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Multi-object pedestrian tracking plays a crucial role in autonomous driving systems, enabling accurate perception of the surrounding environment. In this paper, we propose a comprehensive approach for pedestrian tracking, combining the improved YOLOv8 object detection algorithm with the OC-SORT tracking algorithm. First, we train the improved YOLOv8 model on the Crowdhuman dataset for accurate pedestrian detection. The integration of advanced techniques such as softNMS, GhostConv, and C3Ghost Modules results in a remarkable precision increase of 3.38% and an mAP@0.5:0.95 increase of 3.07%. Furthermore, we achieve a significant reduction of 39.98% in parameters, leading to a 37.1% reduction in model size. These improvements contribute to more efficient and lightweight pedestrian detection. Next, we apply our enhanced YOLOv8 model for pedestrian tracking on the MOT17 and MOT20 datasets. On the MOT17 dataset, we achieve outstanding results with the highest HOTA score reaching 49.92% and the highest MOTA score reaching 56.55%. Similarly, on the MOT20 dataset, our approach demonstrates exceptional performance, achieving a peak HOTA score of 48.326% and a peak MOTA score of 61.077%. These results validate the effectiveness of our approach in challenging real-world tracking scenarios.

When PINNs solve the Navier–Stokes equations, the loss function has a gradient imbalance problem during training. It is one of the reasons why the efficiency of PINNs is limited. This paper proposes a novel method of adaptively adjusting the weights of loss terms, which can balance the gradients of each loss term during training. The weight is updated by the idea of the minmax algorithm. The neural network identifies which types of training data are harder to train and forces it to focus on those data before training the next step. Specifically, it adjusts the weight of the data that are difficult to train to maximize the objective function. On this basis, one can adjust the network parameters to minimize the objective function and do this alternately until the objective function converges. We demonstrate that the dynamic weights are monotonically non-decreasing and convergent during training. This method can not only accelerate the convergence of the loss, but also reduce the generalization error, and the computational efficiency outperformed other state-of-the-art PINNs algorithms. The validity of the method is verified by solving the forward and inverse problems of the Navier–Stokes equation.

In this paper, a grid-free deep learning method based on a physics-informed neural network is proposed for solving coupled Stokes–Darcy equations with Bever–Joseph–Saffman interface conditions. This method has the advantage of avoiding grid generation and can greatly reduce the amount of computation when solving complex problems. Although original physical neural network algorithms have been used to solve many differential equations, we find that the direct use of physical neural networks to solve coupled Stokes–Darcy equations does not provide accurate solutions in some cases, such as rigid terms due to small parameters and interface discontinuity problems. In order to improve the approximation ability of a physics-informed neural network, we propose a loss-function-weighted function strategy, a parallel network structure strategy, and a local adaptive activation function strategy. In addition, the physical information neural network with an added strategy provides inspiration for solving other more complicated problems of multi-physical field coupling. Finally, the effectiveness of the proposed strategy is verified by numerical experiments.

Physics-informed neural networks (PINNs) are effective for solving partial differential equations (PDEs). This method of embedding partial differential equations and their initial boundary conditions into the loss functions of neural networks has successfully solved forward and inverse PDE problems. In this study, we considered a parametric light wave equation, discretized it using the central difference, and, through this difference scheme, constructed a new neural network structure named the second-order neural network structure. Additionally, we used the adaptive activation function strategy and gradient-enhanced strategy to improve the performance of the neural network and used the deep mixed residual method (MIM) to reduce the high computational cost caused by the enhanced gradient. At the end of this paper, we give some numerical examples of nonlinear parabolic partial differential equations to verify the effectiveness of the method.

In this paper, the radial basis function finite difference method is used to solve two-dimensional steady incompressible Navier–Stokes equations. First, the radial basis function finite difference method with polynomial is used to discretize the spatial operator. Then, the Oseen iterative scheme is used to deal with the nonlinear term, constructing the discrete scheme for Navier–Stokes equation based on the finite difference method of the radial basis function. This method does not require complete matrix reorganization in each nonlinear iteration, which simplifies the calculation process and obtains high-precision numerical solutions. Finally, several numerical examples are obtained to verify the convergence and effectiveness of the radial basis function finite difference method based on Oseen Iteration.

With the remarkable development of deep learning in the field of science, deep neural networks provide a new way to solve the Stefan problem. In this paper, deep neural networks combined with small sample learning and a general deep learning framework are proposed to solve the two-dimensional Stefan problem. In the case of adding less sample data, the model can be modified and the prediction accuracy can be improved. In addition, by solving the forward and inverse problems of the two-dimensional single-phase Stefan problem, it is verified that the improved method can accurately predict the solutions of the partial differential equations of the moving boundary and the dynamic interface.

In this paper, an effective numerical algorithm for the Stokes equation of a curved surface is presented and analyzed. The velocity field was decoupled from the pressure by the standard velocity correction projection method, and the penalty term was introduced to make the velocity satisfy the tangential condition. The first-order backward Euler scheme and second-order BDF scheme are used to discretize the time separately, and the stability of the two schemes is analyzed. The mixed finite element pair (P2,P1) is applied to discretization of space. Finally, numerical examples are given to verify the accuracy and effectiveness of the proposed method.

Enhancing crop water productivity is crucial for regional water resource management and agricultural sustainability, particularly in arid regions. However, evaluating the spatial heterogeneity and temporal dynamics of crop water productivity in face of data limitations poses a challenge. In this study, we propose a framework that integrates remote sensing data, time series generative adversarial network (TimeGAN), dynamic Bayesian network (DBN), and optimization model to assess crop water productivity and optimize crop planting structure under limited water resources allocation in the Qira oasis. The results demonstrate that the combination of TimeGAN and DBN better improves the accuracy of the model for the dynamic prediction, particularly for short-term predictions with 4 years as the optimal timescale (R 2  > 0.8). Based on the spatial distribution of crop suitability analysis, wheat and corn are most suitable for cultivation in the central and eastern parts of Qira oasis while cotton is unsuitable for planting in the western region. The walnuts and Chinese dates are mainly unsuitable in the southeastern part of the oasis. Maximizing crop water productivity while ensuring food security has led to increased acreage for cotton, Chinese dates and walnuts. Under the combined action of the five optimization objectives, the average increase of crop water productivity is 14.97%, and the average increase of ecological benefit is 3.61%, which is much higher than the growth rate of irrigation water consumption of cultivated land. It will produce a planting structure that relatively reduced irrigation water requirement of cultivated land and improved crop water productivity. This proposed framework can serve as an effective reference tool for decision-makers when determining future cropping plans.

The predator-prey model is powerful mathematical tool to describe the dynamics of biological systems and promote research on biological populations. In this paper, we present a lumped mass finite element method for solving the predator-prey models on surfaces. The main purpose of the proposed method is to overcome the difficulty of the positivity preservation of the solutions. Based on positivity preservation results, we investigate the stabilities of semi-discrete and fully discrete approximations. Besides, numerical simulations are considered to illustrate the feasibility of the numerical method by convergence tests. Two classical phenomena of the predator-prey model are simulated on three different implicit surfaces.

We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative (0<α<1) with mixed finite element methods (P1b–P1 and P2–P1) for spatial discretization of velocity and pressure. This approach addresses the key challenges of fractional models, including nonlocality and memory effects, while maintaining stability in the presence of the nonlinear damping term γ|u|r−2u, for r≥2. We prove unconditional stability for both semi-discrete and fully discrete schemes and derive optimal error estimates for the velocity and pressure components. Numerical experiments validate the theoretical results. Convergence tests using exact solutions, along with benchmark problems such as backward-facing channel flow and lid-driven cavity flow, confirm the accuracy and reliability of the method. The computed velocity contours and streamlines show close agreement with analytical expectations. This scheme is particularly effective for capturing anomalous diffusion in Newtonian and turbulent flows, and it offers a strong foundation for future extensions to viscoelastic and biological fluid models.