# Spatiotemporal wall pressure forecast of a rectangular cylinder with physics-aware DeepU-Fourier neural network

arXiv: https://arxiv.org/abs/2508.03183 · HTML: https://mlsanity.com/paper/2508.03183/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 125 references that are in the map.
- First posted on arXiv 5 Aug 2025.

## Builds on (most cited first, 29 of 29)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Inception-v4, Inception-ResNet and the Impact of Residual Connections on Learning](https://mlsanity.com/paper/1602.07261/) (2016)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Swin-Unet: Unet-like Pure Transformer for Medical Image Segmentation](https://mlsanity.com/paper/2105.05537/) (2021)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [How Does Batch Normalization Help Optimization?](https://mlsanity.com/paper/1805.11604/) (2018)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [VM-UNet: Vision Mamba UNet for Medical Image Segmentation](https://mlsanity.com/paper/2402.02491/) (2024)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Understanding Batch Normalization](https://mlsanity.com/paper/1806.02375/) (2018)
- [Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning](https://mlsanity.com/paper/2101.00554/) (2021)
- [AFTer-UNet: Axial Fusion Transformer UNet for Medical Image Segmentation](https://mlsanity.com/paper/2110.10403/) (2021)
- [Theory of the Frequency Principle for General Deep Neural Networks](https://mlsanity.com/paper/1906.09235/) (2019)
- [Deep frequency principle towards understanding why deeper learning is faster](https://mlsanity.com/paper/2007.14313/) (2020)
- [RecFNO: a resolution-invariant flow and heat field reconstruction method from sparse observations via Fourier neural operator](https://mlsanity.com/paper/2302.09808/) (2023)
- [Multi-fidelity prediction of fluid flow and temperature field based on transfer learning using Fourier Neural Operator](https://mlsanity.com/paper/2304.06972/) (2023)
- [Multi-scale data reconstruction of turbulent rotating flows with Gappy POD, Extended POD and Generative Adversarial Networks](https://mlsanity.com/paper/2210.11921/) (2022)
- [Enhancing Graph U-Nets for Mesh-Agnostic Spatio-Temporal Flow Prediction](https://mlsanity.com/paper/2406.03789/) (2024)
- [Low-Order Flow Reconstruction and Uncertainty Quantification in Disturbed Aerodynamics Using Sparse Pressure Measurements](https://mlsanity.com/paper/2501.03406/) (2025)

## Built on by (most cited first, 2 of 2)

- [LatentFlow: Cross-Frequency Experimental Flow Reconstruction from Sparse Pressure via Latent Mapping](https://mlsanity.com/paper/2508.16648/) (2025)
- [Deep Wave Network for Modeling Multi-Scale Physical Dynamics](https://mlsanity.com/paper/2605.04198/) (2026)

## Abstract (arXiv)

The wall pressure is of great importance in understanding the forces and structural responses induced by fluid. Recent works have investigated the potential of deep learning techniques in predicting mean pressure coefficients and fluctuating pressure coefficients, but most of existing deep learning frameworks are limited to predicting a single snapshot using full spatial information. To forecast spatiotemporal wall pressure of flow past a rectangular cylinder, this study develops a physics-aware DeepU-Fourier neural Network (DeepUFNet) deep learning model. DeepUFNet comprises the UNet structure and the Fourier neural network, with physical high-frequency loss control embedded in the model training stage to optimize model performance. Wind tunnel testing was performed to collect wall pressures on two-dimensional rectangular cylinders using high-frequency pressure scanning, thereby constructing a database for DeepUFNet training and testing. The DeepUFNet model is found capable of forecasting spatiotemporal wall pressure information with high accuracy on the rectangular cylinder with side ratio 1.5. The comparison between forecast results and experimental data presents agreement in statistical information and physical interpretation. It is also found that embedding a physical high-frequency loss control coefficient b in the DeepUFNet model can significantly improve model performance in forecasting spatiotemporal wall pressure information, particularly, high-order frequency fluctuation and wall pressure variance. Furthermore, the DeepUFNet extrapolation capability is tested with sparse spatial information input, and the model presents a satisfactory extrapolation ability. Last, the DeepUFNet is tested for generalization in unseen cases, rectangular cylinders with side ratio 4 and 3.75, and the model presents satisfactory generalization ability.