Guest Editorial: Special issue on recurrent dynamic neural networks: Theory and applications

Recurrent dynamic neural network has been proven to be a powerful tool in the online solving of problems with considerable complexity and has been applied to various fields. In recent years, various recurrent dynamic neural networks have been developed to solve complex time-varying problems, such as time-varying matrix inversion, time-varying nonlinear optimisation, motion control of manipulators and so on. However, some thorny issues remain, including, but not limited to, sensitivity to noises, slow convergent speed, and high computational complexity.

We envisioned this Special Issue could provide a platform for researchers in this area to publish their latest research ideas. This call received 25 high-quality submissions. After passing through the peer review process, eight high-quality papers were accepted for publication.

In the first paper (Ren et al.), the authors give an overview of the latest process of weakly supervised learning in medical image analysis, including incomplete, inexact and inaccurate supervision, and introduce the related works on different applications for medical image analysis. Related concepts are illustrated to help readers get an overview ranging from supervised to unsupervised learning within the scope of machine learning. Furthermore, the challenges and future works of weakly supervised learning in medical image analysis are discussed.

In the second paper (Gheisari et al.), the ways, advantages, drawbacks, architectures, and methods of deep learning (DL) are investigated in order to have a straightforward and clear understanding of it from different views. Moreover, the existing related methods are compared with each other, and the application of DL is described in some applications, such as medical image analysis, handwriting recognition and so on.

In the third paper (Shi et al.), an advanced continuous-time recurrent neural network (RNN) model based on a double integral RNN design formula is proposed for solving continuous time-varying matrix inversion, which has an incomparable disturbance-suppression property. For digital hardware applications, the corresponding advanced discrete-time RNN model is proposed based on the discretisation formulas. As a result of theoretical analysis, it is demonstrated that the advanced continuous-time RNN model and the corresponding advanced discrete-time RNN model have global and exponential convergence performance, and they are excellent for suppressing different disturbances. Finally, inspiring experiments, including two numerical experiments and a practical experiment, are presented to demonstrate the effectiveness and superiority of the advanced discrete-time RNN model for solving discrete time-varying matrix inversion with disturbance-suppression.

In the fourth paper (Li Z. and Li S.), for the first time, a novel recursive recurrent network model is proposed to solve the kinematic control issue for manipulators with different levels of physical constraints, and the proposed recursive RNN can be formulated as a new manifold system to ensure control solution within all of the joint constraints in different orders. The theoretical analysis shows the stability of the proposed recursive RNN and its convergence to solution. Simulation results further demonstrate the effectiveness of the proposed method in end-effector path tracking control under different levels of joint constraints based on the Kuka manipulator system. Comparisons with other methods such as the pseudoinverse-based method and conventional RNN method substantiate the superiority of the proposed method.

In the fifth paper (Zhao et al.), for investigating tensegrity form-finding problems, the authors established a concise and efficient dynamic relaxation-noise tolerant zeroing neural network (DR-NTZNN) form-finding algorithm by analysing the physical properties of tensegrity structures. In addition, the non-linear constrained optimisation problem which transformed from the form-finding problem is solved by a sequential quadratic programming algorithm. Moreover, for the purpose of suppressing the noise items, a noise tolerant zeroing neural network is presented to solve the search direction, which can endow the anti-noise capability of the form-finding model and enhance the calculation capability. Besides, the dynamic relaxation method is proposed to calculate the nodal coordinates rapidly when the search direction is acquired. The numerical results show that the form-finding model has a huge capability for high-dimensional free form cable-strut mechanisms with complicated topology. Furthermore, comparing with other existing form-finding methods, the contrast simulation results reveal the excellent anti-noise performance and calculation capacity of DR-NTZNN form-finding algorithm. Eventually, in the future, how to construct a general dynamics relaxation form-finding model for engineering applications is the main concern.

In the sixth paper (Wei et al.), an open-closed-loop iterative learning control (ILC) strategy is developed for linear time-varying multiple input multiple output systems with a vector relative degree, where the time interval of operation is iteration-dependent. To compensate the missing tracking signal caused by iteration-dependent interval, the feedback control is introduced in ILC design. As the tracking signal of many continuous iterations is lost in a certain interval, the feedback control part can employ the tracking signal of current iteration for compensation. Under the assumption that the initial state vibrates around the desired initial state uniformly in a mathematical expectation sense, the expectation of the ILC tracking error can converge to zero as the number of iterations tends to infinity. Under the circumstance that the initial state varies around the desired initial state with a bound, as the number of iterations tends to infinity, the expectation of the ILC tracking error can be driven to a bounded range, whose upper bound is proportional to the fluctuation. It is revealed that the convergence condition is dependent on the feedforward control gains, while the feedback control can accelerate convergence speed by selecting appropriate feedback control gains. As a special case, the controlled system with an integrated high relative degree is also addressed by proposing a simplified iteration dependent interval-based open-closed-loop ILC method. Finally, the effectiveness of the developed iteration dependent interval-based open-closed-loop ILC is illustrated by a simulation example with two cases on the initial state.

In the seventh paper (Wang S. et al.), a method for underwater acoustic sensor network (UASN) localisation is proposed based on zeroing neurodynamics methodology to preferably locate moving underwater nodes. A zeroing neurodynamics model specifically designed for UASN localisation is constructed with rigorous theoretical analyses of its effectiveness. The proposed zeroing neurodynamics model is compatible with some localisation algorithms, which can be utilised to eliminate errors in non-ideal situations, thus further improving its effectiveness. Finally, the effectiveness and compatibility of the proposed zeroing neurodynamics model are substantiated by examples and computer simulations.

In the eighth paper (Wang G. et al.), a novel gradient-based neural network model with an activated variable parameter, namely the activated variable parameter gradient-based neural network (AVPGNN) model, is proposed to solve time-varying constrained quadratic programming problems. With a variable parameter, the AVPGNN model can avoid the limitations caused by the matrix inversion and achieve zero residual error. Moreover, various activation functions are exploited to improve the convergence rate of the AVPGNN model. The accuracy and convergence rate of the AVPGNN model are rigorously analysed in theory and verified by numerical experiments. Finally, to explore the feasibility of the AVPGNN model, applications to the motion planning of a robotic manipulator and the portfolio selection of marketed securities are illustrated.

We appreciate all the authors for their submissions and all the reviewers for their valuable reviews and comments. We hope that this Special Issue will inspire new outcomes for the research community in recurrent dynamic neural networks.