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In this paper, we introduce the finite difference weighted essentially non-oscillatory (WENO) scheme based on the neural network for hyperbolic conservation laws. We employ the supervised learning and design two loss functions, one with the mean squared error and the other with the mean squared logarithmic error, where the WENO3-JS weights are computed as the labels. Each loss function consists of two components where the first component compares the difference between the weights from the neural network and WENO3-JS weights, while the second component matches the output weights of the neural network and the linear weights. The former of the loss function enforces the neural network to follow the WENO properties, implying that there is no need for the post-processing layer. Additionally the latter leads to better performance around discontinuities. As a neural network structure, we choose the shallow neural network (SNN) for computational efficiency with the Delta layer consisting of the normalized undivided differences. These constructed WENO3-SNN schemes show the outperformed results in one-dimensional examples and improved behavior in two-dimensional examples, compared with the simulations from WENO3-JS and WENO3-Z.

The aim of this paper is to design the explicit radial basis function (RBF) Runge-Kutta methods for the initial value problem. We construct the two-, three- and four-stage RBF Runge-Kutta methods based on the Gaussian RBF Euler method with the shape parameter, where the analysis of the local truncation error shows that the s-stage RBF Runge-Kutta method could formally achieve order s+1. The proof for the convergence of those RBF Runge-Kutta methods follows. We then plot the stability region of each RBF Runge-Kutta method proposed and compare with the one of the correspondent Runge-Kutta method. Numerical experiments are provided to exhibit the improved behavior of the RBF Runge-Kutta methods over the standard ones.

In this paper we analyze the weighted essentially non-oscillatory (WENO) schemes in the finite volume framework by examining the first step of the explicit third-order total variation diminishing Runge-Kutta method. The rationale for the improved performance of the finite volume WENO-M, WENO-Z and WENO-ZR schemes over WENO-JS in the first time step is that the nonlinear weights corresponding to large errors are adjusted to increase the accuracy of numerical solutions. Based on this analysis, we propose novel Z-type nonlinear weights of the finite volume WENO scheme for hyperbolic conservation laws. Instead of taking the difference of the smoothness indicators for the global smoothness indicator, we employ the logarithmic function with tuners to ensure that the numerical dissipation is reduced around discontinuities while the essentially non-oscillatory property is preserved. The proposed scheme does not necessitate substantial extra computational expenses. Numerical examples are presented to demonstrate the capability of the proposed WENO scheme in shock capturing.

In this paper we propose new Z-type nonlinear weights of the fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme for hyperbolic conservation laws. Instead of employing the classical smoothness indicators for the nonlinear weights, we take the $p$th root of the smoothness indicators and follow the form of Z-type nonlinear weights, leading to fifth order accuracy in smooth regions, even at the critical points, and sharper approximations around the discontinuities. We also prove that the proposed nonlinear weights converge to the linear weights as $p \to \infty$, implying the convergence of the resulting WENO numerical flux to the finite difference numerical flux. Finally, numerical examples are presented by comparing with other WENO schemes, such as WENO-JS, WENO-M and WENO-Z, to demonstrate that the proposed WENO scheme performs better in shock capturing.

Estimating failure probabilities of engineering systems is an important problem in many engineering fields. In this work we consider such problems where the failure probability is extremely small (e.g $\leq10^{-10}$). In this case, standard Monte Carlo methods are not feasible due to the extraordinarily large number of samples required. To address these problems, we propose an algorithm that combines the main ideas of two very powerful failure probability estimation approaches: the subset simulation (SS) and the multicanonical Monte Carlo (MMC) methods. Unlike the standard MMC which samples in the entire domain of the input parameter in each iteration, the proposed subset MMC algorithm adaptively performs MMC simulations in a subset of the state space and thus improves the sampling efficiency. With numerical examples we demonstrate that the proposed method is significantly more efficient than both of the SS and the MMC methods. Moreover, the proposed algorithm can reconstruct the complete distribution function of the parameter of interest and thus can provide more information than just the failure probabilities of the systems.