Traditional gradient-based inversion methods usually suffer from the problems of falling into local minima and relying heavily on initial guesses. Deep-learning methods have received increasing attention due to their excellent nonlinear fitting ability. However, given the recent application of deep-learning methods in the field of magnetotelluric (MT) inversion, there are currently challenges associated with achieving high inversion resolution and extracting sufficient features. We develop a neural network model (called MT2DInv-Unet) based on the deformable convolution for 2D MT inversion to approximate the nonlinear mapping from the MT response data to the resistivity model. The deformable convolution is achieved by adding an offset to each sample point of the conventional convolution operation, which extracts hidden relationships and allows the flexible adjustment of the size and shape of the feature region. Meanwhile, we design the network structure with multiscale residual blocks, which effectively extract the multiscale features of the MT response data. This design not only enhances the network performance but also alleviates issues such as vanishing gradients and network degradation. The results of synthetic models indicate that our network inversion method has stable convergence, good robustness, and generalization performance, and it performs better than the fully convolutional neural network and U-Net network. Finally, the inversion results of field data show that MT2DInv-Unet can effectively obtain a reliable underground resistivity structure and has a good application prospect in MT inversion.

The conventional magnetotelluric inversion method is subject to the influence of the initial model, which leads to an unstable inversion process and a tendency to get trapped at local optimal solutions. In contrast, deep learning technology relies on its powerful non-linear fitting capability and can construct complex non-linear mappings directly from observation data (input) to model (output). In recent years, it has received extensive attention from researchers. Due to the difficulties in creating a sufficiently large dataset and performing extensive neural network training, most current magnetotelluric inversion methods for geophysical exploration remain limited to one-dimensional (1D) or two-dimensional (2D) scenarios. To the best of our knowledge, for deep learning-based three-dimensional (3D) magnetotelluric inversion, currently there is no reported work in the literature. In this work, we propose a 3D magnetotelluric inversion method based on deep learning technology. By designing a neural network architecture for 3D structures (MT3D-Net), we achieve an end-to-end mapping from the network input to output. To alleviate the excessive dependence of the network on the training set, we introduce a joint weighted loss function based on data-driven and physics-driven method, allowing the network to follow the physical constraints of magnetotelluric data during the training process and thus more reasonably guide the update of network parameters. Numerical experiments show that this method combines the advantages of traditional and data-driven inversions, significantly improving the stability and accuracy of magnetotelluric inversion. The proposed method has been successfully applied to synthetic models and measured field data, and has good application prospects.

By using differential equations with discontinuous right-hand sides, a dynamic model for vector-borne infectious disease under the discontinuous removal of infected trees was established after understanding the transmission mechanism of Huanglongbing (HLB) disease in citrus trees. Through calculation, the basic reproductive number of the model can be attained and the properties of the model are discussed. On this basis, the existence and global stability of the calculated equilibria are verified. Moreover, it was found that different $I_0$ in the control strategy cannot change the dynamic properties of HLB disease. However, the lower the value of $I_0$, the fewer HLB-infected citrus trees, which provides a theoretical basis for controlling HLB disease and reducing expenditure.

The Poisson–Nernst–Planck (PNP) system is a nonlinear coupled system that describes the motion of ionic particles. As the exact solution of the system is not available, numerical investigations are essentially important, and there are quite a lot of numerical methods proposed in the existing literature. However, the theoretical analysis is usually neglected due to the complicated nature of the PNP system. In this paper, a theoretical investigation for a symmetrical finite difference method proposed in the previous literature was conducted. An $L^2$ error estimate of $O(\tau +h^2)$ was derived for the numerical scheme in 1D, where $\tau$ denotes the time step size and h denotes the spatial mesh size, respectively. Numerical results confirm the theoretical analysis. More importantly, a positivity-preserving condition for the scheme is provided with rigorously theoretical justification.

The convection-diffusion equation describes a convection and diffusion process, which is the cornerstone of electrochemistry. The process always takes place in a porous medium or on an uneven boundary, and an abnormal diffusion occurs, which will lead to deviations in prediction of the convection-diffusion process. To overcome the problem, a fractal modification is suggested to deal with the ?abnormal? problem, and a 2-D steady-state convection-diffusion equation with fractal derivatives in the fractal space is established. Furthermore, its fractal variational principle is obtained by the semi-inverse method. The fractal variational formula can not only provide the conservation law in the fractal space in the form of energy, but also give the possible solution structure of the equation.