This paper addresses the problem of controlling multiple unmanned aerial vehicles (UAVs) cooperating in a formation to carry out a complex task such as surface inspection. We first use the virtual leader-follower model to determine the topology and trajectory of the formation. A double-loop control system combining backstepping and sliding mode control techniques is then designed for the UAVs to track the trajectory. A radial basis function neural network (RBFNN) capable of estimating external disturbances is developed to enhance the robustness of the controller. The stability of the controller is proven by using the Lyapunov theorem. A number of comparisons and software-in-the-loop (SIL) tests have been conducted to evaluate the performance of the proposed controller. The results show that our controller not only outperforms other state-of-the-art controllers but is also sufficient for complex tasks of UAVs such as collecting surface data for inspection. The source code of our controller can be found at https://github.com/duynamrcv/rbf_bsmc

Machine learning has been successfully applied to various fields of scientific computing in recent years. In this work, we propose a sparse radial basis function neural network method to solve elliptic partial differential equations (PDEs) with multiscale coefficients. Inspired by the deep mixed residual method, we rewrite the second-order problem into a first-order system and employ multiple radial basis function neural networks (RBFNNs) to approximate unknown functions in the system. To aviod the overfitting due to the simplicity of RBFNN, an additional regularization is introduced in the loss function. Thus the loss function contains two parts: the $L_2$ loss for the residual of the first-order system and boundary conditions, and the $\ell_1$ regularization term for the weights of radial basis functions (RBFs). An algorithm for optimizing the specific loss function is introduced to accelerate the training process. The accuracy and effectiveness of the proposed method are demonstrated through a collection of multiscale problems with scale separation, discontinuity and multiple scales from one to three dimensions. Notably, the $\ell_1$ regularization can achieve the goal of representing the solution by fewer RBFs. As a consequence, the total number of RBFs scales like $\mathcal{O}(\varepsilon^{-n\tau})$, where $\varepsilon$ is the smallest scale, $n$ is the dimensionality, and $\tau$ is typically smaller than $1$. It is worth mentioning that the proposed method not only has the numerical convergence and thus provides a reliable numerical solution in three dimensions when a classical method is typically not affordable, but also outperforms most other available machine learning methods in terms of accuracy and robustness.

In this paper we consider a new class of RBF (Radial Basis Function) neural networks, in which smoothing factors are replaced with shifts. We prove under certain conditions on the activation function that these networks are capable of approximating any continuous multivariate function on any compact subset of the $d$-dimensional Euclidean space. For RBF networks with finitely many fixed centroids we describe conditions guaranteeing approximation with arbitrary precision.

We develop a sequential adaptation algorithm for radial basis function (RBF) neural networks of Gaussian nodes, based on the method of succes(cid:173) sive F-Projections. This method makes use of each observation efficiently in that the network mapping function so obtained is consistent with that information and is also optimal in the least L 2-norm sense. The RBF network with the F-Projections adaptation algorithm was used for pre(cid:173) dicting a chaotic time-series. We compare its performance to an adapta(cid:173) tion scheme based on the method of stochastic approximation, and show that the F-Projections algorithm converges to the underlying model much faster.

This paper presents a multivariate functional data statistical approach, for spatiotemporal prediction of COVID-19 mortality counts. Specifically, spatial heterogeneous nonlinear parametric functional regression trend model fitting is first implemented. Classical and Bayesian infinite-dimensional log-Gaussian linear residual correlation analysis is then applied. The nonlinear regression predictor of the mortality risk is combined with the plug-in predictor of the multiplicative error term. An empirical model ranking, based on random K-fold validation, is established for COVID-19 mortality risk forecasting and assessment, involving Machine Learning (ML) models, and the adopted Classical and Bayesian semilinear estimation approach. This empirical analysis also determines the ML models favored by the spatial multivariate Functional Data Analysis (FDA) framework. The results could be extrapolated to other countries.

F. Fallside We develop a sequential adaptation algorithm for radial basis function (RBF) neural networks of Gaussian nodes, based on the method of successive F-Projections.This method makes use of each observation efficiently in that the network mapping function so obtained is consistent with that information and is also optimal in the least L

We develop a sequential adaptation algorithm for radial basis function (RBF) neural networks of Gaussian nodes, based on the method of successive F-Projections.

We develop a sequential adaptation algorithm for radial basis function (RBF) neural networks of Gaussian nodes, based on the method of successive F-Projections.