelty is rather limited. I wonder in particular what is the effect of the grouped reversible GNN relatively to a more traditional solution such as chaining layers of GAT/GIN/GEN?\n\n- The model seems to use a simulation bias to obtain a first approximation of the magnitude of the velocity farfield (the fact that the simulation uses a null pressure field at the boundary). So I have three questions: (1) To what extent could this approximation be performed on real data? (2) How far is this first approximation from the true velocity farfield (this could be checked by plotting figure 5a for this approximation)? and (3) what is the contribution of this first estimation in the performance of the models? in other words, does the performance deteriorate noticeably without this approximation?\n\nI nevertheless point out the relevance of the experiments carried out, in particular table 2 and figure 5 which correctly analyze the important points of the model. I noted several citation errors in the references, and 11 pre-prints among the 33 citations.\n\n[1] Thuerey, N., Weißenow, K., Prantl, L., & Hu, X. (2020). Deep learning methods for Reynolds-averaged Navier–Stokes simulations of airfoil flows. AIAA Journal, 58(1), 25-36.\n\n[2] Ahmed, S., Kamal, K., Ratlamwala, T. A. H., Mathavan, S., Hussain, G., Alkahtani, M., & Alsultan, M. B. M. (2022). Aerodynamic Analyzes of Airfoils Using Machine Learning as an Alternative to RANS Simulation. Applied Sciences, 12(10), 5194.\n\n[3] Schillaci, A., Quadrio, M., Pipolo, C., Restelli, M., & Boracchi, G. (2021, January). Inferring functional properties from fluid dynamics features. In 2020 25th International Conference on Pattern Recognition (ICPR) (pp. 4091-4098). IEEE.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "nzZRwsCTHky", "reviewer": "Reviewer_N3A3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The manuscript is describing how to estimate the flow of air around an airfoil only from the pressure and velocity distribution on the airfoil for high Reynolds number flows. Graph based structure is used to define a \"mesh\" has some intelligence, i.e. features at each vertex. The paper combines far-field information with contextual qualities of the incoming field. A digital ensemble of training data is created with OpenFoam and RANS turbulence models.\n\nThe far-field conditions are initialised with Bernoulli flow and the pressure with a random sample with a mean and standard deviation of the values on the airflow. The pressure values are propagated with about 20 feature propagation steps. Different architectures are studied.\n\nThe trained Graph model do give quite good velocity profiles, although the turbulence and flow detachment point seem to diverge. From global features, especially the far field velocity, is estimated very well.\n\n. ", "summary_of_the_review": "The paper reproduces the state of the art of standard CFD codes, with the addition that the inflow velocity is kept open. The benefits of this solution should be brought more clearly forward - like how much faster this compared to a normal CFD analysis of the problem - especially as the turbulence prediction seems to be off.", "strength_and_weaknesses": "Strengths: Original problem, and a solution that provides quite good inference. \n\nWeaknesses:  The update scheme in the graph could be seen in terms of the Lattice Boltzmann models, except the flow update is defined in an a way that does not automatically lead to zero divergence in summing up the velocities to each vertex to zero. Perhaps relaxation has to be done for a faster convergence. What happens, if the solution is just propagated with from the airfoil surface to the outer edge with the Boltzmann lattice update rules? Also, I think more work should be done on calculating the training data with a LES solver that goes to sub graph network \"mesh\" sizes, so that the complexity of the features could be used to find a closure for the turbulence and give more reliable solutions for the turbulence, as this is the fundamental problem at high Reynolds numbers.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "FvooVrVg5q3", "reviewer": "Reviewer_up38", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper considers the problem of extrapolating and reconstructing the flow around 2-D airfoils using measurements on the surface. A deep reversible graph neural network is trained to perform this reconstruction ", "summary_of_the_review": "Overall, the paper takes a standard GNN approach and uses that for CFD analysis. Some of the doubts of the reviewer are attempting to ascertain the value of this work in the CFD domain, since the novelty from the point of view of ML is limited to application. The paper appears to be a work in progress.", "strength_and_weaknesses": "Strengths\n\n- A rich dataset is generated with graph representations of fluid around the airfoil taking 60,000 nodes, different inflow conditions (flow velocity, turbulence, and angle of attack) and different airfoil shapes from the UUIC database.\n\n- The study not only explores different shapes but does a decent sweep of inflow conditions ranging across a large interval of Reynolds number (2 * 10^5 to 6.5 * 10^6). Non-laminar flow (beyond 5 * 10^5) makes flow reconstruction difficult. \n\n- The idea of using global context features in addition to node features is interesting. Incorporating the length of the boundary between cells as an edge feature is also neat. But could authors discuss why the analysis was limited to 1 chord circle centered on the manifold? Could the scalability challenge be elaborated with some quantitative analysis? \n\nWeaknesses\n\n- In absence of access to code, it is not feasible to determine if the meshing by Gmsh and using OpenFOAM has yielded trustworthy simulation data. There is some mention of details such as sizing field but the reviewer is unsure about fidelity of the dataset. Typically, tools such as Ansys Fluent and Comsol are used because of their ability to provide guidance in running CFD on different shapes. Even then, human expertise is critical in CFD analysis. Since the paper mentions dataset as a critical component, the absence of code and dataset for review makes it difficult to determine the quality and hence significant of this part of the paper.\n\n- The experimental results appear to be mixed. The proposed approach fails on farfield turbulence intensity prediction and struggles with prediction of AoA.  It appears the training set was 896 large and the testing was done on 112 airfoils. It is important to make sure that the training and test set had airfoils that were sufficiently different from each other. It will be useful to describe how this split was done. Considering many different splits will also be important. Using a parametric airfoil shape generator can enable scaling the experiments further. Also, is this method restricted to 2D? Why not test the method also on 3D examples? \n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "1qKOcBlIsF", "reviewer": "Reviewer_FKyW", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes to use MPNNs to reconstruct the (time-averaged) pressure and velocity fields around an airfoil based on measurements of the pressure distribution at the airfoil's surface. Some global parameters of the flow (farfield velocity, angle of attack, turbulence intensity) are also predicted. The authors use Grouped Reversible GNNs with three different layer types and with Feature Propagation. The models are trained in a supervised manner using a set of over 1k FVM simulations, and in evaluations, the GIN and GAT layers perform better than the GEN layer.\n", "summary_of_the_review": "This is an interesting application paper. By tackling a wider range of conditions in a single model, this paper moves towards more realistic scenarios than prior work. However, with all simulations restricted to 2d steady-state conditions, this still feels quite far from the practical applications with MEMS sensors mentioned in the abstract as motivation. With no ML innovations, and lack of clarity whether the relative performance of the different GNN layers in this context has any wider applications, I think that this paper could be better suited to a more specialized venue.\n\n\n", "strength_and_weaknesses": "Strengths of the paper include the use of varying experimental conditions (airfoil shapes, angles of attack, Re) and high Reynolds numbers. The paper also contributes the dataset of simulations used to train the model, and the code used for the experiments is available.\n\nA limitation of the experiments is that only 2d, steady state cases were evaluated. Real settings of interest where pressure sensors are mounted on a physical airfoil would generally be 3d and time-dependent. The experimental exploration done is also limited to testing the 3 layers -- there is no report of testing of the different hyperparameters of the model (L, N, C). The physics-based divergence-free loss component is mentioned, but then not discussed further, so it's unclear what role it plays.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject