Physics-informed neural networks (PINNs) are an efficient tool for solving forward and inverse problems for fractional diffusion equations. However, since the automatic differentiation is not applicable to fractional derivatives, solving fractional diffusion equations by PINNs meets a number of challenges. To deal with the arising problems, we propose an extension of PINNs called the Laplace-based fractional physics-informed neural networks (Laplace-fPINNs). It can effectively solve forward and inverse problems for fractional diffusion equations. Note that this approach avoids introducing a mass of auxiliary points and simplifies the loss function. We validate the effectiveness of using the Laplace-fPINNs by several examples. The numerical results demonstrate that the Laplace-fPINNs method can effectively solve the forward and inverse problems for fractional diffusion equations.

The present work studies the double-poles and triple-poles soliton solutions of the Kundu-type equation with zero boundary conditions (ZBCs) and non-zero boundary conditions (NZBCs) via the Riemann-Hilbert (RH) method. We construct the RH problem with ZBCs and NZBCs both analyzing the discrete spectral and combining with the analyticity, symmetries, as well as asymptotic behavior of the modified Jost function and the scattering matrix. In the case that the reflection coefficient is double-poles and triple-poles, the inverse scattering transformation (IST) are established and solved by the RH problem with ZBCs and NZBCs, and the reconstruction formula, trace formula and theta conditions. The general formulas of double-poles and triple-poles soliton solutions with ZBCs and NZBCs are explicitly realized through expresses of determinants. The dynamic analysis for the double-poles and triple-poles soliton solutions of ZBCs/NZBCs are vividly described in the form of images.

In this paper, we consider the electromagnetically and thermally driven flow which is modeled by evolutionary magnetohydrodynamic equations and heat equation coupled through generalized Boussinesq approximation with temperature-dependent coefficients. Based on a third-order backward differential formula for temporal discretization, mixed finite element approximation for spatial discretization and extrapolated treatments in linearization for nonlinear terms, a linearized backward differentiation formula type scheme for the considered equations is proposed and analysed. Optimal $L^2$-error estimates for the proposed fully discretized scheme are obtained by the temporal-spatial error splitting technique. Numerical examples are presented to check the accuracy and efficiency of the scheme.

High order neural networks have strong nonlinear mapping ability, but the network structure is more complex, which restricts the efficiency of the network, and the relevant theoretical analysis is still not perfect up to now. To solve these problems, an online gradient learning algorithm model of Pi-Sigma neural network with a smooth set lasso regular term is proposed. Since the original lasso regular term contains absolute values and is not differentiable at the origin, it causes experiment oscillations and poses a great challenge to the convergence analysis of the algorithm. We use grinding technology to overcome this deficiency. The main contribution of this paper lies in the adoption of online learning algorithm, which effectively improves the efficiency of the algorithm. At the same time, strict theoretical proofs are presented, including strong convergence and weak convergence. Finally, the effectiveness of the algorithm and the correctness of the theoretical results are verified by numerical experiments.

In this article we introduce a novel numerical method to solve the problem of optimal transport and the related elliptic Monge-Ampère equation using neural networks. It is one of the few numerical algorithms capable of solving this problem efficiently with the proper transport boundary condition. Unlike the traditional deep learning solution of partial differential equations (PDEs) attributed to an optimization problem, in this paper we adopt a relaxation algorithm to split the problem into three sub-optimization problems, making each subproblem easy to solve. The algorithm not only obtains the mapping that solves the optimal mass transport problem, but also can find the unique convex solution of the related elliptic Monge-Ampère equation from the mapping using deep input convex neural networks, where second-order partial derivatives can be avoided. It can be solved for high-dimensional problems, and has the additional advantage that the target domain may be non-convex. We present the method and several numerical experiments.

Error estimates of $L1$ scheme for delay fractional equations are derived by discrete Laplace transform method. Theoretical result shows that the convergence order is ${\rm min}\{(k+1)α, 1\}$ at $(k\tau)^+,$ where $k ∈ \mathbb{N},$ $\tau$ is delay factor, $α ∈ (0, 1)$ is the order of Caputo fractional derivative. At the points without derivative discontinuities, first order convergence is achieved. The uniqueness of the inverse problem, the reaction coefficient, and the delay factor are established by employing asymptotic expansions and the monotonicity of the Mittag-Leffler function. An inversion algorithm based on the Tikhonov regularization method is given.

Singularly perturbed problems present inherent difficulty due to the presence of thin layers in their solutions. To overcome this difficulty, we propose using deep operator networks (DeepONets), a method previously shown to be effective in approximating nonlinear operators between infinite-dimensional Banach spaces. In this paper, we demonstrate for the first time the application of DeepONets to one-dimensional singularly perturbed problems, achieving promising results that suggest their potential as a robust tool for solving this class of problems. We consider the convergence rate of the approximation error incurred by the operator networks in approximating the solution operator, and examine the generalization gap and empirical risk, all of which are shown to converge uniformly with respect to the perturbation parameter. By utilizing Shishkin mesh points as locations of the loss function, we conduct several numerical experiments that provide further support for the effectiveness of operator networks in capturing the singular layer behavior.

For the large-scale linear discrete ill-posed problems with multiple right-hand sides, the global Krylov subspace iterative methods have received a lot of attention. In this paper, we analyze the regularizing properties of the global generalized minimum error method (GMERR), and develop a regularized global GMERR method for solving linear discrete ill-posed problems with multiple right-hand sides. The efficiency of the proposed method is confirmed by the numerical experiments on test matrices.

In this paper, we investigate the phenomenon of entropy generation of natural convection Jeffrey fluid flow through a sloping channel under Navier-Slip conditions. The study considers the influences of the Soret effect, inclined magnetic field, and Hall current. Employing appropriate transformations, we express the main equations in dimensionless form. Numerical solutions are implemented using the spectral quasi-linearization method (SQLM). Graphical representations are utilized to assess the impacts of diverse thermophysical parameters. Empirical findings indicate that augmenting the channel inclination angle, inclination angle, Soret number, and magnetic parameter induces a proportional rise in entropy generation. Conversely, a surge in Hall current yields a reduction in entropy generation. When the Soret parameter, magnetic parameter, channel inclination angle, and Jeffrey fluid parameter increase, it increases flow and cross-flow velocity. A contrary trend is observed for the Hall parameter.