High-fidelity computational fluid dynamics (CFD) is crucial to vehicle aerodynamic analysis, but its cost still constrains early-stage design exploration. Machine-learning-based surface-field prediction offers a faster alternative if the model can efficiently capture both global flow context and local geometric detail. This work proposes a machine-learning-based method, named the geometry-aware triplane field network (GTF-Net), for vehicle aerodynamic pressure and wall shear stress prediction. GTF-Net constructs triplane features directly from sampled surface points through a shared multilayer perceptron (MLP) and smooth bilinear rasterization. The planes are then processed by a dual-stream backbone that combines adaptive Fourier neural operator (AFNO) spectral mixing with convolutional neural network (CNN) refinement, so long-range aerodynamic coupling and local geometry-induced variations are modeled in the same representation. At query stage, sampled triplane features are combined with vehicle-aligned directional coordinates, normal-projection features, and a voxel-based curvature proxy. GTF-Net is compared with Transolver, geometry-informed neural operator (GINO), and TripNet, a triplane-based surrogate model. GTF-Net improves the relative L2 error from the strongest baseline value of 0.157 to 0.145 for pressure prediction and from 0.237 to 0.226 for wall shear stress prediction. Ablation results show that AFNO mixing, local CNN refinement, and query-side geometric encoding each contribute to accuracy, supporting the proposed mechanism of combining structured triplane representation with explicit aerodynamic geometry cues.
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.
To comprehend complex systems with multiple states, it is imperative to reveal the identity of these states by system outputs. Nevertheless, the mathematical models describing these systems often exhibit nonlinearity so that render the resolution of the parameter inverse problem from the observed spatiotemporal data a challenging endeavor. Starting from the observed data obtained from such systems, we propose a novel framework that facilitates the investigation of parameter identification for multi-state systems governed by spatiotemporal varying parametric partial differential equations. Our framework consists of two integral components: a constrained self-adaptive physics-informed neural network, encompassing a sub-network, as our methodology for parameter identification, and a finite mixture model approach to detect regions of probable parameter variations. Through our scheme, we can precisely ascertain the unknown varying parameters of the complex multi-state system, thereby accomplishing the inversion of the varying parameters. Furthermore, we have showcased the efficacy of our framework on two numerical cases: the 1D Burgers' equation with time-varying parameters and the 2D wave equation with a space-varying parameter.
Rapid aerodynamic evaluation is crucial for modern vehicle design, yet existing neural operators struggle to capture intricate spatial correlations. We propose the rotary-enhanced transformer operator (RETO), a novel neural solver featuring a dual-stage spatial awareness mechanism: sinusoidal-cosine encodings for global referencing and rotary positional encodings (RoPE) for relative displacements. RoPE encodes spatial relations via unitary rotations, enforcing translation invariance and enhancing local gradient resolution. RETO is validated on ShapeNet and the high-fidelity DrivAerML benchmark. On ShapeNet, RETO achieves a relative $L_2$ error of 0.063, outperforming RegDGCNN at 0.125 and representing a 16\% improvement over the Transolver baseline, which yields an error of 0.075. These performance gains are further amplified on the DrivAerML dataset, where RETO achieves relative $L_2$ errors of 0.089 for surface pressure and 0.097 for velocity. In comparison, Transolver results in errors of 0.116 and 0.121 for the same metrics, indicating that RETO achieves precision enhancements of 23\% and 19\%, respectively. For comprehensive comparison, the surface pressure and velocity errors for AB-UBT are 0.102 and 0.124, while RegDGCNN yields 0.235 and 0.312, respectively. Information-theoretical analysis shows that the entropy peak of RETO at 0.35 is significantly lower than that of Transolver at 0.75 under $10^4$ resolution, indicating a focused attentional mechanism capable of preserving localized gradients against global diffusion.