Solving Multi-Dimensional Schr\"{o}dinger Equations Based on EPINNs

Due to the good performance of neural networks in high-dimensional and nonlinear problems, machine learning is replacing traditional methods and becoming a better approach for eigenvalue and wave function solutions of multi-dimensional Schr\"{o}dinger equations. This paper proposes a numerical method based on neural networks to solve multiple excited states of multi-dimensional stationary Schr\"{o}dinger equation. We introduce the orthogonal normalization condition into the loss function, use the frequency principle of neural networks to automatically obtain multiple excited state eigenfunctions and eigenvalues of the equation from low to high energy levels, and propose a degenerate level processing method. The use of equation residuals and energy uncertainty makes the error of each energy level converge to 0, which effectively avoids the order of magnitude interference of error convergence, improves the accuracy of wave functions, and improves the accuracy of eigenvalues as well. Comparing our results to the previous work, the accuracy of the harmonic oscillator problem is at least an order of magnitude higher with fewer training epochs. We complete numerical experiments on typical analytically solvable Schr\"{o}dinger equations, e.g., harmonic oscillators and hydrogen-like atoms, and propose calculation and evaluation methods for each physical quantity, which prove the effectiveness of our method on eigenvalue problems. Our successful solution of the excited states of the hydrogen atom problem provides a potential idea for solving the stationary Schr\"{o}dinger equation for multi-electron atomic molecules

Persistence of nonautonomous logistic system with time-varying delays and impulsive perturbations

In this paper, we develop the impulsive control theory to nonautonomous logistic system with time-varying delays. Some sufficient conditions ensuring the persistence of nonautonomous logistic system with time-varying delays and impulsive perturbations are derived. It is shown that the persistence of the considered system is heavily dependent on the impulsive perturbations. The proposed method of this paper is completely new. Two examples and the simulations are given to illustrate the proposed method and results

Positive almost periodicity on SICNNs incorporating mixed delays and D operator

This article involves a kind of shunting inhibitory cellular neural networks incorporating D operator and mixed delays. First of all, we demonstrate that, under appropriate external input conditions, some positive solutions of the addressed system exist globally. Secondly, with the help of the differential inequality techniques and exploiting Lyapunov functional approach, some criteria are established to evidence the globally exponential stability on the positive almost periodic solutions. Eventually, a numerical case is provided to test and verify the correctness and reliability of the proposed findings

Multi/Single-stage structured zero-gradient-sum approach for prescribed-time optimization

Prescribed-time convergence mechanism has become a prominent research focus in the current field of optimization and control due to its ability to precisely control the target completion time. The recently arisen prescribed-time algorithms for distributed optimization, currently necessitate multi-stage structures to achieve global convergence. This paper introduces two modified zero-gradient-sum algorithms, each based on a multi-stage and a single-stage structural frameworks established in this work. These algorithms are designed to achieve prescribed-time convergence and relax two common yet stringent conditions. This work also bridges the gap in current research on single-stage structured PTDO algorithm. The excellent convergence performance of the proposed algorithms is validated through a case study

The Kirchhoff Index of Toroidal Meshes and Variant Networks

The resistance distance is a novel distance function on electrical network theory proposed by Klein and Randić. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices in G. In this paper, we established the relationships between the toroidal meshes network Tm×n and its variant networks in terms of the Kirchhoff index via spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes of L(Tm×n), S(Tm×n), T(Tm×n), and C(Tm×n) were proposed, respectively. Finally, the asymptotic behavior of Kirchhoff indexes in those networks is obtained by utilizing the applications of analysis approach

On Training Traffic Predictors via Broad Learning Structures:A Benchmark Study

A fast architecture for real-time (i.e., minute-based) training of a traffic predictor is studied, based on the so-called broad learning system (BLS) paradigm. The study uses various traffic datasets by the California Department of Transportation, and employs a variety of standard algorithms (LASSO regression, shallow and deep neural networks, stacked autoencoders, convolutional, and recurrent neural networks) for comparison purposes: all algorithms are implemented in MATLAB on the same computing platform. The study demonstrates a BLS training process two-three orders of magnitude faster (tens of seconds against tens-hundreds of thousands of seconds), allowing unprecedented real-time capabilities. Additional comparisons with the extreme learning machine architecture, a learning algorithm sharing some features with BLS, confirm the fast training of least-square training as compared to gradient training

Effect of dynamic route guidance on urban traffic network under Connected Vehicle environment

Although Connected Vehicle technology is developing rapidly, connected vehicles (CV) are going to mix with the traditional vehicles (i.e., non-connected vehicles) for a long time. The effects of deploying CV on urban traffic systems are actually not clear. The main objective of this study is to evaluate the potential effects of route guidance under connected vehicle environment on an urban traffic network in terms of traffic mobility and safety. Microscopic simulation approach is used to conduct CV environment simulation and the rolling horizon approach is used for information updating among the connected vehicles. Meanwhile, driving behavior is modeled through aggressiveness and awareness of drivers. Traffic mobility for the road network was measured by average trip time and average vehicle trip speed. A surrogate measure, i.e., the time-to-collision involved incident rate for one kilometer driven, was used to assess the safety of the road network. Based on a real urban traffic network, the impacts of market penetration levels of connected vehicles and information updating intervals were studied. Simulation results showed that market penetration level of connected vehicles has little impact on the mobility and safety of road network. In addition, according to the simulation conducted in this paper, shorter updating interval is shown to be likely to lead to better mobility, while the safety of road network is likely to decline, under the assumptions embraced in the simulation. By contrast, the simulation also showed that longer updating interval is likely to lead to better safety and decreased mobility

Razumikhin-type theorem for stochastic functional differential systems via vector Lyapunov function

This paper is concerned with input-to-state stability of SFDSs. By using stochastic analysis techniques, Razumikhin techniques and vector Lyapunov function method, vector Razumikhin-type theorem has been established on input-to-state stability for SFDSs. Novel sufficient criteria on the pth moment exponential input-to-state stability are obtained by the established vector Razumikhin-type theorem. When input is zero, an improved criterion on exponential stability is obtained. Two examples are provided to demonstrate validity of the obtained results

On the Incidence Energy of Some Toroidal Lattices

The incidence energy IE(G), defined as the sum of the singular values of the incidence matrix of G, is a much studied quantity with well known applications in chemical physics. In this paper, we derived the closed-form formulae expressing the incidence energy of the 3.12.12 lattice, triangular kagomé lattice, and S(m,n) lattice, respectively. Simultaneously, the explicit asymptotic values of the incidence energy in these lattices are obtained by utilizing the applications of analysis method with the help of software calculation