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

TL;DR

This work tackles flow reconstruction around arbitrary 2D objects from sparse sensors by employing Schwarz-Christoffel mappings to map the computational domain to an annulus, enabling geometry-invariant learning. It introduces SMGFR for instantaneous reconstruction and STMGFR for short-horizon predictions, demonstrates substantial accuracy gains over Cartesian sampling across pressure, velocity, and especially vorticity fields, and shows that a spatial model paired with a temporal FNO can predict future flow states robustly. The approach achieves $Re \approx 300$ with 64 training geometries and 16 validation geometries, and indicates practical potential for experiments and shape optimization where full-field sensing is limited. Limitations and future directions include extending to a range of Reynolds numbers, 3D flows, physics-informed losses, and possibly lift/drag prediction, with code repositories provided for reproducibility.

Abstract

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.

Figures (15)

• Figure 1: A random geometry (left) and its annular preimage (right). Blue and green contours depict the norm and argument in the $w$ domain, respectively. The outer boundary of the domain is smaller than the ones used in the actual study for illustrative purposes.
• Figure 2: Twelve geometries used in this study, among a total of 80.
• Figure 3: Illustration of the medium sensor setup for one of the geometries used in the study.
• Figure 4: Diagrams of the SD-UNet (top), an FNO layer (middle), and the SD-FNO (bottom). The values 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.
• Figure 5: Summary of the two-step spatio-temporal reconstruction approach.