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Forward–backward doubly stochastic differential equations (FBDSDEs) serve as a probabilistic interpretation of stochastic partial differential equations (SPDEs) with diverse applications. Coupled FBDSDEs encounter numerous challenges in numerical approximation compared to forward–backward stochastic differential equations (FBSDEs) and decoupled FBDSDEs, including ensuring the measurability of the numerical solutions, accounting for the mutual influences between forward and backward processes, and considering the relationship with respect to SPDEs rather than PDEs. This paper introduces, for the first time, a numerical method for solving multi-dimensional coupled FBDSDEs. By integrating an optimal control-based approach with deep neural networks, it effectively addresses the coupling-related challenges between forward and backward equations. Computational examples of coupled FBDSDEs with explicit solutions demonstrate that the proposed deep learning-based numerical algorithm achieves commendable performance in terms of both accuracy and efficiency. [ABSTRACT FROM AUTHOR]

Abstract: The discrete delayed Hopfield neural networks is an extension of the discrete Hopfield neural networks. In this paper, the convergence of discrete delayed Hopfield neural networks is mainly studied, and some results on the convergence are obtained by using Lyapunov function. Several new sufficient conditions for the delayed networks converging towards a limit cycle with period at most 2 are proved in parallel updating mode. Also, some conditions for the delayed networks converging towards a limit cycle with 2-period are investigated in parallel updating mode. All results established in this paper extend the previous results on the convergence of both the discrete Hopfield neural networks, and the discrete delayed Hopfield neural networks in parallel updating mode. [Copyright &y& Elsevier]

Abstract: Employing the matrix measure approach and Lyapunov function, the author studies the global exponential stability of delayed neural networks with impulses in this paper. Some novel and sufficient conditions are given to guarantee the global exponential stability of the equilibrium point for such delayed neural networks with impulses. Finally, three examples are given to show the effectiveness of the results obtained here. [Copyright &y& Elsevier]

Abstract: There have been many studies on the simultaneous approximation capability of feed-forward neural networks (FNNs). Most of these, however, are only concerned with the density or feasibility of performing simultaneous approximations. This paper considers the simultaneous approximation of algebraic polynomials, employing Taylor expansion and an algebraic constructive approach, to construct a class of FNNs which realize the simultaneous approximation of any smooth multivariate function and all of its derivatives. We also present an upper bound on the approximation accuracy of the FNNs, expressed in terms of the modulus of continuity of the functions to be approximated. [Copyright &y& Elsevier]

Abstract: In this paper, the existence and global exponential stability of periodic solutions for a class of numerical discretization neural networks are considered. Using coincidence degree theory and the Lyapunov method, sufficient conditions for the existence and global exponential stability of periodic solutions are obtained. A numerical simulation is given to illustrate the results. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate a class of impulsive BAM neural networks with time-varying delays. By establishing the delay differential inequality with impulsive initial conditions and employing -matrix theory, we find some new sufficient conditions ensuring the existence, uniqueness and global exponential stability of the equilibrium point for impulsive BAM neural networks with time-varying delays. In particular, the estimate of the exponential convergence rate is also provided, which depends on the system parameters. An example is given to show the effectiveness of the results obtained here. [Copyright &y& Elsevier]

Abstract: Minor component analysis (MCA) is an important statistical tool for signal processing and data analysis. Neural networks can be used to extract online minor component from input data. Compared with traditional algebraic approaches, a neural network method has a lower computational complexity. Stability of neural networks learning algorithms is crucial to practical applications. In this paper, we propose a stable MCA neural networks learning algorithm, which has a more satisfactory numerical stability than some existing MCA algorithms. Dynamical behaviors of the proposed algorithm are analyzed via deterministic discrete time (DDT) method and the conditions are obtained to guarantee convergence. Simulations are carried out to illustrate the theoretical results achieved. [Copyright &y& Elsevier]

Abstract: In this paper, we study simple algorithms for tracking objects in a video sequence, based on the selection of landmark points representative of the moving objects in the first frame of the sequence to be analyzed. The movement of these points is estimated using a sparse optical-flow method. Methods of this kind are fast, but they are not very robust. Particularly, they are not able to handle the occlusion of the moving objects in the video. To improve the performance of optical flow-based methods, we propose the use of adaptive filters and neural networks to predict the expected instantaneous velocities of the objects, using the predicted velocities as indicators of the performance of the tracking algorithm. The efficiency of these strategies in handling occlusion problems are tested with a set of synthetic and real video sequences. [Copyright &y& Elsevier]

Abstract: In this paper, the problem of asymptotic synchronization for a class of neural networks with reaction-diffusion terms and time-varying delays is investigated. Using the drive-response concept, a control law is derived to achieve the state synchronization of two identical neural networks with reaction-diffusion terms. Moreover, we derive a sufficient asymptotic synchronization condition for the neural networks with reaction-diffusion terms if reaction-diffusion terms satisfy a weaker condition. The synchronization condition is easy to verify and relies on the connection matrix in the driven networks and the suitable designed controller gain matrix in the response networks. [Copyright &y& Elsevier]

Abstract: This paper presents a financial distress prediction model that combines the approaches of neural network learning and logit analysis. This combination can retain the advantages and avoid the disadvantages of the two kinds of approaches in solving such a problem. The radial basis function network (RBFN) is adopted to construct the prediction model. The architecture of RBFN allows the grouping of similar firms in the hidden layer of the network and then performs a logit analysis on these groups instead of directly on the firms. Such a manner can remedy the problem of nominal variables in the input space. The performance of the proposed RBFN is compared to the traditional logit analysis and a backpropagation neural network and demonstrates superior results to both the counterparts in predictive accuracy for unseen data. [Copyright &y& Elsevier]

This paper develops a general purpose numerical method to compute the feedback Nash equilibria in dynamic games. Players'' feedback strategies are first approximated by neural networks which are then trained online by parallel genetic algorithms to search over all time-invariant equilibrium strategies synchronously. To eliminate the dependence of training on the initial conditions of the game, the players use the same stationary feedback policies (the same networks), to repeatedly play the game from a number of initial states at any generation. The fitness of a given feedback strategy is then computed as the sum of payoffs over all initial states. The evolutionary equilibrium of the game between the genetic algorithms is the feedback Nash equilibrium of the dynamic game. An oligopoly model with investment is approximated as a numerical example. [Copyright &y& Elsevier]

When analysing software failure data, many software reliability models are available and in particular, nonhomogeneous Poisson process (NHPP) models are commonly used. However, difficulties posed by the assumptions, their validity, and relevance of these assumptions to the real testing environment have limited their usefulness. The connectionist approach using neural network models are more flexible and with less restrictive assumptions. This model-free technique requires only the failure history as inputs and then develops its own internal model of failure process. Their ability to model nonlinear patterns and learn from the data makes it a valuable alternative methodology for characterising the failure process. In this paper, a modified Elman recurrent neural network in modeling and predicting software failures is investigated. The effects of different feedback weights in the proposed model are also studied. A comparative study between the proposed recurrent architecture, with the more popular feedforward neural network, the Jordan recurrent model, and some traditional parametric software reliability growth models are carried out. [Copyright &y& Elsevier]