Super-exponential Convergence Research Articles

Novel method of particular solutions for second-order variable coefficient differential equations on a square

In this work, a novel method of particular solutions (MPS, for short) is proposed with Legendre polynomials to solve second-order partial differential equations (PDEs) with a variable coefficient on a square. Specially, we prove the uniqueness of the solution for the given model, which is equipped with a sufficiently smooth variable coefficient excluding elliptic properties. Based on the characteristics of Legendre polynomials, particular solutions, which satisfy the natural boundary condition, are constructed for the variable coefficient within the PDEs. Meanwhile we investigate the a-priori error estimates in $ H^1 $-norm and $ L^2 $-norm of the MPS approximations, which depict the asymptotic super-exponential convergence rates of the numerical approximations for sufficiently smooth solutions. Some numerical examples and convergence rates are listed to depict the highlights of this proposed meshless method.

A Novel Finite-Time-Gain-Adjustment Controller Design Method for UAVs Tracking Time-Varying Targets

As for time-varying tracking control problems, many neural-dynamics-based control methods have been proposed because of their high efficiency. The varying-parameter convergent neural dynamics design method with the characteristic of super-exponential convergence has been applied to design controllers for unmanned aerial vehicles. Although the varying-parameter convergent neural dynamics controller has a fast convergence speed, it still needs long enough time to achieve tracking theoretically. By combining the finite-time activation function with the varying-parameter convergent neural dynamics design method, a finite-time-gain-adjustment design method is proposed and proved theoretically in this paper. This controller can make state variables of the system converge to their time-varying targets in finite time. Compared with some traditional methods, contrastive experiments and application to a multi-rotor unmanned aerial vehicle system illustrate that the proposed controller has finite convergence time, better anti-noise performance, and faster convergence speed, which enable the multi-rotor unmanned aerial vehicles to track time-varying targets more quickly and accurately, so as to achieve more complex and efficient control tasks.

A Vary-Parameter Convergence-Accelerated Recurrent Neural Network for Online Solving Dynamic Matrix Pseudoinverse and its Robot Application

Among this study, a vary-parameter convergence-accelerated neural network (VPCANN) model is generalized to solving dynamic matrix pseudoinverse, which can achieve super exponential convergence and noise-resistant, compared to the traditional Zhang neural network (ZNN) designed for dynamic problems. Simulative experiments reveal that the neural state solutions synthesized by the VPCANN can quickly approach to the theoretical pseudoinverse. Moreover, based on three types of noise disturbance including constant noise, random noise and dynamic noise, comparisons between the VPCANN and ZNN model are also investigated, verifying noise-resistant of the VPCANN model is better than the ZNN. In addition, to show the potential application of the VPCANN in practice, the kinematic motion planning of a six-links robot manipulator is considered, further substantiating the efficacy of the VPCANN in the dynamic matrix pseudoinverse.

A generalized varying-parameter recurrent neural network for super solution of quadratic programming problem

To obtain superior convergent speed, a generalized varying-parameter recurrent neural network (GVP-RNN) is established and analyzed for solving quadratic programming (QP) problems. Different from the classical fix-parameter neural network, such as Zhang neural network (ZNN) and finite-time convergent differential neural network (FT-CDNN), GVP-RNN dynamics can achieve exponential convergent results and possess high precision. By Lagrange theorem, QP problems can be transformed to a time-varying matrix equation, which is prepared for the GVP-RNN model. Theoretical analysis has shown the GVP-RNN model takes a super-exponential convergence comparing the corresponding simulated results of ZNN and FT-CDNN. Simulation comparisons are presented to evaluate the theoretical performance of the GVP-RNN with three types of activation functions. Furthermore, the proposed GVP-RNN model is analyzed and applied to control kinematic trajectory of redundant manipulators.

A Parameter-Changing and Complex-Valued Zeroing Neural-Network for Finding Solution of Time-Varying Complex Linear Matrix Equations in Finite Time

For solving complex-valued linear matrix equations with time-varying coefficients (CV-LME-TVC) in the complex field, this article proposes a parameter-changing and complex-valued zeroing neural network (PC-CVZNN) model through integrating a new parameter-changing function. As compared to previous complex-valued zeroing neural networks (CVZNNs) with fixed parameters and existing parameter-changing functions, the PC-CVZNN model can achieve superior performance due to the accelerated role of the new parameter-changing function. In parts of theoretical analysis, we take advantage of Lyapunov methodology to prove that the proposed PC-CVZNN model can acquire the global and super-exponential convergence when the linear activation function is adopted, and even acquire super finite-time convergence when the new sign-bi-power activation function and its modified one are used. In parts of numerical comparison experiments, it is shown that the PC-CVZNN model possesses faster convergence rate than fixed-parameter CVZNN models and other analogy neural networks with parameter-changing function, when applied to finding the solution of CV-LME-TVC. Importantly, an application of the proposed method to the mobile manipulator control provides the potential practical value of the PC-CVZNN model in the industrial field.

Comprehensive Analysis of a New Varying Parameter Zeroing Neural Network for Time Varying Matrix Inversion

The matrix inversion problem plays a very important role in mathematics as well as practical engineering applications. In this article, unlike the traditional fixed-parameter zeroing neural network (ZNN) model, on the basis of the original varying-parameter ZNN (VPZNN) model, an improved VPZNN (IVPZNN) model is established and researched to solve time-varying matrix inversion (TVMI). Specifically, the value of the proposed novel time-varying parameter in the IVPZNN model can grow rapidly over time, which can better meet the needs of ZNN in hardware implementation. In addition, theoretical analyses of the novel time varying parameter and the proposed IVPZNN model are given to guarantee the global superexponential convergence and finite-time convergence. Numerical calculation results verify the superior property of the established IVPZNN model for addressing the TVMI problem, as compared with the existing fixed-parameter ZNN and VPZNN models.

Legendre-spectral Dyson equation solver with super-exponential convergence.

Quantum many-body systems in thermal equilibrium can be described by the imaginary time Green's function formalism. However, the treatment of large molecular or solid ab initio problems with a fully realistic Hamiltonian in large basis sets is hampered by the storage of the Green's function and the precision of the solution of the Dyson equation. We present a Legendre-spectral algorithm for solving the Dyson equation that addresses both of these issues. By formulating the algorithm in Legendre coefficient space, our method inherits the known faster-than-exponential convergence of the Green's function's Legendre series expansion. In this basis, the fast recursive method for Legendre polynomial convolution enables us to develop a Dyson equation solver with quadratic scaling. We present benchmarks of the algorithm by computing the dissociation energy of the helium dimer He2 within dressed second-order perturbation theory. For this system, the application of the Legendre spectral algorithm allows us to achieve an energy accuracy of 10-9Eh with only a few hundred expansion coefficients.

Varying-Parameter Convergent-Differential Neural Solution to Time-Varying Overdetermined System of Linear Equations

To solve a time-varying overdetermined problem, a novel varying-parameter convergent-differential neural network (VP-CDNN) is proposed, designed, and discussed. Specifically, a vector-error function is first defined. Second, according to neural dynamic design method, an implicit-dynamic equation with a time-varying parameter is derived. Mathematics analysis and theoretical proof verify that the VP-CDNN can obtain the least-squares solution with a super exponential convergence rate. In addition, it is also proved that VP-CDNN can restrain the noise efficiently. Simulations among the VP-CDNN, gradient-based recurrent neural network and zeroing neural network verify that the VP-CDNN has faster speed, higher accuracy, and stronger robustness. At last, applications to data fitting and system identification further verify the high effectiveness and efficiency of the VP-CDNN.

A varying-gain recurrent neural-network with super exponential convergence rate for solving nonlinear time-varying systems

In order to solve a nonlinear time-varying system, a novel varying-gain recurrent neural network (termed as VG-RNN) is proposed and analyzed. To achieve a fast convergent performance, a vector-based unbounded error function is first defined. Second, a varying-gain neural dynamic approach is employed to design the recurrent neural network formula. Being different from the traditional constant-gain recurrent neural networks with fixed design parameters such as the gradient-based neural network (termed as GNN) and the zeroing neural network (termed as ZNN), the gain coefficient of the proposed VG-RNN is time-varying, which can change with time evolves. Otherwise, compared to the previous numerical methods on solving nonlinear time-varying systems, the solution obtained by VG-RNN is more precise. Third, rigorous mathematics analysis proves the super exponential convergence and accuracy of the proposed VG-RNN. Numerical experiments demonstrate the high accuracy, effectiveness and superiority of the VG-RNN compared with the conventional neural networks for solving nonlinear time-varying systems. Furthermore, we hope to apply the theory proposed in this paper to practical nonlinear time-varying automatic control systems, such as robots with nonlinear time-varying systems.

Varying-parameter Zhang neural network for approximating some expressions involving outer inverses

A varying-parameter ZNN (VPZNN) neural design is defined for approximating various generalized inverses and expressions involving generalized inverses of complex matrices. The proposed model is termed as and defined on the basis of the error function which includes three appropriate matrices A,F,G. The evolution design includes so far defined VPZNN models for computing generalized inverses and also generates a number of matrix expressions involving these generalized inverses. Global and super-exponential convergence properties of the proposed model as well as behaviour of its equilibrium state are investigated. Main contribution of the defined model is its generality. Most important particular cases of the defined model are presented in order to show this fact explicitly. Presented simulation results illustrate generality and effectiveness of the discovered ZNN evolution design.

Power-Type Varying-Parameter RNN for Solving TVQP Problems: Design, Analysis, and Applications.

Many practical problems can be solved by being formulated as time-varying quadratic programing (TVQP) problems. In this paper, a novel power-type varying-parameter recurrent neural network (VPNN) is proposed and analyzed to effectively solve the resulting TVQP problems, as well as the original practical problems. For a clear understanding, we introduce this model from three aspects: design, analysis, and applications. Specifically, the reason why and the method we use to design this neural network model for solving online TVQP problems subject to time-varying linear equality/inequality are described in detail. The theoretical analysis confirms that when activated by six commonly used activation functions, VPNN achieves a superexponential convergence rate. In contrast to the traditional zeroing neural network with fixed design parameters, the proposed VPNN has better convergence performance. Comparative simulations with state-of-the-art methods confirm the advantages of VPNN. Furthermore, the application of VPNN to a robot motion planning problem verifies the feasibility, applicability, and efficiency of the proposed method.

Rejecting Chaotic Disturbances Using a Super-Exponential-Zeroing Neurodynamic Approach for Synchronization of Chaotic Sensor Systems.

Due to the existence of time-varying chaotic disturbances in complex applications, the chaotic synchronization of sensor systems becomes a tough issue in industry electronics fields. To accelerate the synchronization process of chaotic sensor systems, this paper proposes a super-exponential-zeroing neurodynamic (SEZN) approach and its associated controller. Unlike the conventional zeroing neurodynamic (CZN) approach with exponential convergence property, the controller designed by the proposed SEZN approach inherently possesses the advantage of super-exponential convergence property, which makes the synchronization process faster and more accurate. Theoretical analyses on the stability and convergence advantages in terms of both faster convergence speed and lower error bound within the task duration are rigorously presented. Moreover, three synchronization examples substantiate the validity of the SEZN approach and the related controller for synchronization of chaotic sensor systems. Comparisons with other approaches such as the CZN approach, show the convergence superiority of the proposed SEZN approach. Finally, extensive tests further investigate the impact on convergence performance by choosing different values of design parameter and initial state.

A New Varying-Parameter Convergent-Differential Neural-Network for Solving Time-Varying Convex QP Problem Constrained by Linear-Equality

To solve online continuous time-varying convex quadratic-programming problems constrained by a time-varying linear-equality, a novel varying-parameter convergent-differential neural network (termed as VP-CDNN) is proposed and analyzed. Different from fixed-parameter convergent-differential neural network (FP-CDNN), such as the gradient-based recurrent neural network, the classic Zhang neural network (ZNN), and the finite-time ZNN (FT-ZNN), VP-CDNN is based on monotonically increasing time-varying design-parameters. Theoretical analysis proves that VP-CDNN has super exponential convergence and the residual errors of VP-CDNN converge to zero even under perturbation situations, which are both better than traditional FP-CDNN and FT-ZNN. Computer simulations based on different activation functions are illustrated to verify the super exponential convergence performance and strong robustness characteristics of the proposed VP-CDNN. A robot tracking example is finally presented to verify the effectiveness and availability of the proposed VP-CDNN.

A new finite-time varying-parameter convergent-differential neural-network for solving nonlinear and nonconvex optimization problems

Abstract To solve nonlinear and nonconvex optimization problems, a novel finite-time varying-parameter convergent-differential neural network (termed as FT-VP-CDNN) is proposed and analyzed. Compared with finite-time fixed-parameter convergent-differential neural networks (FT-FP-CDNNs), the proposed FT-VP-CDNN has super exponential convergence, finite-time convergence and strong robustness. Finite-time convergence property of the FT-VP-CDNN is proved and various computer simulations are presented. Numerical simulations verify the superiority of the FT-VP-CDNN when solving nonlinear and nonconvex optimization problem.

A Varying-Parameter Convergent Neural Dynamic Controller of Multirotor UAVs for Tracking Time-Varying Tasks

To stably control the position and attitude angles of an unmanned aerial vehicle (UAV), a varying-parameter convergent neural dynamic (VP-CND) method is proposed and applied. First, the dynamic models of multirotor UAVs are presented. Second, to meet the requirements of high accuracy and real-time control, a VP-CND method is proposed based on an error function to derive the position and attitude angle controllers. The existing fixed-parameter CND methods (e.g., the triple-Zhang dynamics or the Zhang dynamics and gradient dynamics) and the corresponding controllers are presented, and their limitations are analyzed. The proposed VP-CND control method not only can track time-varying target values but also possesses super-exponential convergence performance. Third, simulation comparisons verify the effectiveness, stability, and fast convergence of the VP-CND controllers.

Behavior of the Roots of the Taylor Polynomials of Riemann’s $$\xi $$ ξ Function with Growing Degree

We establish a uniform approximation result for the Taylor polynomials of Riemann’s \(\xi \) function valid in the entire complex plane as the degree grows. In particular, we identify a domain growing with the degree of the polynomials on which they converge to Riemann’s \(\xi \) function. Using this approximation, we obtain an estimate of the number of “spurious zeros” of the Taylor polynomial that lie outside of the critical strip, which leads to a Riemann–von Mangoldt type formula for the number of zeros of the Taylor polynomials within the critical strip. Super-exponential convergence of Hurwitz zeros of the Taylor polynomials to bounded zeros of the \(\xi \) function are also established. Finally, we explain how our approximation techniques can be extended to a collection of analytic L-functions.

Convergence and Robustness Analysis of the Exponential-Type Varying Gain Recurrent Neural Network for Solving Matrix-Type Linear Time-Varying Equation

To solve matrix-type linear time-varying equation more efficiently, a novel exponential-type varying gain recurrent neural network (EVG-RNN) is proposed in this paper. Being distinguished from the traditional fixed-parameter gain recurrent neural network (FG-RNN), the proposed EVG-RNN is derived from a vector- or matrix-based unbounded error function by a varying-parameter neural dynamic approach. With four different kinds of activation functions, the super-exponential convergence performance of EVG-RNN is proved theoretically in details, of which the error convergence rate is much faster than that of FG-RNN. In addition, mathematics proves that the computation errors of EVG-RNN can converge to zero, and it possesses the capability of restraining external interference. Finally, series of computer simulations verify and illustrate the better performance of convergence and robustness of EVG-RNN than that of FG-RNN and FTZNN when solving the identical linear time-varying equation.

Super-Exponentially Convergent Parallel Algorithm for a Fractional Eigenvalue Problem of Jacobi-Type

Abstract A new algorithm for eigenvalue problems for the fractional Jacobi-type ODE is proposed. The algorithm is based on piecewise approximation of the coefficients of the differential equation with subsequent recursive procedure adapted from some homotopy considerations. As a result, the eigenvalue problem (which is in fact nonlinear) is replaced by a sequence of linear boundary value problems (besides the first one) with a singular linear operator called the exact functional discrete scheme (EFDS). A finite subsequence of m terms, called truncated functional discrete scheme (TFDS), is the basis for our algorithm. The approach provides super-exponential convergence rate as m → ∞ {m\to\infty} . The eigenpairs can be computed in parallel for all given indexes. The algorithm is based on some recurrence procedures including the basic arithmetical operations with the coefficients of some expansions only. This is an exact symbolic algorithm (ESA) for m = ∞ {m=\infty} and a truncated symbolic algorithm (TSA) for a finite m. Numerical examples are presented to support the theory.

Superexponentially Convergent Algorithm for an Abstract Eigenvalue Problem with Applications to Ordinary Differential Equations

A new algorithm for the solution of eigenvalue problems for linear operators of the form A = A + B (with a special application to high-order ordinary differential equations) is proposed and justified. The algorithm is based on the approximation of A by an operator $$ \overline{A}=A+\overline{B} $$ such that the eigenvalue problem for Ā is supposed to be simpler than for A: The algorithm for this eigenvalue problem is based on the homotopy idea and, for a given eigenpair number, recursively computes a sequence of approximate eigenpairs that converges to the exact eigenpair with a superexponential convergence rate. The eigenpairs can be computed in parallel for all prescribed indexes. The case of multiple eigenvalues of the operator Ā is emphasized. Examples of eigenvalue problems for the high-order ordinary differential operators are presented to support the theory.

Super-Exponentially Convergent Parallel Algorithm for Eigenvalue Problems with Fractional Derivatives

Abstract A new algorithm for eigenvalue problems for linear differential operators with fractional derivatives is proposed and justified. The algorithm is based on the approximation (perturbation) of the coefficients of a part of the differential operator by piecewise constant functions where the eigenvalue problem for the last one is supposed to be simpler than the original one. Another milestone of the algorithm is the homotopy idea which results at the possibility for a given eigenpair number to compute recursively a sequence of the approximate eigenpairs. This sequence converges to the exact eigenpair with a super-exponential convergence rate. The eigenpairs can be computed in parallel for all prescribed indexes. The proposed method possesses the following principal property: its convergence rate increases together with the index of the eigenpair. Numerical examples confirm the theory.