Approximate Function Research Articles

Federated quantum long short-term memory (FedQLSTM)

Quantum federated learning (QFL) can facilitate collaborative learning across multiple clients using quantum machine learning (QML) models, while preserving data privacy. Although recent advances in QFL span different tasks like classification while leveraging several data types, no prior work has focused on developing a QFL framework that utilizes temporal data to approximate functions useful to analyze the performance of distributed quantum sensing networks. In this paper, a novel QFL framework that is the first to integrate quantum long short-term memory (QLSTM) models with temporal data is proposed. The proposed federated QLSTM (FedQLSTM) framework is exploited for performing the task of function approximation. In this regard, three key use cases are presented: Bessel function approximation, sinusoidal delayed quantum feedback control function approximation, and Struve function approximation. Simulation results confirm that, for all considered use cases, the proposed FedQLSTM framework achieves a faster convergence rate under one local training epoch, minimizing the overall computations, and saving 25–33% of the number of communication rounds needed until convergence compared to an FL framework with classical LSTM models.

Approximate Functionals for Multistate Density Functional Theory.

Recently Lu and Gao [J. Phys. Chem. Lett. 2022, 13 (33), 7762-7769] published a new, rigorous density functional theory for excited states and proved that the projection of the kinetic and electron-repulsion operators into the subspace of the lowest electronic states are a universal functional of the matrix density D(r). This is the first attempt to find an approximation to the multistate universal functional . It is shown that (i) does not explicitly depend on the number of electronic states and (ii) is an analytic matrix functional. The Thomas-Fermi-Dirac-von Weizsäcker model and the correlation energy of the homogeneous electron gas are turned into matrix functionals guided by two principles: that each matrix functional should transform properly under basis set transformations and that the ground state functional should be recovered for a single electronic state. Lieb-Oxford-like bounds on the average kinetic and electron-repulsion energies in the subspace are given. When evaluated on the numerically exact matrix density of LiF, this simple approximation reproduces the matrix elements of the electron-repulsion operator in the basis of the exact eigenstates accurately for all bond lengths. In particular the off-diagonal elements of the effective Hamiltonian that come from the interactions of different electronic states can be calculated with the same or better accuracy than the diagonal elements. Unsurprisingly, the largest error comes from the kinetic energy functional. More exact conditions that constrain the functional form of are needed to go beyond the local density approximation.

Inference and diagnostics for censored linear regression model with skewed generalized t distribution

In recent years, some interested data can be recorded only if the values fall within an interval range, and the responses are often subject to censoring. Attempting to perform effective statistical analysis with censored, especially heavy-tailed and asymmetric data, can be difficult. In this paper, we develop a novel linear regression model based on the proposed skewed generalized t distribution for censored data. The likelihood-based inference and diagnostic analysis are established using the Expectation/Conditional Maximization Either algorithm in conjunction with smoothing approximate functions. We derive relevant measures to perform global influence for this novel model and develop local influence analysis based on the conditional expectation of the complete-data log-likelihood function. Some useful perturbation schemes are discussed. We illustrate the finite sample performance and the robustness of the proposed method by simulation studies. The proposed model is compared with other procedures based on a real dataset, and a sensitivity analysis is also conducted.

Experimental study on the mechanical characteristics of weakly cemented mudstone under different loading rates

With the gradual shift of coal mining to the western coal mining region of China, floor heave in weakly cemented mudstone roadways has become an issue affecting the safety and efficiency of coal mine production. Additionally, different mining rates can lead to fluctuating support stresses on the roof and floor of weakly cemented mudstone roadways. Therefore, obtaining a comprehensive understanding of the mechanical properties of weakly cemented mudstone at different loading rates is conducive to improving the issue of floor heave in such roadways and provides a theoretical basis for further study. In this context, a series of uniaxial mechanical tests with concurrent acoustic emission monitoring were conducted on specimens of weakly cemented mudstone under various loading rates (0.005, 0.01, 0.05, and 0.1 mm/s). The stress‒strain and acoustic emission response curves were obtained to effectively characterize the strength, deformation, damage, macroscale instability, and crack propagation characteristics of the mudstone under the influence of loading rate effects. The research results support the following findings: (1) With increasing loading rate, the peak strength and elastic modulus of weakly cemented mudstone significantly increase, while the peak axial strain and peak radial deformation significantly decrease. (2) With increasing loading rate, the stress required to trigger the expansion of weakly cemented mudstone gradually increases, and a significant power-law relationship arises between the strain of the mudstone at the start of expansion and the loading rate. (3) With increasing loading rate, the acoustic emission ringing count of weakly cemented mudstone increases: The failure of weakly cemented mudstone changes from small-range progressive failure to sudden failure, and the failure mode transitions from shear failure to tensile‒shear composite failure. (4) The studied mudstone damage variables increase with increasing loading rate, following an approximate exponential function. The conclusions obtained in this work can provide a theoretical basis for the evolution mechanism and control of floor heave in deep roadway mining.

A hierarchical Bayesian approach for parameter identification in structural deterioration models using probabilistic surrogate models

Most structural systems currently in operation are subjected to potentially multiple deterioration processes, for which different models exist to describe their evolution in time. The quantities of interest (QoIs) involved in them are often directly unobservable. However, advances in Structural Health Monitoring (SHM) have enabled gathering structural response data which can be used to indirectly make inferences on unobservable (state) quantities through Bayesian inference. Obtaining updated estimates of deterioration model parameters, given observed data, enables them to be used to predict future health states of interest for the structure, which constitutes a core tenet of condition-based maintenance (CBM). Key to obtaining robust parameter estimates is accounting for different uncertain quantities involved in the model that transforms unobservable quantities to observable ones, which in a Bayesian context is achieved through the likelihood function. This work presents an offline, flexible hierarchical framework to tackle this problem that employs probabilistic reduced order models (ROMs) in the construction of the likelihood function, that enable marginalizing out these uncertain quantities. The approximate likelihood function is then used within a typical Markov Chain Monte Carlo (MCMC) setting for parameter estimation. The framework is demonstrated in applications related to deterioration processes common to ship structures.

System response curve based first‐order optimization algorithms for cyber‐physical‐social intelligence

AbstractThe continuous enhancement of optimization algorithms and their parameters has spurred the expansion of AI into novel application domains such as image recognition and smart home technology. This paper employs the system response curve (SRC) to the adaptive learning rate optimizer, addressing challenges associated with the establishment of the optimizer control model and parameter adjustments affecting the dynamic performance of the system. These insights offer theoretical support for the optimizer's application in deep learning models. To begin, the adaptive learning rate optimizer is a time‐varying system. Based on the intrinsic relationship between the network optimization and the control system, the time domain expression and approximate transfer function of the adaptive learning rate optimizer are derived, and the system dynamic model is established. Furthermore, based on the system control model of the optimizer, it is proposed to explain the performance impacts of different optimizers and their hyperparameters on the deep learning model through the SRC. Finally, experiments are performed on the MNIST, CIFAR‐10, UTKinect‐Action3D, and Florence3D‐Action datasets to validate the control theory of explaining optimizers through system response curves. The experimental results show that the recognition performance of the Adaptive Moment Estimate (Adam) is better than that of the Adaptive Gradient (AdaGrad) and Root Mean Square Propagation (RMSprop). Additionally, the learning rate affects the model training speed, and the practical application aligns with the theoretical analysis.

Analysis of the dynamic response for Kirchhoff plates by the element-free Galerkin method

Dynamic response analysis involves examining structural behavior under dynamic loading conditions. Transient dynamic analysis emerges as a method employed to assess the responses of deformable bodies. Due to the depth analysis of transient responses for thin plates, the associated error analysis work becomes particularly important. In this work, we conduct stability and convergence analysis of the element-free Galerkin (EFG) method for the dynamic response of Kirchhoff plate model. We start by discretizing the thin plate dynamics using the EFG method for the spatial domain and the central difference scheme for the temporal domain. Stability and convergence analysis for transient responses, including deflection and moment responses, are derived similarly to those in two-dimensional transient structural dynamic analysis. The key to the analysis is transforming the governing equations and clamped boundary conditions into a weak formulation with penalty terms using the penalty method. The main results demonstrate a significant correlation between the error estimate and both nodal spacing and time step size. Additionally, the continuity of the approximation functions and the selection of penalty factors significantly affect numerical accuracy. To validate the theoretical results, we conduct numerical experiments involving clamped square and L-shaped plates with uniform and nonuniform node distributions, as well as a perforated plate model. Furthermore, we investigate the impact of node influence domains and penalty factors on numerical accuracy, facilitating parameter selection for practical engineering applications.

Toward the Next Generation of Density Functionals: Escaping the Zero-Sum Game by Using the Exact-Exchange Energy Density.

ConspectusKohn-Sham density functional theory (KS DFT) is arguably the most widely applied electronic-structure method with tens of thousands of publications each year in a wide variety of fields. Its importance and usefulness can thus hardly be overstated. The central quantity that determines the accuracy of KS DFT calculations is the exchange-correlation functional. Its exact form is unknown, or better "unknowable", and therefore the derivation of ever more accurate yet efficiently applicable approximate functionals is the "holy grail" in the field. In this context, the simultaneous minimization of so-called delocalization errors and static correlation errors is the greatest challenge that needs to be overcome as we move toward more accurate yet computationally efficient methods. In many cases, an improvement on one of these two aspects (also often termed fractional-charge and fractional-spin errors, respectively) generates a deterioration in the other one. Here we report on recent notable progress in escaping this so-called "zero-sum-game" by constructing new functionals based on the exact-exchange energy density. In particular, local hybrid and range-separated local hybrid functionals are discussed that incorporate additional terms that deal with static correlation as well as with delocalization errors. Taking hints from other coordinate-space models of nondynamical and strong electron correlations (the B13 and KP16/B13 models), position-dependent functions that cover these aspects in real space have been devised and incorporated into the local-mixing functions determining the position-dependence of exact-exchange admixture of local hybrids as well as into the treatment of range separation in range-separated local hybrids. While initial functionals followed closely the B13 and KP16/B13 frameworks, meanwhile simpler real-space functions based on ratios of semilocal and exact-exchange energy densities have been found, providing a basis for relatively simple and numerically convenient functionals. Notably, the correction terms can either increase or decrease exact-exchange admixture locally in real space (and in interelectronic-distance space), leading even to regions with negative admixture in cases of particularly strong static correlations. Efficient implementations into a fast computer code (Turbomole) using seminumerical integration techniques make such local hybrid and range-separated local hybrid functionals promising new tools for complicated composite systems in many research areas, where simultaneously small delocalization errors and static correlation errors are crucial. First real-world application examples of the new functionals are provided, including stretched bonds, symmetry-breaking and hyperfine coupling in open-shell transition-metal complexes, as well as a reduction of static correlation errors in the computation of nuclear shieldings and magnetizabilities. The newest versions of range-separated local hybrids (e.g., ωLH23tdE) retain the excellent frontier-orbital energies and correct asymptotic exchange-correlation potential of the underlying ωLH22t functional while improving substantially on strong-correlation cases. The form of these functionals can be further linked to the performance of the recent impactful deep-neural-network "black-box" functional DM21, which itself may be viewed as a range-separated local hybrid.

Optimal Piecewise Polynomial Approximation for Minimum Computing Cost by Using Constrained Least Squares.

In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed.

Chemical Potentials and the One-Electron Hamiltonian of the Second-Order Perturbation Theory from the Functional Derivative Approach.

We develop a functional derivative approach to calculate the chemical potentials of second-order perturbation theory (MP2). In the functional derivative approach, the correlation part of the MP2 chemical potential, which is the derivative of the MP2 correlation energy with respect to the occupation number of frontier orbitals, is obtained from the chain rule via the noninteracting Green's function. First, the MP2 correlation energy is expressed in terms of the noninteracting Green's function, and its functional derivative to the noninteracting Green's function is the second-order self-energy. Then, the derivative of the noninteracting Green's function to the occupation number is obtained by including the orbital relaxation effect. We show that the MP2 chemical potentials obtained from the functional derivative approach agree with that obtained from the finite difference approach. The one-electron Hamiltonian, defined as the derivative of the MP2 energy with respect to the one particle density matrix, is also derived using the functional derivative approach, which can be used in the self-consistent calculations of MP2 and double-hybrid density functionals. The developed functional derivative approach is promising for calculating the chemical potentials and the one-electron Hamiltonian of approximate functionals and many-body perturbation approaches dependent explicitly on the noninteracting Green's function.

Solving multi-dimensional European option pricing problems by integrals of the inverse quadratic radial basis function on non-uniform meshes

This paper explores multi-asset options as a means to diversify portfolios, mitigating risk across various assets. We present a numerical method using radial basis function-generated finite difference solvers via integrals of the inverse quadratic kernel. Our method introduces new weights for the task we are dealing with. We derive and compute analytical solutions to approximate function derivatives on three-node stencils with non-uniform and uniform distances. Our findings highlight the convergence order of the proposed analytical weights. Numerical examples illustrate the theory.

Dynamic modeling and identification of a reluctance synchronous machine parameters

This article discusses an identification and modeling approach of a reluctance synchronous motor (RSM) based on the running rotor technique. The applied flux linkage approximation functions reflect the self-saturation and cross-saturation effects, and the applied mathematical model is continuous and differentiable. The proper design of the experiment is discussed, and relevant recommendations are made to ensure the mitigation of procedural mistakes in the experiments. A detailed analysis of the impact of configuration faults on the obtained experimental data is provided, considering distortions in the obtained flux linkage and inductance surfaces. Considering the achieved model accuracy, a novel model evaluation considering the achieved model accuracy technique based on transient current response is proposed.

A note on the generalized Babuška–Brezzi theory: Revisiting the proof of the associated Strang error estimates

In this note we simplify the derivation of the error estimates for the generalized Babuška–Brezzi theory with Galerkin schemes defined in terms of approximate bilinear forms and functionals. More precisely, we provide a straight proof that makes no use of any translated continuous or discrete kernel nor of the distance between them, but of suitable upper bounds of the distances of each component of the Galerkin solution to any other member of the respective finite element subspace. In this way, the Strang error estimates are obtained simply by applying the aforementioned bounds along with the triangle inequality, so that they become cleaner and with fully explicit constants. The case in which the discrete bilinear forms can be evaluated at the continuous solution is also considered, which yields the consistency terms to appear separately from the distances to the subspaces, thus allowing the former to be handled independently from the latter.

Analysis of Na2CuP ternary semiconductor compound for optoelectronic application by first-principles methods using GGA and mGGA functionals

An ab initio study of Zintl Na2CuP ternary semiconductor compounds was carried out by applying first-principles methods to calculate the structural, electronic, elastic, mechanical, and optical properties using generalised gradient approximation (GGA) and metaGGA exchange-correlation functionals. The bandgap was determined to be 0.7523 eV and 0.7848 eV using GGA with Wu-Cohen and Perdew-Burke-Ernzerhof (PBE) functionals, respectively. The bandgap was re-approximated using the more accurate metaGGA functionals as 1.078 eV and 1.084 eV using the Tran-Blaha modified Becke Johnson exchange and correlation functional (TB-mBJ) and the strongly constrained and appropriately normed (SCAN) functional, respectively. The projected density of states using the GGA revealed that the conduction band formation was mainly by Cu 2p, P 2p, and Na 2s orbitals with the rest of the orbitals making a minor contribution. In contrast, the valence band formation was mainly formed by Cu 3d and P 2p, with the rest of the orbital playing a minor role in the formation. The material was found to be mechanically brittle with a covalent bond, which is a characteristic of Zintl-phased materials. The Na2CuP material was also observed to have a strong absorption coefficient between 1.15 eV and 15 eV, a characteristic suitable for photovoltaic applications.

Impact of variable electrical conductivity, viscosity on convective heat and mass transfer flow of CuO- and Al2O3-water nanofluids in cylindrical annulus

In the food industry, electrical conductivity is essential for heating processes. The dependence on temperature conductivity of electricity on the outermost layers flow of the nanofluid is the main topic of this paper. Variable electrical conductivity, viscosity, thermo diffusion, thermal radiation and radiation absorption on convective heat and mass transfer flow Cuo and Al2O3-water nano-fluids confined in cylindrical annulus. The non-linear governing equations have been solved by finite element technique with quadratic approximation functions. For various parametric adjustments, the temperature, speed, and nanoconcentration have all been examined. Similar to the cylindrical wall, quantitative evaluations have been made of the surface resistance, temperature rate and mass transport. It is discovered that for both types of nanofluids, a higher thermo-diffusion effect leads to a lower concentration and Sherwood digits on the cylinders. An augment in Q1 enriches the rapidity in CuO-water nanofluidic system as well as decreases in Al2O3-water nanofluidic. Increased Q1 lowers the real temperature and nanoconcentration in both types of nanofluids.

A nonconvex sparse recovery method for DOA estimation based on the trimmed lasso

Sparse direction-of-arrival (DOA) estimation methods can be formulated as a group-sparse optimization problem. Meanwhile, sparse recovery methods based on nonconvex penalty terms have been a hot topic in recent years due to their several appealing properties. Herein, this paper studies a new nonconvex regularized approach called the trimmed lasso for DOA estimation. We define the penalty term of the trimmed lasso in the multiple measurement vector model by ℓ2,1-norm. First, we use the smooth approximation function to change the nonconvex objective function to the convex weighted problem. Next, we derive sparse recovery guarantees based on the extended Restricted Isometry Property and regularization parameter for the trimmed lasso in the multiple measurement vector problem. Our proposed method can control the desired level of sparsity of estimators exactly and give a more precise solution to the DOA estimation problem. Numerical simulations show that our proposed method overperforms traditional approaches, which is more close to the Cramer-Rao bound.

GGT-SNN: Graph learning and Gaussian prior integrated spiking graph neural network for event-driven tactile object recognition

Spiking neural networks possess asynchronous, discrete, and sparse characteristics that enable direct processing of event-based tactile sensor data and more efficient transmission of information. These networks have been widely applied in the tactile perception field. However, due to the sparsity of tactile data, these models are prone to generalization issues. Additionally, existing methods for tactile graph construction are constrained when spatially connecting high-dimensional tactile data, making it challenging for the model to fuse and process such information effectively. To address these challenges, we propose the GGT-SNN, which introduces a parameter estimation method with Gaussian priors to alleviate generalization problems caused by sparse tactile data. Furthermore, we design M-tree and Z-tree tactile graph construction methods to compensate for deficiencies of other approaches when handling high-dimensional tactile spatial connections. This approach enhances the model’s ability to efficiently process high-dimensional event-driven tactile information. We introduce an approximate LIF neuron activation function to enable backpropagation within the model and comprehensively compare and evaluate its performance against different approximate functions. The experimental results demonstrate that our proposed method performs significantly better than SOTA methods, exhibiting a 10.83 % improvement over TactileSGNet on the EvTouch-Containers dataset and a 2.92 % improvement over TactileSGNet on the EvTouch-Objects dataset.

An Efficient Algorithm for Basic Elementary Matrix Functions with Specified Accuracy and Application

If the matrix function f(At) posses the properties of f(At)=gf(tkA, then the recurrence formula fi−1=gfi,i=N,N−1,⋯,1,f(tA)=f0, can be established. Here, fN=f(AN)=∑j=0majANj,AN=tkNA. This provides an algorithm for computing the matrix function f(At). By specifying the calculation accuracy p, a method is presented to determine m and N in a way that minimizes the time of the above algorithm, thus providing a fast algorithm for f(At). It is important to note that m only depends on the calculation accuracy p and is independent of the matrix A and t. Therefore, f(AN) has a fixed calculation format that is easily computed. On the other hand, N depends not only on A, but also on t. This provides a means to select t such that N is equal to 0, a property of significance. In summary, the algorithm proposed in this article enables users to establish a desired level of accuracy and then utilize it to select the appropriate values for m and N to minimize computation time. This approach ensures that both accuracy and efficiency are addressed concurrently. We develop a general algorithm, then apply it to the exponential, trigonometric, and logarithmic matrix functions, and compare the performance with that of the internal system functions of Mathematica and Pade approximation. In the last section, an example is provided to illustrate the rapid computation of numerical solutions for linear differential equations.

Alternative CNDOL Fockians for fast and accurate description of molecular exciton properties.

CNDOL is an a priori, approximate Fockian for molecular wave functions. In this study, we employ several modes of singly excited configuration interaction (CIS) to model molecular excitation properties by using four combinations of the one electron operator terms. Those options are compared to the experimental and theoretical data for a carefully selected set of molecules. The resulting excitons are represented by CIS wave functions that encompass all valence electrons in the system for each excited state energy. The Coulomb-exchange term associated to the calculated excitation energies is rationalized to evaluate theoretical exciton binding energies. This property is shown to be useful for discriminating the charge donation ability of molecular and supermolecular systems. Multielectronic 3D maps of exciton formal charges are showcased, demonstrating the applicability of these approximate wave functions for modeling properties of large molecules and clusters at nanoscales. This modeling proves useful in designing molecular photovoltaic devices. Our methodology holds potential applications in systematic evaluations of such systems and the development of fundamental artificial intelligence databases for predicting related properties.

Approximate analytical solutions for three-dimensional ascent trajectory of a solid-fuel launch vehicle with time-varying mass flow rate

In the scenario that a solid-fuel launch vehicle maneuvers in outer space at high angles of attack and sideslip for energy management, Approximate Analytical Solutions (AAS) for the three-dimensional (3D) ascent flight states are derived, which are the only solutions capable of considering time-varying Mass Flow Rate (MFR) at present. The uneven MFR makes the thrust vary nonlinearly and thus increases the difficulty of the problem greatly. The AAS are derived based on a 3D Generalized Ascent Dynamics Model (GADM) with a normalized mass as the independent variable. To simplify some highly nonlinear terms in the GADM, several approximate functions are introduced carefully, while the errors of the approximations relative to the original terms are regarded as minor perturbations. Notably, a finite series with positive and negative exponents, called Exponent-Symmetry Series (ESS), is proposed for function approximation to decrease the highest exponent in the AAS so as to reduce computer round-off errors. To calculate the ESS coefficients, a method of seeking the Optimal Interpolation Points (OIP) is proposed using the least-squares-approximation theory. Due to the artful design of the approximations, the GADM can be decomposed into two analytically solvable subsystems by a perturbation method, and thus the AAS are obtained successfully. Finally, to help implement the AAS, two indirect methods for measuring the remaining mass and predicting the burnout time in flight are put forward using information from accelerometers. Simulation results verify the superiority of the AAS under the condition of time-varying MFR.