To combine a feedforward neural network (FNN) and Lie group (symmetry) theory of differential equations (DEs), an alternative artificial NN approach is proposed to solve the initial value problems (IVPs) of ordinary DEs (ODEs). Introducing the Lie group expressions of the solution, the trial solution of ODEs is split into two parts. The first part is a solution of other ODEs with initial values of original IVP. This is easily solved using the Lie group and known symbolic or numerical methods without any network parameters (weights and biases). The second part consists of an FNN with adjustable parameters. This is trained using the error back propagation method by minimizing an error (loss) function and updating the parameters. The method significantly reduces the number of the trainable parameters and can more quickly and accurately learn the real solution, compared to the existing similar methods. The numerical method is applied to several cases, including physical oscillation problems. The results have been graphically represented, and some conclusions have been made.

In practice, several e-commerce platforms offering online channels not only act as resellers but also serve as the marketplace. However, the existing literature rarely explores the impact of the offline service effort strategy with the showrooming effect on the platform’s optimal sales mode. Considering a supply chain consisting of a manufacturer and a platform, we examine the interplay between the manufacturer’s offline service effort strategy and the platform’s online sales modes. We derive conditions under which each of the four scenarios (adopting the service effort strategy under the agency or reselling modes, not adopting the service effort strategy under the agency or reselling modes) emerges in equilibrium. Our results show that the service effort strategy with the showrooming effect can induce the platform’s sales mode selection. Specifically, when the referral fee is low and the showrooming effect is moderate, the platform may choose the agency mode instead of the reselling mode, while when the referral fee is sufficiently high and the showrooming effect is moderate, the platform may adopt the reselling mode instead of the agency mode. Furthermore, when the competition intensity and showrooming effect are sufficiently small, the service effort strategy will be beneficial to the manufacturer and the platform, creating a win-win situation. When the competition intensity or showrooming effect is sufficiently large, the service effort strategy may cause a prisoner’s dilemma for the manufacturer and the platform. In addition, the supply chain consisting of a manufacturer, an offline store and an online platform is also studied in the extension section, and we find that our main results are valid.

In this paper, with the aid of symbolic computation system Python and based on the deep neural network (DNN), automatic differentiation (AD), and limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization algorithms, we discussed the modified Korteweg-de Vries (mkdv) equation to obtain numerical solutions. From the predicted solution and the expected solution, the resulting prediction error reaches . The method that we used in this paper had demonstrated the powerful mathematical and physical ability of deep learning to flexibly simulate the physical dynamic state represented by differential equations and also opens the way for us to understand more physical phenomena later.

The study of wave-like propagation of information in nonlinear and dispersive media is a complex phenomenon. In this paper, we provide a new approach to studying this phenomenon, paying special attention to the nonlinear solitary wave problem of the Korteweg–De Vries (KdV) equation. Our proposed algorithm is based on the traveling wave transformation of the KdV equation, which reduces the dimensionality of the system, enabling us to obtain a highly accurate solution with fewer data. The proposed algorithm uses a Lie-group-based neural network trained via the Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimization method. Our experimental results demonstrate that the proposed Lie-group-based neural network algorithm can simulate the behavior of the KdV equation with high accuracy while using fewer data. The effectiveness of our method is proved by examples.

Today, most of the real physical world problems can be best modelled with fractional telegraph equation. Besides modelling, the solution techniques and their reliability are the most important. Therefore, high accuracy solutions are always needed. As we all know, reproducing kernel method (RKM) has been successfully presented for solving various ordinary differential equations. However, the numerical results are not perfectly satisfactory when we directly use the traditional RKM for solving fractional partial differential equation. The aim of this paper is to fill this gap. In this paper, a new method is provided for solving fractional telegraph equation in the reproducing kernel space by piecewise technique, which can obtain more accurate solution than traditional method. Three experiments are given to demonstrate the effectiveness of the present method.