Gaussian Basis Functions Research Articles

Vibroacoustic behaviour of a sound cavity enclosed by an acoustic black hole beam

In this paper, we propose a numerical approach to characterize the vibroacoustic coupling between a two-dimensional air cavity and an acoustic black hole (ABH) beam. Displacement variables expanded in terms of Gaussian basis functions are used for both the fluid and the solid, and the coupling condition at the interface between the beam and the cavity is imposed following the null space method (NSM). The approach turns out to be free of numerical instabilities. Concerning the simulations, it is shown that the noise inside the cavity is significantly reduced when the excitation is exerted on the ABH beam as compared to a uniform one. However, if a sound source is placed inside the cavity, no significant noise reduction is achieved by coupling the cavity to an ABH beam instead of a uniform one.

Quantum Chemical Molecular Dynamics Simulations for Methane‐Water Cages

ABSTRACTThe dynamic stability of various methane‐water clathrate‐like cages (CH4@(H2O)n, n = 16, 18, 20, 22), has been analyzed explicitly considering thermal effects by means of ab initio M06‐2X/6–31+G*/PCM calculations, which make use of Gaussian basis functions. Starting from the equilibrium filled cage structures, classical, dynamic reaction coordinate (DRC) on the Born–Oppenheimer surface, and semiclassical, Born–Oppenheimer plus harmonic zero‐point energy surface (BOMD), molecular dynamics have been carried out. Water molecules have a high tendency to orient covalent O–H bonds tangentially to the hydrophobic surface, thus clathrate‐like arrangements are an acceptable model to fully hydrate methane. If the cage size is such as to minimize core repulsion, due to electron cloud overlap, and to maximize host–guest van der Waals attractions, the clathrate‐like structures have a life‐time of two picoseconds in classical DRC simulations. The inclusion of quantum kinetic energy in BOMD simulations results in less structured cages with a reduced amount of hydrogen bond network. The preferential tangential orientation of the O‐H bonds is largely maintained, although few of them point toward the methane for a very short time in BOMD simulations. The reduced configurational space of water molecules hydrating hydrophobic moiety is highlighted, thus any satisfactory molecular modeling has to account for it.

A nearly complete treatment of the effect of non-adiabaticity on rovibrational energies of H3+ (Part III).

In this article, significant contributions of non-adiabaticity for the rovibrational bound states up to 25 000 cm-1 and total angular momentum J = 0-20 of H3+ are investigated. A coupled-perturbed full configuration interaction (CP-FCI) treatment is applied to calculate all couplings between electronic states caused by the nuclear motion. These derivative couplings were evaluated up to the second order by means of a perturbation treatment and include all nuclear Cartesian first and second derivatives of the electronic wavefunctions. In particular, the coupling of special derivatives with respect to r and R in the Jacobi coordinate representation is more significant than thought. The perturbation approach is especially optimal for the treatment of weak non-adiabaticity in case of rovibrational energies in H3+ and had not been available before for H3+ or other triatomics. Using exclusively Gaussian basis functions for CP-FCI appears to be sufficient, because explicit correlated wavefunctions are already used for all other potential energy contributions. Our work is an extension of earlier non-adiabatic investigations based on first derivative couplings of electronic states that led to the concept of geometry-dependent effective nuclear masses and which needs only a single potential energy surface for the dynamics. The implementation allows us to include all non-adiabatic effects up to the order of O(μ-2), μ being the reduced nuclear mass. Our treatment works for any isotopologue and for the whole potential energy curve or surface. By this treatment, a further reduction in deviations to experimental data for most rovibrational levels to less than 0.1 cm-1 is possible. For the related transition frequencies, 1366 of 1720 known rovibrational transitions in H3+ have deviations less than 0.1 cm-1 without using any empirically adjustable parameters or optimizing the nuclear mass for a specific transition. For many questionable assignments (deviations >0.3 cm-1) of observed transitions in H3+, a new labeling is proposed.

GBasis: A Python library for evaluating functions, functionals, and integrals expressed with Gaussian basis functions.

GBasis is a free and open-source Python library for molecular property computations based on Gaussian basis functions in quantum chemistry. Specifically, GBasis allows one to evaluate functions expanded in Gaussian basis functions (including molecular orbitals, electron density, and reduced density matrices) and to compute functionals of Gaussian basis functions (overlap integrals, one-electron integrals, and two-electron integrals). Unique features of GBasis include supporting evaluation and analytical integration of arbitrary-order derivatives of the density (matrices), computation of a broad range of (screened) Coulomb interactions, and evaluation of overlap integrals of arbitrary numbers of Gaussians in arbitrarily high dimensions. For circumstances where the flexibility of GBasis is less important than high performance, a seamless Python interface to the Libcint C package is provided. GBasis is designed to be easy to use, maintain, and extend following many standards of sustainable software development, including code-quality assurance through continuous integration protocols, extensive testing, comprehensive documentation, up-to-date package management, and continuous delivery. This article marks the official release of the GBasis library, outlining its features, examples, and development.

A semi-analytical electromechanical model for energy harvesting of plate with acoustic black hole indentations

Acoustic black hole (ABH) has been proved to be helpful for the energy harvesting of beam. Since thin plates are more widely used in practical engineering, it is of great significance to explore the energy harvesting with ABH effect applying to thin plate. Therefore, a piezoelectric plate with ABH indentations is designed and a standard full-wave rectifier circuit connected to the energy harvester in this paper. A semi-analytical electromechanical model is proposed for the energy harvesting system and Gaussian basis functions are used to solve the coupled electromechanical equations. The harvested power is calculated with the proposed method and the acceptable accuracy could be found through the validation against finite element simulation. Finally, a comprehensive analysis has been performed and numerical results show the system parameters (such as piezoelectric layer parameters, ABH parameters, and circuit parameters) have a significant influence on the energy harvesting.

Safe dynamic sparse training of modified RBF networks for joint feature selection and classification

Developing an efficient embedded feature selection method for both binary and multiclass classification problems is a fundamental topic to be further studied. In this paper, the conventional radial basis function (RBF) network is modified for joint feature selection and classification. By utilizing the anisotropic Gaussian basis function, an individual weight parameter is assigned to each feature for feature weighting. The L1-regularized loss function is designed by combining the softmax cross entropy loss with L1 regularization terms imposing on feature and output weights. A specialized active-set limited-memory projected Quasi-Newton (SAL-PQN) algorithm is proposed to minimize the L1-regularized loss function, which can optimize all the adjustable parameters and zero out the redundant feature and output weights simultaneously. Furthermore, a safe dynamic pruning strategy is integrated into the SAL-PQN algorithm for dynamic sparse training of the modified RBF network, which continuously excludes the dynamically zeroed-out feature weights from the training process. It is theoretically assured of yielding the same gradient updates as SAL-PQN, thus the relevant computation of the zeroed-out feature weights in the forward and backward propagations can be safely eliminated. The experimental results demonstrate the effectiveness and efficiency of the proposed methods for joint feature selection and classification, and illustrate the practical utility of the inherent feature weighting capability.

Optimal Transport Distances to Characterize Electronic Excitations.

Understanding the character of electronic excitations is important in computational and reaction mechanistic studies, but their classification from simulations remains an open problem. Distances based on optimal transport have proven very useful in a plethora of classification problems and, therefore, seem a natural tool to try to tackle this challenge. We propose and investigate a new diagnostic Θ based on the Sinkhorn divergence from optimal transport. We evaluate a k-NN classification algorithm on Θ, the popular Λ diagnostic, and their combination, and assess their performance in labeling excitations, finding that (i) the combination only slightly improves the classification, (ii) Rydberg excitations are not separated well in any setting, and (iii) Θ breaks down for charge transfer in small molecules. We then define a length-scale-normalized version of Θ and show that the result correlates closely with Λ for results obtained with Gaussian basis functions. Finally, we discuss the orbital dependence of our approach and explore an orbital-independent version. Using an optimized combination of the optimal transport and overlap diagnostics together with a different metric is in our opinion the most promising for future classification studies.

Command Filter-Based Adaptive Neural Control for Nonstrict-Feedback Nonlinear Systems with Prescribed Performance

In this paper, a new command filter-based adaptive NN control strategy is developed to address the prescribed tracking performance issue for a class of nonstrict-feedback nonlinear systems. Compared with the existing performance functions, a new performance function, the fixed-time performance function, which does not depend on the accurate initial value of the error signal and has the ability of fixed-time convergence, is proposed for the first time. A radial basis function neural network is introduced to identify unknown nonlinear functions, and the characteristic of Gaussian basis functions is utilized to overcome the difficulties of the nonstrict-feedback structure. Moreover, in contrast to the traditional Backstepping technique, a command filter-based adaptive control algorithm is constructed, which solves the “explosion of complexity” problem and relaxes the assumption on the reference signal. Additionally, it is guaranteed that the tracking error falls within a prescribed small neighborhood by the designed performance functions in fixed time, and the closed-loop system is semi-globally uniformly ultimately bounded (SGUUB). The effectiveness of the proposed control scheme is verified by numerical simulation.

Response of Gaussian white noise excited oscillators with inertia nonlinearity based on the RBFNN method

Although stochastic averaging methods have proven effective in solving the responses of nonlinear oscillators with a strong stiffness term under broadband noise excitations, these methods appear to be ineffective when dealing with oscillators that have a strong inertial nonlinearity term (also known as coordinate-dependent mass) or multiple potential wells. To address this limitation, a radial basis function neural network (RBFNN) algorithm is applied to predict the responses of oscillators with both a strong inertia nonlinearity term and multiple potential wells. The well-known Gaussian functions are chosen as radial basis functions in the model. Then, the approximate stationary probability density function (PDF) is expressed as the sum of Gaussian basis functions (GBFs) with weights. The squared error of the approximate solution for the Fokker-Plank-Kolmogorov (FPK) function is minimized using the Lagrange multiplier method to determine optimal weight coefficients. Three examples are presented to demonstrate how inertia nonlinearity terms and potential wells affect the responses. The mean square errors between Monte Carlo simulations (MCS) and RBFNN predictions are provided. The results indicate that RBFNN predictions align perfectly with those obtained from MCS.

Gaussians for Electronic and Rovibrational Quantum Dynamics.

The assumptions underpinning the adiabatic Born-Oppenheimer (BO) approximation are broken for molecules interacting with attosecond laser pulses, which generate complicated coupled electronic-nuclear wave packets that generally will have components of electronic and dissociation continua as well as bound-state contributions. The conceptually most straightforward way to overcome this challenge is to treat the electronic and nuclear degrees of freedom on equal quantum-mechanical footing by not invoking the BO approximation at all. Explicitly correlated Gaussian (ECG) basis functions have proved successful for non-BO calculations of stationary molecular states and energies, reproducing rovibrational absorption spectra with very high accuracy. In this Article, we present a proof-of-principle study of the ability of fully flexible ECGs (FFECGs) to capture the intricate electronic and rovibrational dynamics generated by short, high-intensity laser pulses. By fitting linear combinations of FFECGs to accurate wave function histories obtained on a large real-space grid for a regularized 2D model of the hydrogen atom and for the 2D Morse potential, we demonstrate that FFECGs provide a very compact description of laser-driven electronic and rovibrational dynamics.

Command filter-based event-triggered fixed-time control for nonlinear systems with prescribed performance and actuator faults

This paper investigates command filter-based adaptive neural network (NN) minimal learning practical fixed-time control for stochastic nonlinear systems with prescribed performance and actuator faults. The considered system is in a high-order nonstrict-feedback stochastic structure with unknown dynamics and external disturbances. By combining NN with minimal learning parameter method, the need for prior knowledge of nonlinear functions is eliminated, and the number of weights’ updating laws for the NN is reduced to one, regardless of the system's order and number of neural nodes. This reduction significantly decreases the computational burden. Additionally, a unique property of the Gaussian basis function of NNs is applied to solve the algebraic loop problem of the nonstrict-feedback structure. A novel event-triggered control mechanism is proposed to save communication resources. In order to surmount the "explosion of complexity" and "singularity" problems, a novel fixed-time command filter is suggested, and then, a modified compensation mechanism is proposed to mitigate the errors that emerge from command filters. Furthermore, to improve the transient and steady-state performance of the tracking error, a prescribed performance function is taken into account. Via the Lyapunov stability theory, it will be shown that the developed adaptive backstepping control scheme, guarantees that the closed-loop system signals are bounded in probability in a fixed time, that the convergence time is independent of the initial value, and the tracking error remains within the decaying prescribed performance bounds all the time. Finally, the effectiveness and practicability of the theoretical results are verified by a practical simulation example.

An efficient impact force identification methodology via a single sensor utilizing the concept of generalized transmissibility

The issue of localizing and reconstructing the impact force through double-inverse approaches is widely recognized as an ill-posed and computationally demanding problem. Such a challenge has emerged as a prominent research topic of interest and great concern in the field of structural health monitoring. This study proposes a novel and efficient hierarchical methodology that is adept at ascertaining both the location and time history of an impact force through data obtained from a single accelerometer. The proposed methodology entails an efficient retrieval of uncorrected modal constants with a ratio function referred to as generalized transmissibility. The generalized transmissibility symbolizes the ratio of responses from identical accelerometer subjected to two distinct impacting scenarios, one excited at a yet-to-be-identified location and the other at a pre-determined reference location. The theoretical disclosure of the correlation between the generalized transmissibility at pole frequencies and retrieved uncorrected modal constants is demonstrated in details herein. These modal constants are then served as a signature to identify the impact location. Following localization, the force reconstruction problem is tackled by fitting a parametric model, wherein a gaussian basis function is employed to effectively approximate the time history of impact force. Experimental demonstrations on a metallic plate and a composite wing were carried out to validate the efficient and accurate localization as well as the rapid reconstruction capabilities in response to an impact force applied anywhere on a 2D structure.

Algebraic model studies of the structure and reactions of 12C and 16O

The role of algebraic models and geometric cluster models of alpha clusters in the structure of ground and first excited 0+ and second excited 2+ states in 12C are evaluated with the aim of establishing if the rotational bands are compatible with rigid structures or if they are instead overlaps of different configurations. A rigid equilateral triangle shape is assumed in Ref.[2, 3]: the energy spectrum, electromagnetic observables are calculated and the transition densities are compared with elastic and inelastic cross-sections for various processes. In a different approach [4, 5], we solve the three-body quantum problem for 3 alphas using the orthogonality condition and the stochastic variational method with correlated Gaussian basis functions. The two-body density distributions indicate that the main configurations of both the second 0+ and 2+ states are acute iscosceles triangle shapes with a 8Be(0+)+α configuration. We also find clues that the second 2+ state is probably not a member of the rigid rotational band built on the Hoyle state.

Adaptive radial basis function neural network sliding mode control of robot manipulator based on improved genetic algorithm

ABSTRACT Since the trajectory-tracking control performance of multi-joint robot manipulator may be degraded due to modeling errors and external disturbances, this paper designs a new adaptive robot manipulator trajectory tracking control method through improved genetic algorithm and radial basis function neural network sliding mode control (IGA-RBFNNSMC). Firstly, the genetic algorithm (GA) is improved by establishing superior populations centered on individuals with high fitness values and selecting individuals in the superior populations for crossover and variation. Secondly, the improved genetic algorithm (IGA) is used for the optimization of the center vector and width vector of the Gaussian basis function in radial basis function (RBF) neural network. Then, based on the dynamics model of the robot manipulator, the modeling errors are approximated by RBF neural network and eliminated by sliding mode control (SMC), and the Lyapunov theorem is used to prove the stability and convergence of the control system. Finally, a two-joint robot manipulator is taken as the research objective and the simulation results show that IGA can significantly reduce the solution time on the basis of guaranteed accuracy and IGA-RBFNNSMC can make the trajectory tracking control accurate and more efficient, which proves the effectiveness of the proposed control method.

Gaussian representation of coarse-grained interactions of liquids: Theory, parametrization, and transferability.

Coarse-grained (CG) interactions determined via bottom-up methodologies can faithfully reproduce the structural correlations observed in fine-grained (atomistic resolution) systems, yet they can suffer from limited extensibility due to complex many-body correlations. As part of an ongoing effort to understand and improve the applicability of bottom-up CG models, we propose an alternative approach to address both accuracy and transferability. Our main idea draws from classical perturbation theory to partition the hard sphere repulsive term from effective CG interactions. We then introduce Gaussian basis functions corresponding to the system's characteristic length by linking these Gaussian sub-interactions to the local particle densities at each coordination shell. The remaining perturbative long-range interaction can be treated as a collective solvation interaction, which we show exhibits a Gaussian form derived from integral equation theories. By applying this numerical parametrization protocol to CG liquid systems, our microscopic theory elucidates the emergence of Gaussian interactions in common phenomenological CG models. To facilitate transferability for these reduced descriptions, we further infer equations of state to determine the sub-interaction parameter as a function of the system variables. The reduced models exhibit excellent transferability across the thermodynamic state points. Furthermore, we propose a new strategy to design the cross-interactions between distinct CG sites in liquid mixtures. This involves combining each Gaussian in the proper radial domain, yielding accurate CG potentials of mean force and structural correlations for multi-component systems. Overall, our findings establish a solid foundation for constructing transferable bottom-up CG models of liquids with enhanced extensibility.

Adaptive control of a class of uncertain nonlinear systems using brain emotional learning and Legendre polynomials

In this paper, an adaptive controller for a class of uncertain nonlinear systems is presented using a combination of Legendre polynomials and brain emotional learning-based intelligent controller (BELBIC). Recently, some versions of BELBIC have been presented with the aim of satisfying the universal approximation property using Gaussian basis function. However, the size of regressor vector is too large that imposes a heavy computational load to the processor. The novelty of this paper is presenting a new version of BELBIC with less computational burden using Legendre polynomials. Moreover, there are very few tuning parameters in Legendre polynomials. Another contribution of this paper is editing the stability analysis presented in recent related works. Due to the intrinsic non-differentiability of the adaptation rules of BELBIC, the second time derivative of Lyapunov function is undefined and thus, the Barbalat’s lemma cannot be applied to verify the asymptotic convergence of the error function. Therefore, bounded-input-bounded-output (BIBO) stability can only be claimed for this controller. Simulation results on different case studies show that Legendre polynomials can improve the universal approximation property of BELBIC with less tuning parameters. Moreover, in the absence of the robust control term in the control law, the performance Legendre polynomials will not deteriorate, while the performance degrade in Gaussian basis function is quite considerable.

Sound waves in continuum models of periodic sonic black holes

In this work we address the study of continuum models of periodic sonic black holes (SBHs) at duct terminations. A conventional SBH combines a decreasing power-law profile with a wall admittance in such a way that sound waves entering the SBH slow down, while their amplitude increases and their wavelength decreases as they approach the duct termination. In an ideal scenario, the wave is trapped and dissipated inside the SBH so that no reflection occurs. Instead of dealing with conventional SBHs, we are interested here in periodic SBHs. These have the potential to extend the application range of SBHs below the cut-on frequency thanks to the formation of stopbands, among other significant aspects. First, the case of an SBH cell placed between semi-infinite ducts is analyzed and it is shown how it reduces not only sound reflection but also sound transmission. The acoustic pressure inside the SBH is governed by a modified Webster equation that is solved in weak form using a basis of Gaussian functions. The dependence of the reflection, transmission and absorption coefficients for different SBH parameters and boundary conditions is analyzed in detail. Next, the case of an infinite periodic SBH lattice is considered. The nullspace method (NSM) is used to impose the periodic boundary conditions of the problem. Complex dispersion curves are computed and the separate roles played by the SBH power-law profile and wall admittance on the dispersion curves and pressure distribution inside the cells are examined. Bragg scattering turns out to be the main mechanism behind bandgap formation and wall admittance is essential to achieve the SBH effect. A parametric study follows, showing the influence of the damping, the residual radius, the SBH order and its length on the imaginary part of the complex dispersion curves. Finally, the reflection, transmission and absorption coefficients for a finite 3-cell SBH are investigated. Although most of the research on SBHs has focused on practical application designs, it is believed that a thorough understanding of the performance of continuum models of SBHs could be very useful for the former.

Machine Learning Techniques to Evaluate the Approximation of Utilization Power in Circuits

The need for products that are more streamlined, more useful, and have longer battery lives is rising in today's culture. More components are being integrated onto smaller, more complex chips in order to do this. The outcome is higher total power consumption as a result of increased power dissipation brought on by dynamic and static currents in integrated circuits (ICs). For effective power planning and the precise application of power pads and strips by floor plan engineers, estimating power dissipation at an early stage is essential. With more information about the design attributes, power estimation accuracy increases. For a variety of applications, including function approximation, regularization, noisy interpolation, classification, and density estimation, they offer a coherent framework. RBFNN training is also quicker than training multi-layer perceptron networks. RBFNN learning typically comprises of a linear supervised phase for computing weights, followed by an unsupervised phase for determining the centers and widths of the Gaussian basis functions. This study investigates several learning techniques for estimating the synaptic weights, widths, and centers of RBFNNs. In this study, RBF networks—a traditional family of supervised learning algorithms—are examined. Using centers found using k-means clustering and the square norm of the network coefficients, respectively, two popular regularization techniques are examined. It is demonstrated that each of these RBF techniques are capable of being rewritten as data-dependent kernels. Due to their adaptability and quicker training time when compared to multi-layer perceptron networks, RBFNNs present a compelling option to conventional neural network models. Along with experimental data, the research offers a theoretical analysis of these techniques, indicating competitive performance and a few advantages over traditional kernel techniques in terms of adaptability (ability to take into account unlabeled data) and computing complexity. The research also discusses current achievements in using soft k-means features for image identification and other tasks.

Domain decomposition method with generalized scattering matrix for radiowave propagation analysis

In this paper, we present a novel method based on generalized scattering matrix representation to analyse radiowave propagation. Problem domain is characterized as cascaded two port networks, of which the scattering relations are obtained by the use of Hermite–Gaussian (HG) basis functions. Then the field variation at the boundaries of the problem domain can be calculated by simple matrix operations. The approach is shown to yield accurate results in proposed numerical examples. The method proposed here can be extended to more complex and large domain electromagnetic problems with its systematic methodology.

Phase retrieval using Gaussian basis functions

A wavefront map of arbitrary aperture can be represented by a sum of Gaussian basis functions. Its performance is compared here to other representation methods such as orthonormal polynomials and Fourier modes. Gaussian basis functions can be applied to phase retrieval of high-frequency wavefront maps. Experiments show good agreement with direct measurements by a wavefront sensor. Optimum measurement conditions are discussed based on the statistics of wavefront properties.