Deep learning fluid flow reconstruction around arbitrary two-dimensional objects from sparse sensors using conformal mappings

The usage of neural networks (NNs) for flow reconstruction (FR) tasks from a limited number of sensors is attracting strong research interest, owing to NNs' ability to replicate high dimensional relationships. Trained on a single flow case for a given Reynolds number or over a reduced range of Reynolds numbers, these models are unfortunately not able to handle flows around different objects without re-training. We propose a new framework called Spatial Multi-Geometry FR (SMGFR) task, capable of reconstructing fluid flows around different two-dimensional objects without re-training, mapping the computational domain as an annulus. Different NNs for different sensor setups (where information about the flow is collected) are trained with high-fidelity simulation data for a Reynolds number equal to approximately $300$ for 64 objects randomly generated using Bezier curves. The performance of the models and sensor setups are then assessed for the flow around 16 unseen objects. It is shown that our mapping approach improves percentage errors by up to 15\% in SMGFR when compared to a more conventional approach where the models are trained on a Cartesian grid, and achieves errors under 3\%, 10\% and 30\% for pressure, velocity and vorticity fields predictions, respectively. Finally, SMGFR is extended to predictions of snapshots in the future, introducing the Spatio-temporal MGFR (STMGFR) task. A novel approach is developed for STMGFR involving splitting DNNs into a spatial and a temporal component. We demonstrate that this approach is able to reproduce, in time and in space, the main features of flows around arbitrary objects.


I. INTRODUCTION
Most fluid dynamics experiments have access to only sparse measurements, due to the intrusive (i.e. flow-altering) nature of pitot tubes and pressure probes used for measurements. Although noninvasive methods to obtain full flow fields in experiments such as particle imaging velocimetry 1 (PIV), magnetic resonance velocimetry 2 (MRV) and laser doppler flowmetry 3 (LDF) exist, their usage can be limited by practicality, cost or safety constraints; for instance PIV systems often require 'Class IV' lasers that can gravely harm human eyes and cost thousands of dollars. Despite these practical limitations in experiments, knowledge of the full flow fields is often critical to understanding the dynamics of many complex fluid flows. Flow reconstruction (FR) methodologies can offer reliable estimation of a full flow field from only sparse measurements.
Callaham et al. 4 describe the FR task in terms of a high-and a low-dimensional state vector x ∈ R m , s ∈ R p , m >> p, where x represents the 'full' flow field and s are sparse sensor measurements.
The two state vectors are linked through measurement and reconstruction operators H : R m → R p and P : R p → R m such that x = P(s) The goal of the FR task is to find an approximation mapping R : R p → R m such that some measure of error, typically L 2 , betweenx = R(s) and x = P(s) is minimized. In practice, F is a statistical or deep learning algortihm with a parameter set w optimized to fit some dataset; i.e. R(s) = R(s, w).
Deviating from the formulation by Callaham et al. 4 to use some generic objective function L instead of the L 2 norm, the flow reconstruction task can be expressed as a minimization problem of the following form arg min w L(x, R(s, w)) Historically, methods such as Gappy PODs 5,6 and Linear Stochastic Estimation 7 are some of the main methodologies investigated for FR, but they are often unsuitable for multi-geometry FR (the reconstruction of the flow field generated past an arbitrary geometry), as detailed in Section II. Research in this field has recently intensified, and a new wave of studies -the vast majority focused on using neural networks (NNs) -have been published, see 4,[8][9][10] to name a few. As a starting point for neural network based flow reconstruction, these publications largely focus on obtaining models that work on a single fluid flow case, typically predicting vorticity fields for incompressible flow past a circular cylinder at a single (or over a narrow range) of Reynolds numbers. As a result, such approaches do not have the potential yet to be used in wind tunnel testing driven shape optimization, as training such NNs first requires collecting large datasets of the flow past specific objects.
Training NNs for multi-geometry FR from sparse sensors is not straightforward. Within the aforementioned setting of vorticity field reconstruction on Cartesian grids for two-dimensional (2D) incompressible flows past arbitrary objects, naively augmenting the dataset with multiple objects results in models that fail to reproduce key flow features; concentrations of vorticity in boundary layers and near stagnation points often disappear, and the objects themselves are engulfed by amorphous blobs of non-physical vorticity concentrations. The root cause of these issues is that, within these settings, the models lack information regarding the shape of the object they are making predictions on, as the need for such information is obviated by the single-shape nature of the datasets.
Overcoming these issues requires a representation of the flow field in a way that removes the necessity for the model to predict the shape of the object immersed in the fluid. An effective tool for this is a mapping, whereby all possible geometries are mapped to a single shape. For 2D cases, this can be achieved via the Schwarz-Christoffel (S-C) conformal mapping, which can be used to map any l-connected domain to a disc with l holes 11 . Thus, the fluid domain in any bluff body flow with a single object can be mapped to an annulus.
In this work, the Spatial Multi-Geometry Flow Reconstruction (SMGFR) task is introduced, with the objective of reconstructing pressure, velocity and vorticity fields surrounding randomly generated objects immersed in a fluid flow from sparse sensor measurements. S-C mappings are utilized to map the fluid computational domains surrounding the said randomly generated bluff bodies to annuli, and a dense field sampling approach based on grids uniformly spaced in angular and radial directions in the annular domains is developed. Over a comprehensive set of 24 experiments (different models and sensor arrangements), the mapping approach is compared to reconstruction based on uniformly spaced Cartesian grids, and the performance of different sensor setups and NN architectures (feed forward, U-Net 12 and Fourier Neural Operator 13 ) in SMGFR are investigated.
As a further step towards spatio-temporal reconstruction, a modified version of the SMGFR is proposed whereby the model is expected to construct snapshots at future times given sensor readings at a present time. In this task, dubbed Spatio-temporal Multi-Geometry Flow Reconstruction (STMGFR), a model obtained from the spatial-only SMGFR task (called the spatial model) is coupled with a second neural network model. This second model, called the temporal model, accepts the reconstructed dense field as its input and predicts the state of the full flow field k time-steps in the future. The resulting system, composed of the spatial and the temporal models, is thus able to reconstruct the full flow fields k time-steps in the future given current sensor measurements.
The work is organized as follows: first, a brief overview of recent and historical approaches to the FR task is provided in Section II. Subsequently, an introduction to the S-C mapping and details of its novel application to the FR task are presented in Section III. The dataset constructed to take advantage of the S-C mapping is described in Section IV, and the models chosen to fit this dataset as well as their training procedures are described in Section V. Finally, the results are showcased in Section VI and a summary of this work and plans for future investigations are provided in Section VII.

II. RELATED WORK
The FR task, as introduced in Section I, falls under the broad umbrella of inverse problems 14 .
Commonly encountered in a wide range of scientific and engineering fields including fluid dynamics, inverse problems often lack well-defined, unique solutions and rely on minimization of some objective such as the L 2 norm, as encountered e.g. in the Moore-Penrose pseudo-inverse for linear least-squares problems. Dubois et. al. 8  In comparison, the body of works investigating deep learning based FR for the reconstruction of the flow past arbitrary objects without re-training is small. One notable recent work in this area is by Chen et al. 34 , using graph convolutional neural networks (GCNNs) for reconstructing steady flow fields around random objects. Training a GCNN to predict the velocity and pressure fields around 1600 random objects generated via Bezier curves, they applied the model to predict the pressure and velocity fields around 400 test geometries, which permitted the estimation of drag and lift coefficients with very small mean percentage error levels. The present work differs from Chen et al. 34 as it focuses on the usage of a novel mapping approach to achieve the geometry invariance as opposed to graph convolutions which permits the usage of traditional NN architectures. Here the aim is to reconstruct instantaneous snapshots as opposed to steady fields, and to explore predicting future instantaneous snapshots from current measurements. Additionally, this work is conducted at a substantially higher Re = 300 as opposed to Re = 10 in Chen et al. 34 which leads to the emergence of unsteady rotational flows. To assess the performance of the models, the focus is on reconstructing the vorticity fields (which are difficult to predict from pressure and velocity sensors, as shown in Section VI A). Reconstructed data for the pressure and velocity fields are also briefly presented for completeness.

III. SCHWARZ-CHRISTOFFEL MAPPINGS
Conformal transformations have been used extensively in fluid dynamics, especially the wellknown Joukowsky and Kármán-Trefftz 35 (K-T) transformations which map the unit circle to airfoil shapes. However, the usefulness of these two transformations can be limited due to their inability to generalize to arbitrary shapes. A more flexible alternative is the Schwarz-Christoffel (S-C) mapping, which is a conformal transformation historically used to map polygonal simply connected domains to the unit disc. The S-C mapping has been extended in the recent decades to multiply connected domains. Although the existence of a conformal mapping between any given two l-connected domains is guaranteed 11 , the practical computation of such an S-C mapping typically requires the use of numerical methods to determine a number of parameters in the mapping expression, referred to as the Schwarz-Christoffel parameter problem. Numerically implementing the S-C mapping for doubly (or higher) connected domains is not a trivial undertaking. In this work, the DSCPACK code 36 , which is a Fortran package aimed at computing S-C mappings between doubly connected domains bounded by polygons to annuli, was utilized to solve the parameter problem.
A rigorous treatment of the methodology used in this package lies beyond the scope of this work, for which the reader is directed to established works in S-C mapping literature 11,37,38 , though the general strategy can be summarized as follows: denoting z as the complex coordinates in the original domain and w as the complex coordinates in the annulus domain, DSCPACK uses an expression of the form as the mapping, where C is some complex valued constant; µ is the radius of the inner ring of the annulus in the w-domain; M, α 0q , w 0q and N, α 1r , w 1r are the number of vertices, the turning angles and prevertices 39 of the outer and the inner polygon, respectively. Of these variables, C, µ, w 0q and w 1r are unknowns ('accessory parameters' of the mapping) and must be computed by solving a series of nonlinear integral equations where z 0q and z 1r denote the complex coordinates of the polygon vertices in the original domain.
The DSCPACK code solves this nonlinear system using a Newton iteration scheme. The resulting expression maps the outer ring (with unity radius) of an annulus to the outer polygonal boundary, while the inner ring is mapped to the inner polygon.
Once the forward mapping g is known, the inverse mapping f can be approximated for any where Equation 14 follows from the application of the fundamental theorem of calculus to Equation 4.
To ensure smooth interoperability of DSCPACK with modern machine learning packages, a set of Python bindings to a modified form of the code were developed, dubbed pydscpack. Furthermore, a number of enhancements to the original code were made to parallelize performance-critical sections with OpenMP. Figure 1 depicts an S-C mapping computed using pydscpack, for a geometry used in this study.

B. Ground truth values
Using uniform Dirichlet velocity boundary conditions (u, v) = (1.0, 0.0) along the external edges of the domain, the flow around each object was computed for Re = uL m /ν = 300 (where ν is the kinematic viscosity) using the PyFR solver 41 , which is a flux reconstruction 42 based advectiondiffusion equation solver using the artificial compressibility approach to solve the incompressible Navier-Stokes equations. It was chosen for its Python interface and GPU acceleration capabilities.
The simulations were performed on two Nvidia V100 GPUs. Normalizing the physical time τ by the large eddy turnover time to obtain τ * = uτ/L m , t ∈ [0..600] snapshots (containing the pressure and velocity field components p t,i , u t,i and v t,i ) were recorded per geometry for a total of 48080 snapshots between τ * = 3.333 and τ * = 23.333.
Following the simulations, referring to the fluid domain around each G i as F i , the forward and inverse mappings g i and f i between F i and the corresponding annuli A i were computed using pydscpack. 64 × 256 grids uniformly spaced in the radial and angular directions with coordinates w A,i were generated for each A i . Subsequently, w A,i were mapped back to the original domains F i using the computed mappings g i to obtain the annular grid coordinates in the original domain z A,i = g i (w A,i ). The velocity, pressure and vorticity fields u t,i , v t,i , p t,i and ω t,i from the highfidelity simulation data were interpolated to z A,i to obtain the interpolated fieldsũ t,i ,ṽ t,i ,p t,i and ω t,i , which form the ground truth values of the Annular dataset.
This sampling strategy and grid resolution provide a high grid density near the object, scaling with a factor of 1/r based on distance to the center of the annulus. It ensures that the regions of the flow with high vorticity concentrations have enough grid points for a correct representation of the vortical structures. Additionally, as a baseline case, a further collection of ground truth values sampled naively on a 128 × 128 uniformly spaced Cartesian grid was also produced, with the same number of grid point as for the mapping approach. Note that these grids are used solely for the interpolation of flow variables, not to perform the fluid simulations.

C. Inputs
The inputs of the dataset are vectors of pressure and/or velocity values s t,i at a sparse number of sensor locations, obtained via interpolation of the PyFR solution to the sensor locations. The sensor setup to build the inputs of the dataset is split into two sensor types, chosen to represent a setup that can be practically implemented in a laboratory environment: • Pressure: Placed on the surface of each G i , with equal angular spacing along the inner ring of each A i .
• Velocity: Positioned on a rectangular grid spanning a 2L m /3 × 4L m /3 region, the left edge of which is L m /6 units behind the rearmost point of each G i and the centroid of which is vertically level with each G i .
Based on this general template three setups with varying sensor quantities were considered, summarized in Table I. Figure 3 depicts the medium sensor setup for a sample geometry.

D. Normalization
Normalizing inputs and/or outputs plays an important role in obtaining good performance from deep learning algorithms, as it permits better conditioning of the gradients within the optimization landscape during training, by keeping the per-layer statistical distribution of the gradients  43 . A variety of data normalization methodologies, including mean centering, standardization and min-max scaling were tried in a preliminary study. Denoting X as the dataset inputs and T as the target values, X + and X − as the maximum and minimum values of X, respectively, and µ and σ as the mean and standard deviation, respectively, the three normalization methods can be summarized as follows: Mean centering: Mean centering both inputs and ground truth values based on the ground truth mean values was chosen as the data normalization method, as it provides the results with the lowest validation loss levels for models trained using either the Cartesian and Annulus datasets.

V. EXPERIMENTAL SETUP
Two related tasks have been investigated, with the dataset detailed in Section IV. The first is identical to spatial flow reconstruction tasks from sparse sensors in previous literature 4,8,33,44 , but with the inclusion of snapshots taken from a multitude of geometries in the training and validation datasets. As a reminder the name Spatial Multi-Geometry Flow Reconstruction (SMGFR) is used to describe the FR task in this specific configuration. The second task is a generalization of SMGFR where target snapshots are in the future relative to the sensor measurements by a fixed amount of time ∆τ * , as opposed to SMGFR where the target snapshots are contemporaneous with the measurements, dubbed Spatio-temporal Multi-Geometry Flow Reconstruction (STMGFR).
For both tasks, the dataset from Section IV was split by randomly choosing the data associated with 64 geometries as the training set; the remaining 16 geometries constituted the validation set.
Training in all experiments was conducted using the Adam 45 algorithm using an initial learning rate (LR) of 10 −3 , reduced by 90% each time the loss values plateaued. A. Spatial multi-geometry flow reconstruction (SMGFR) Using the notation in Section IV, the SMGFR task can be summarized as predictingp t,i ,ũ t,i , v t,i orω t,i given s t,i . The experiments investigate the performance of four different models, all implemented using Tensorflow 46 v2.5.1, with the parameter counts available in Table II. The latter two models are described schematically in details in Figure 4. Below is an overview of the models investigated, with some arguments to justify their use in the present study: ...  in the parentheses indicate the shape of each block's output tensor; D1=64 and D2=256 for the Annulus dataset, and D1=D2=128 for the Cartesian dataset. In the SD-UNet, each convolution block is formed of two convolution layers, each preceded by a batch normalization layer and followed by a dropout layer, each deconvolution block is formed of a batch normalization layer followed by a deconvolution with stride 2.

B. Spatio-temporal multi-geometry flow reconstruction (STMGFR)
The spatio-temporal multi-geometry flow reconstruction (STMGFR) task extends the purely spatial SMGFR task presented under Section V A. Whereas SMGFR focuses on obtaining a reconstructionx t,i of the (interpolated) target fieldx t,i (e.g. Training the temporal model is done in a supervised manner. Since the temporal model would be expected to perform well given reconstructed inputsx t,i from the spatial model in an inference scenario, in each epoch the input associated with every sample is randomly chosen to be either a reconstructed vorticity fieldx t,i or a ground truth fieldx t,i , with a 50% chance for each. Using separate spatial and temporal models with parameter counts P s and P t (versus a larger single model with parameter count P s + P t directly predictingx t+k,i from s t,i ) is highly computationally efficient as it permits easy re-training of multiple temporal models for different values of k. The results from the spatial model can be easily cached and re-used when training a new temporal model, which translates to computational speedups as well as model accuracy benefits owing to the possibility of using larger batch sizes given a fixed pool of memory.

VI. RESULTS
The results of a series of SMGFR and STMGFR experiments are detailed in this section, the setups of which are detailed in the previous chapters. First, to compare the accuracy and quality of our reconstruction methodology to the previous work by Chen et al. 34 , we briefly present results on reconstructing pressure and velocity fields in Section VI A. Additionally, we numerically demonstrate that the reconstruction of vorticity presents a greater challenge than the reconstruction of pressure and velocity from pressure and velocity sensors.
Subsequently, we proceed to detailed comparisons of vorticity reconstruction performance, where the differences between the various setups are visible more acutely. Section VI B details the results from the spatial reconstruction task with four models and three sensor setups, as detailed in Section IV and Section V A. With the best performing configuration for the spatial task identi- A. Spatial multi-geometry pressure and velocity reconstruction In order to compare the quality of our reconstruction methodology with the previous multigeometry reconstruction work by Chen et al. 34 , we briefly present the results of training the k = 0 SD-UNet+FNO combination from Section V B on pressure and velocity data in Table III, using the large sensor setup.
Additionally, to demonstrate that a fairly complicated and non-linear reconstruction relationship P is present between the sensor measurements and the vorticity fields investigated in Sections Section V A and Section V B, we include two difficulty measures D and M in Table III. These measures are based on the Frobenius norms of the Spearman rank correlation coefficient matrix 49 (SRCC) and Mutual Information 50 (MI). Defining ψ t,i = [s t,i x t,i ] ∈ R p+m as a vector concatenating the sensor measurements s ∈ R p and the full field x ∈ R m (pressure, vorticity etc.) for a particular snapshot i at a particular time t, we construct a large matrix Ψ containing the entirety of the data in our dataset: The difficulty metrics clearly show the reason for lower performance when predicting ω with all setups. The D scores display greater correlation between the sensor inputs and pressure/velocity fields compared to the vorticity field. This translates to, on average, greater monotonicity in the relations mapping the sensor data to the full field data for the pressure and velocity fields compared to the vorticity field. M , meanwhile, demonstrates that the probability distributions of the sensor inputs and full field values are substantially more alike (i.e. have lower relative entropy). Both of these have profound effects on the accuracy of the neural networks, which manifests in the difference in the MAPE scores when predicting different target fields. Since the higher difficulty associated with predicting the vorticity field is illustrated, we move forward to comparing the quality of our pressure and velocity results with previous works. Chen et al. 34 reported reconstruction errors amounting to 7.70 × 10 − 3 in a similar setup, but at a substantially lower Re, predicting on flow cases within steady, laminar flow regimes only. Thus, considering the substantially higher Re in this work which results in the creation of unsteady vortical structures, the differences in data generation methodologies, and the different objective involving the prediction of instantaneous as opposed to steady fields, the error levels exhibited are in line with previous works. The MAPE levels, under 3% and 10% respectively for pressure and the velocity components, clearly demonstrate that our work is a clear step forward for SMGFR. A gallery of sample velocity and pressure predictions is provided in Appendix A.

B. Spatial multi-geometry vorticity reconstruction
Considering the higher difficulty of predicting the vorticity field from pressure/velocity sensors, and to push the boundaries of the neural networks for flow reconstruction tasks, a comprehensive set of 24 experiments for the vorticity SMGFR have been conducted to highlight the differences between the combinations of sensor setups, model architectures and sampling strategies. Table IV summarizes the performance of all combinations.  To provide deeper insight into the performance of the models with the different sensor and sampling setups beyond the overall error values, a gallery of predictions is provided. Figures 6,   7 and 8 depict the predictions for a bluff body-like shape, dubbed 'Shape A', using the large, medium and small sensor setups, respectively. Furthermore, Figure 9 and Figure  Moving forward to Shape B in Figure 9, the difference is most striking for the SD and SD-Large, where the Cartesian versions of the models predict a high concentration of negative vorticity entirely engulfing the object, which is highly non physical. Additionally, the location of the high positive vorticity blob near the 'leading edge' of the object is predicted as detached from the object surface. In contrast, the same models with annular sampling correctly predict these key The trends for the other shapes continue in the case of Shape C in Figure 10. Similar to both feedforward models for Shape B, the SD-Large model with Cartesian sampling spuriously predicts a large concentration of very high vorticity surrounding the object while also substantially underestimating the intensity of the downstream vortices. The SD-UNet with annular sampling performs the best but with Cartesian sampling the error is much higher, also due to an underestimation of the intensity of the two counter-rotating vortices behind the object. Finally, the SD-FNO is the trendbreaker, with Cartesian sampling managing a narrow quantitative win thanks to a better estimation of the vortex intensity. However, from a qualitative perspective, both SD-FNO predictions are noisy and do not accurately reproduce the smoothness of the vorticity field unlike the SD-UNet.
As a final remark, we draw attention to the presence of high error along the same contours for different sampling strategies and models among the images for each snapshot. This is due to the presence of very high percentage error (despite low absolute error) near the zero contours of the target field due to very small denominators. Visible in low vorticity areas across all geometries, it is ultimately caused by the objective function as explained above. It is also the main reason why the MAPE and HV-MAPE may appear high with values consistently between 20% and 60%.

C. Spatio-temporal multi-geometry vorticity reconstruction
Since the SD-UNet model, used alongside annular sampling and the large sensor setup, was identified as the best performing combination in Section VI B, this combination was chosen for the spatial model in this work's approach to the STMGFR task. The performance of the temporal model is summarized in Table V for different values of the temporal gap k (refer to Table IV   As expected, the error declines as the temporal gap k is reduced and the model performs better when ground truth snapshots are provided as the inputs as opposed to reconstructed inputs. Surprisingly, however, MAPE levels for this task are substantially lower than results for the purely spatial task in Table IV despite  Finally, Shape F in Figure 13 is a thick airfoil like shape, also set at a high incidence angle The good performance of the temporal model when given ground truth snapshots as inputs is largely consistent with the previous literature on the FNO 13 , where the capability of the FNO to time-march the vorticity field for turbulence-in-a-box settings were demonstrated with low error levels. The additional insight, however, is that these results demonstrate that the FNO model coupled with our training methodology (whereby 50% of the inputs are randomly replaced with spatial model predictions) is robust to noisy inputs, with a loss of accuracy of the order of 2% (as shown in Table V) despite average errors in the input approaching 40%. In fact, in two of the three cases displayed in the figures, the MAPE of the temporal model prediction from the spatial model reconstruction relative to the ground truth at time t + k is lower than the MAPE of the spatial model reconstruction relative to the ground truth at time t. In a physical experimental setting, where measurement noise is a real concern, this is a key capability as the impact of the measurement errors on the spatial reconstruction will not catastrophically degrade the accuracy of the temporal reconstruction.

D. Training and inference time
The final topic of discussion for comparing the relative merits of the different model architectures in the previous sections is the computational cost of training and using each model. Table VI outlines the wall-clock runtimes for running training (conducted on an IBM AC922 system with two 20-core POWER9 CPUs and two Nvidia V100 GPUs) and inference (conducted with an AMD EPYC 7443 CPU and a single Nvidia A100 GPU) using each model, using the single-precision floating point format. While the SD-UNet and SD-FNO consistently displayed better performance in terms of numer- The runtime costs for the (temporal) FNO model used as the temporal component of the approach to STMGFR are even higher than the spatial models, though it is likely that an implementationspecific issue was present as GPU power draw was observed to be substantially lower for this model during inference than the four spatial architectures despite high utilization statistics.

VII. CONCLUSION AND FUTURE WORK
This work introduced the Spatial and Spatio-temporal Multi-Geometry Flow Reconstruction (SMGFR, STMGFR) tasks for reconstructing dense contemporaneous or future vorticity fields of flows past arbitrary objects from current sparse sensor measurements, respectively, without geometry-specific training. To achieve optimal performance in these tasks, the use of Schwarz-Christoffel mappings to choose the sampling points of the dense fields was explored.
The performance of four different models was investigated on the SMGFR task, using datasets For the novel STMGFR task involving the prediction of future snapshots given current sensor measurements, an innovative approach separating the task to spatial and temporal components was developed, whereby a spatial model first reconstructs the current snapshot given the current measurements (equivalent to the SMGFR task) and subsequently a temporal model predicts the future snapshot given the predicted current snapshot. A stack of Fourier Neural Operator 13 layers acting as the temporal model was coupled with the best performing configuration from the SMGFR experiments used as the spatial model. The temporal model was trained to be robust to input noise caused by inaccuracies in spatial model predictions, by randomly providing it spatial model predictions or ground truth snapshots for each sample during training. Experimentation indicated that this approach is capable of accurately reconstructing future vorticity snapshots with mean absolute percentage error levels on the order of 30%. Furthermore, using a temporal gap of zero, the same two model setup can be used to improve accuracy of models used in SMGFR, also bringing their MAPE levels below 30%.
We hope to expand the investigations in this work in the future by: • Developing models that perform well over a range of Reynolds numbers. The present work focused on predictions for flows at Re ≈ 300; the models presented are not expected to perform well for other Re and likely require re-training to adjust to different Reynolds numbers. Two potential ways of overcoming this are using a dataset containing snapshots from a range of Reynolds numbers, and using 'physics-informed' loss functions.
• Experimentation with more advanced neural network architectures. While this work focused on the relatively simple case of supervised training of feedforward and convolutional architectures to focus on investigating the mapping approach presented, techniques such as generative adversarial networks 54 and variational autoencoders 44 are gaining traction in FR literature. Adoption of such techniques may lead to higher accuracy in SMGFR and STMGFR tasks.
• Extending the methodology to 3D fluid flows. The mapping approach in this work relies on the Schwarz-Christoffel mapping, which is defined for the complex plane C only. A version of this work for R 3 will require an alternative mapping approach.
• Prediction of lift and drag coefficients. Although this work focused on the reconstruction of vorticity fields for the ease of comparison with previous works, changing the target fields to velocity and pressure can permit the prediction of the lift and drag coefficients. The mapping approach is especially conducive to this as, unlike Cartesian sampling, it eliminates the need to further process the vorticity and pressure fields to obtain values at the object boundary.

VIII. ACKNOWLEDGEMENTS
This work was supported by a PhD studentship funded by the Department of Aeronautics, Imperial College London and an Academic Hardware Grant provided by Nvidia. The authors would like to thank Sean Chai, Neil Ashton and Thomas Delillo for fruitful discussions at the start of this study.

IX. CODE REPOSITORIES
The pydscpack library for computing Schwarz-Christoffel mappings and the code for replicating the results in this work can be found on GitHub.