Traffic volume is an indispensable ingredient to provide fine-grained information for traffic management and control. However, due to limited deployment of traffic sensors, obtaining full-scale volume information is far from easy. Existing works on this topic primarily focus on improving the overall estimation accuracy of a particular method and ignore the underlying challenges of volume estimation, thereby having inferior performances on some critical tasks. This paper studies two key problems with regard to traffic volume estimation: (1) underdetermined traffic flows caused by undetected movements, and (2) non-equilibrium traffic flows arise from congestion propagation. Here we demonstrate a graph-based deep learning method that can offer a data-driven, model-free and correlation adaptive approach to tackle the above issues and perform accurate network-wide traffic volume estimation. Particularly, in order to quantify the dynamic and nonlinear relationships between traffic speed and volume for the estimation of underdetermined flows, a speed patternadaptive adjacent matrix based on graph attention is developed and integrated into the graph convolution process, to capture non-local correlations between sensors. To measure the impacts of non-equilibrium flows, a temporal masked and clipped attention combined with a gated temporal convolution layer is customized to capture time-asynchronous correlations between upstream and downstream sensors. We then evaluate our model on a real-world highway traffic volume dataset and compare it with several benchmark models. It is demonstrated that the proposed model achieves high estimation accuracy even under 20% sensor coverage rate and outperforms other baselines significantly, especially on underdetermined and non-equilibrium flow locations. Furthermore, comprehensive quantitative model analysis are also carried out to justify the model designs.
Transformer neural operators have recently become an effective approach for surrogate modeling of systems governed by partial differential equations (PDEs). In this paper, we introduce a modified implicit factorized transformer (IFactFormer-m) model which replaces the original chained factorized attention with parallel factorized attention. The IFactFormer-m model successfully performs long-term predictions for turbulent channel flow, whereas the original IFactFormer (IFactFormer-o), Fourier neural operator (FNO), and implicit Fourier neural operator (IFNO) exhibit a poor performance. Turbulent channel flows are simulated by direct numerical simulation using fine grids at friction Reynolds numbers $\text{Re}_τ\approx 180,395,590$, and filtered to coarse grids for training neural operator. The neural operator takes the current flow field as input and predicts the flow field at the next time step, and long-term prediction is achieved in the posterior through an autoregressive approach. The results show that IFactFormer-m, compared to other neural operators and the traditional large eddy simulation (LES) methods including dynamic Smagorinsky model (DSM) and the wall-adapted local eddy-viscosity (WALE) model, reduces prediction errors in the short term, and achieves stable and accurate long-term prediction of various statistical properties and flow structures, including the energy spectrum, mean streamwise velocity, root mean square (rms) values of fluctuating velocities, Reynolds shear stress, and spatial structures of instantaneous velocity. Moreover, the trained IFactFormer-m is much faster than traditional LES methods. By analyzing the attention kernels, we elucidate the reasons why IFactFormer-m converges faster and achieves a stable and accurate long-term prediction compared to IFactFormer-o. Code and data are available at: https://github.com/huiyu-2002/IFactFormer-m.