Aerodynamic design fundamentally shapes aircraft performance, optimizing metrics such as maneuverability, fuel efficiency, and structural safety. These attributes are critical for aerospace engineering applications, including high-speed missiles and commercial aircraft. Traditional computational fluid dynamics methods deliver accurate simulations but rely on extensive computational resources. Their iterative processes, often spanning hours or days, hinder rapid design iterations essential for modern aerospace development. Current deep learning methods for aerodynamic modeling face challenges in processing complex three-dimensional geometries and integrating physical information. Weight imbalance between geometric features and physical information frequently degrades prediction accuracy, limiting model generalizability. This study proposed AeroPointNet, a neural network-based method to predict lift and drag coefficients with high efficiency and accuracy for diverse three-dimensional aircraft geometries under varying flow conditions, addressing these limitations to advance aerodynamic design.

AeroPointNet utilized three-dimensional point cloud representations of aircraft to capture local and global geometric features. Point clouds underwent farthest point sampling to select 2000 points, standardizing input dimensions. A multilayer perceptron mapped coordinates to a high-dimensional feature space. The network architecture employed a point cloud transformation module with local self-attention and learnable positional encoding, dynamically aggregating geometric features based on neighborhood relationships. Alternating downsampling and transformation modules reduced point counts from 2000 to 7 while expanding feature dimensions from 32 to 512. To model diverse flow conditions, AeroPointNet integrated geometric features with physical information, including Mach number, angle of attack, and sideslip angle. Fusion-based and separation-based weighted attention mechanisms, dynamically adjusted feature weights to mitigate weight imbalance arising from data distribution disparities. The fusion mechanism jointly modeled feature dependencies, while the separation mechanism independently computed weights to enhance interaction. The model was trained and tested on a dataset of 196 missile geometries, comprising 179340 samples across Mach numbers from 1.5 to 2.5, sideslip angles from 0° to 40°, and angles of attack from 0° to 60°. Comparative experiments evaluated AeroPointNet against seven established point cloud neural networks, using mean relative error, mean absolute error, mean squared error, and parameter size.

AeroPointNet significantly enhances prediction accuracy and computational efficiency. It achieves a prediction time of 0.11 seconds per sample, over three orders of magnitude faster than traditional computational fluid dynamics simulations, which average 11 minutes. In geometry generalization tests, AeroPointNet attains a mean relative error of 1.55% for lift coefficients and 1.51% for drag coefficients, surpassing baseline models. Its parameter size of 4.634 MB is 73.95% smaller than PointNeXt's 17.786 MB. In flow condition generalization tests across five randomized datasets, AeroPointNet maintains mean relative errors of 1.84% for lift and 4.87% for drag coefficients, demonstrating robust performance under unseen flow conditions. For most samples, predictions closely align with true values, with errors consistently below 0.25 across angles of attack. Ablation studies validate the weighted attention mechanisms, with their combined use reducing lift coefficient mean relative error by 63.79% and drag coefficient mean relative error by 65.68% compared to a no-attention baseline, adding only 2.89% to parameters. Error visualizations show stable predictions, with 95% confidence intervals narrowing beyond 10° angle of attack, indicating high reliability. Sampling size analysis confirms 2000 points as optimal for balancing accuracy and efficiency.

AeroPointNet provides a robust and efficient solution for aerodynamic coefficient prediction, overcoming limitations of traditional computational fluid dynamics and existing deep learning approaches. Its point cloud-based architecture and weighted attention mechanisms effectively address geometric feature extraction and weight imbalance, ensuring high precision and generalizability across diverse geometries and flow conditions. The model's lightweight design and rapid prediction capabilities enable real-time aerodynamic analysis. Future research will incorporate aerodynamic governing equations and turbulence models to enhance physical interpretability. Extending AeroPointNet to unsteady flow scenarios, such as vortex shedding or flow separation, will broaden its scope. Integration with intelligent shape optimization frameworks could streamline aircraft design processes, enabling automated optimization of aerodynamic performance. These advancements position AeroPointNet as a transformative tool for aerospace engineering, with potential to accelerate development of next-generation aircraft.

PDEs (partial differential equations) serve as fundamental mathematical models in diverse scientific and engineering disciplines, including fluid dynamics, heat transfer, electromagnetics, and quantum mechanics. Despite their broad applicability, traditional numerical methods for solving PDEs—such as finite difference, finite element, and spectral methods—often suffer from high computational costs, meshing complexity, and scalability limitations, especially when applied to high-dimensional or real-time simulation scenarios. These challenges hinder the efficient deployment of PDE-based models in practical applications. In recent years, PINNs have emerged as a promising mesh-free, data-efficient framework for solving PDEs by embedding physical laws directly into the training of deep neural networks. However, the accuracy and convergence speed of standard PINNs remain limited due to the difficulty in balancing multiple loss terms, insufficient representation capacity, and weak enforcement of physical constraints. To address these limitations, this study aims to enhance the accuracy and robustness of PINN-based solvers through structural innovation and improved physical regularization. The primary objective is to develop novel neural network architectures that more effectively integrate multi-source physical information, thereby achieving significantly higher solution accuracy compared to conventional PINN approaches.

This work proposes two advanced PINN-based frameworks: EmPINN and DL-PINN. In EmPINN, a novel network architecture is introduced that incorporates a dimension-expanding mechanism combined with residual connections. Specifically, the input features are mapped into a higher-dimensional latent space through a learnable transformation, enabling the network to capture more complex nonlinear mappings inherent in PDE solutions. The residual connections further facilitate gradient flow and improve training stability. In DL-PINN, this dimension expansion is integrated with a multi-physics loss strategy that simultaneously enforces multiple forms of physical consistency. Beyond the standard PDE residual loss, DL-PINN introduces gradient-enhanced terms that penalize deviations in first- and higher-order derivatives, thus improving the smoothness and physical plausibility of the solution. Additionally, a variational physics-informed loss is incorporated, derived from the weak formulation of the PDE, which provides complementary regularization and enhances robustness, particularly in regions with sharp gradients or discontinuities. Both models are trained using a unified loss function that balances the data fidelity, PDE residuals, boundary/initial conditions, and the additional physics-based terms through adaptive weighting strategies. The training process relies solely on collocation points sampled from the domain and boundaries, without requiring labeled solution data, maintaining the unsupervised nature of PINNs.

The proposed methods are evaluated on two canonical PDE benchmarks: the Poisson equation and the Burgers’ equation, both of which exhibit distinct challenges such as nonlinearity, shock formation, and multi-scale behavior. Quantitative comparisons are conducted against standard PINN and other enhanced variants under identical network sizes and training budgets. Experimental results demonstrate that both EmPINN and DL-PINN achieve superior accuracy, with relative L2 errors reduced by one to two orders of magnitude across all test cases. Notably, DL-PINN exhibits the best performance due to its comprehensive integration of gradient enhancement and variational principles, showing particular strength in capturing fine-scale solution features and maintaining stability over long-time simulations. Ablation studies confirm the individual contributions of dimension expansion, residual connections, and multi-physics loss components, validating the effectiveness of each design choice. Furthermore, the models show improved convergence rates, requiring fewer training iterations to reach high-accuracy solutions, which indicates enhanced optimization dynamics.

This study presents two innovative deep learning frameworks—EmPINN and DL-PINN—for solving partial differential equations with significantly improved accuracy and robustness. By introducing dimension expansion and multi-physics loss integration, the proposed methods more effectively leverage the expressive power of neural networks while enforcing physical consistency from multiple perspectives. The results confirm that the fusion of structural enhancements and diversified physical constraints leads to substantial performance gains over traditional PINNs. These advancements contribute to the development of efficient, accurate, and scalable surrogate solvers for complex PDE systems. While the current work focuses on proof-of-concept demonstrations, future research will investigate the interpretability of such models and explore automated neural architecture design tailored to specific PDE types. With ongoing advances in deep learning, PINN-based methods hold great promise for enabling fast, precise, and adaptive solutions to a wide range of scientific computing problems.