Search

In scientific computing, neural networks have been widely used to solve partial differential equations (PDEs). In this paper, we propose a novel RBF-assisted hybrid neural network for approximating solutions to PDEs. Inspired by the tendency of physics-informed neural networks (PINNs) to become local approximations after training, the proposed method utilizes a radial basis function (RBF) to provide the normalization and localization properties to the input data. The objective of this strategy is to assist the network in solving PDEs more effectively. During the RBF-assisted processing part, the method selects the center points and collocation points separately to effectively manage data size and computational complexity. Subsequently, the RBF processed data are put into the network for predicting the solutions to PDEs. Finally, a series of experiments are conducted to evaluate the novel method. The numerical results confirm that the proposed method can accelerate the convergence speed of the loss function and improve predictive accuracy.

Low-rank matrix completion aims to recover an unknown matrix from a subset of observed entries. In this paper, we solve the problem via optimization of the matrix manifold. Specially, we apply QR factorization to retraction during optimization. We devise two fast algorithms based on steepest gradient descent and conjugate gradient descent, and demonstrate their superiority over the promising baseline with the ratio of at least 24%.

In this paper, we propose a deep wavelet neural network (DWNN) model to approximate the natural phenomena that are described by some classical PDEs. Concretely, we introduce wavelets to deep architecture to obtain a fine feature description and extraction. That is, we constructs a wavelet expansion layer based on a family of vanishing momentum wavelets. Second, the Gaussian error function is considered as the activation function owing to its fast convergence rate and zero-centered output. Third, we design the cost function by considering the residual of governing equation, the initial/boundary conditions and an adjustable residual term of observations. The last term is added to deal with the shock wave problems and interface problems, which is conducive to rectify the model. Finally, a variety of numerical experiments are carried out to demonstrate the effectiveness of the proposed approach. The numerical results validate that our proposed method is more accurate than the state-of-the-art approach.

Image registration aims to align two images through a spatial transformation. It plays a significant role in brain imaging analysis. In this research, we propose a symmetric diffeomorphic image registration model based on multi-label segmentation masks to solve the problems in brain MRI registration. We first introduce the similarity metric of the multi-label masks to the energy function, which improves the alignment of the brain region boundaries and the robustness to the noise. Next, we establish the model on the diffeomorphism group through the relaxation method and the inverse consistent constraint. The algorithm is designed through the local linearization and least-squares method. We then give spatially adaptive parameters to coordinate the descent of the energy function in different regions. The results show that our approach, compared with the mainstream methods, has better accuracy and noise resistance, and the transformations are more smooth and more reasonable.

In the field of autonomous driving, carriers are equipped with a variety of sensors, including cameras and LiDARs. However, the camera suffers from problems of illumination and occlusion, and the LiDAR encounters motion distortion, degenerate environment and limited ranging distance. Therefore, fusing the information from these two sensors deserves to be explored. In this paper, we propose a fusion network which robustly captures both the image and point cloud descriptors to solve the place recognition problem. Our contribution can be summarized as: (1) applying the trimmed strategy in the point cloud global feature aggregation to improve the recognition performance, (2) building a compact fusion framework which captures both the robust representation of the image and 3D point cloud, and (3) learning a proper metric to describe the similarity of our fused global feature. The experiments on KITTI and KAIST datasets show that the proposed fused descriptor is more robust and discriminative than the single sensor descriptor.