Order Gradient Research Articles

Online Training of LSTM Networks in Distributed Systems for Variable Length Data Sequences.

In this brief, we investigate online training of long short term memory (LSTM) architectures in a distributed network of nodes, where each node employs an LSTM-based structure for online regression. In particular, each node sequentially receives a variable length data sequence with its label and can only exchange information with its neighbors to train the LSTM architecture. We first provide a generic LSTM-based regression structure for each node. In order to train this structure, we put the LSTM equations in a nonlinear state-space form for each node and then introduce a highly effective and efficient distributed particle filtering (DPF)-based training algorithm. We also introduce a distributed extended Kalman filtering-based training algorithm for comparison. Here, our DPF-based training algorithm guarantees convergence to the performance of the optimal LSTM coefficients in the mean square error sense under certain conditions. We achieve this performance with communication and computational complexity in the order of the first-order gradient-based methods. Through both simulated and real-life examples, we illustrate significant performance improvements with respect to the state-of-the-art methods.

On the role of higher order deformation gradients in necking, spinodal decomposition and neuron firing

This is a modest contribution in memory of Gérard Maugin-a contemporary French scholar of Paris who inspired many young scientists in Greece and maintained special ties with Aristotle University and Thessaloniki. It introduces three particular areas of research in nonlinear science that he defined in collaboration with the third author and presents preliminary one-dimensional results mainly due to previous work of the first author. These pertain to the phenomena of necking in cold drawing, spinodal decomposition in diffusive phase transformations and energy transfer in neuronal microtubules. The common ingredient in all three is the introduction of higher order gradients of deformation in a variational formalism, in order to describe solitary waves and pattern formation that emerge from uniform states when the system enters its nonconvex free energy regime.

Bending analysis of functionally graded nanobeams based on the fractional nonlocal continuum theory by the variational Legendre spectral collocation method

In this article, the size-dependent bending behavior of nanobeams made of functionally graded materials is studied through a numerical variational approach. The nonlocal effects are captured in the context of fractional calculus. The nanobeams are modelled based on the Euler–Bernoulli beam theory whose governing fractional equation is derived utilizing the minimum total potential energy principle. In the solution procedure, which is directly applied to the variational form of governing equation, the truncated Legendre series in conjunction with the Legendre operational matrix of fractional derivatives are employed for numerical integration of fractional differential equation. The strong form of the governing equation is also derived and solved to examine the accuracy and efficiency of the proposed solution approach as well as for the comparison purpose. The influences of fractional order, nonlocality and material gradient index on the bending characteristics of nanobeams subject to different end conditions are investigated.

Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on [formula omitted

We derive sharp Adams inequalities for the Riesz and more general Riesz-like potentials on the whole of Rn. As a consequence, we obtain sharp Moser–Trudinger inequalities for the critical Sobolev spaces Wα,nα(Rn), 0<α<n. These inequalities involve fractional Laplacians, higher order gradients, general homogeneous elliptic operators with constant coefficients, and general trace type Borel measures.

Order Parameter Boundary Conditions for Ferroics: Application to Flexoelectricity

We study boundary conditions for ferroics in the context of the Landau theory with higher order gradient terms. Such thermodynamic description requires additional boundary conditions for the order parameter gradients, that we derive here. The results are of practical importance for materials exhibiting incommensurate phases due to competing interactions and for advanced modeling for nano‐scale ferroic systems. Boundary conditions are derived for the case when the order parameter is fixed at the free surface (blocking boundary condition) as well as for the free boundary case. The application of the boundary conditions is demonstrated using phase field simulations for ferroelectrics where there is a coupling of the polarization order parameter with strain gradients.

Why Are Indian Children So Short? The Role of Birth Order and Son Preference

Child stunting in India exceeds that in poorer regions like sub-Saharan Africa. Data on over 168,000 children show that, relative to Africa, India's height disadvantage increases sharply with birth order. We posit that India’s steep birth order gradient is due to favoritism toward eldest sons, which affects parents' fertility decisions and resource allocation across children. We show that, within India, the gradient is steeper for high-son-preference regions and religions. The gradient also varies with sibling gender as predicted. A back-of-the-envelope calculation suggests that India's steeper birth order gradient can explain over one-half of the India-Africa gap in average child height.

Mean velocity equation for fluctuating flow

Arguments are presented that the momentum equation derived by Zwanzig–Mori technique of N-particle statistical mechanics is identical to the Reynolds equation for the mean velocity field in fluctuating hydrodynamics; thus, calculating the non-linear stress of the momentum equation is perhaps a means to avoid the closure problem of turbulence theory. In this paper, the third-order term of the friction force is calculated. Multilinear mode coupling theory is applied in order to obtain formulas for the third- and fourth-order equilibrium time correlation functions appearing in the expression. The force term is obtained as a multilinear convolution integral containing higher order gradients of the velocity field.As an application, circular jet flow for low Reynolds numbers is investigated. Numerical solutions have been obtained from an iteration starting with the corresponding solution of the Navier–Stokes equation. Deviations become distinct for Re>30.

Modelling turbulent and dispersion heat fluxes in turbulent porous medium flows using the resolved LES data

Resolved large eddy simulations (LESs) of turbulent conjugate heat transfer in porous media are performed by the lattice Boltzmann method (LBM) for modelling turbulent and dispersion heat flux terms of the double-averaged energy equation. The considered porous structures are square rod arrays, staggered cube arrays and body centred cubic foam. In the LBM, the double-distribution function method which solves the distribution functions for the velocity and the internal energy is used. For the velocity and thermal fields, the D3Q27 multiple-relaxation-time method and the regularized D3Q19 single-relaxation-time method are applied, respectively. A priori tests using the LES data suggest that the trends of the sum of the dispersion and volume-averaged turbulent heat fluxes can be well captured by the second order gradient diffusion model.

Construction of second gradient continuum models for random fibrous networks and analysis of size effects

In this work we develop anisotropic first and second order displacement gradient linear elastic continuum models for two-dimensional random fiber networks. The continuum constitutive parameters are evaluated based on the response of the explicit representation of the network in which each fiber is a beam and the fibers are connected at crossing points with welded joints. The scaling of the first and second order moduli with the network parameters, such as the network density and the ratio of the fiber bending to axial stiffness, is determined. We observe that the dependence of the second gradient moduli on these two parameters is similar to the dependence of the classical moduli on the same parameters. The internal length scales associated with the gradient terms of the constitutive equation are also defined in terms of the network parameters. The influence of the model size on the elastic constants is discussed. We observe that if the model size is large enough for the classical moduli to be size effect free. However, there is still a strong size dependency of the computed internal lengths associated to second order gradient effects.

Study on fractional order gradient methods

In this paper, convergence capability of the conventional fractional order gradient methods (FOGMs) is analyzed and a new FOGM with guaranteed and faster convergence ability is proposed. In particular, we further consider the case that the derivative order varies between 1 and 2, and give some discussion on how to determine the derivative order in the algorithm settings. Simulation results from a number of numerical examples are provided to illustrate the approaches.

Identification for Hammerstein nonlinear ARMAX systems based on multi-innovation fractional order stochastic gradient

A multi-innovation fractional order stochastic gradient (MIFOSG) algorithm, which involves a variable initial value scheme, is investigated to identify the Hammerstein nonlinear ARMAX systems in this paper. Firstly, according to an improved fractional order gradient method, the MIFOSG algorithm is proposed. Furthermore, according to the martingale convergence theorem, the convergence analysis of the proposed algorithm is developed. In addition, for the purpose of improving the convergence performance, a forgetting factor on step size and a variable gradient order are introduced. Given a sufficiently large number of independent runs, the effectiveness of the proposed algorithm is demonstrated in two numerical examples finally.

Quasi-continuum study of the buckling behavior of single-walled carbon nanocones subjected to bending under thermal loading

Abstract

Anomalous transport from holography: part II

This is a second study of chiral anomaly-induced transport within a holographic model consisting of anomalous U(1)_Vtimes U(1)_A Maxwell theory in Schwarzschild–AdS_5 spacetime. In the first part, chiral magnetic/separation effects (CME/CSE) are considered in the presence of a static spatially inhomogeneous external magnetic field. Gradient corrections to CME/CSE are analytically evaluated up to third order in the derivative expansion. Some of the third order gradient corrections lead to an anomaly-induced negative B^2-correction to the diffusion constant. We also find modifications to the chiral magnetic wave nonlinear in B. In the second part, we focus on the experimentally interesting case of the axial chemical potential being induced dynamically by a constant magnetic and time-dependent electric fields. Constitutive relations for the vector/axial currents are computed employing two different approximations: (a) derivative expansion (up to third order) but fully nonlinear in the external fields, and (b) weak electric field limit but resuming all orders in the derivative expansion. A non-vanishing nonlinear axial current (CSE) is found in the first case. The dependence on magnetic field and frequency of linear transport coefficient functions is explored in the second.

Accelerated gradient with optimal step size for second-order blind signal separation

This paper proposes an algorithm for the second-order blind signal separation problem with convolutive mixtures. An iterative first order gradient method based on the accelerated gradient is developed for solving the optimization problem. For each search direction, the question becomes how to effectively calculate the optimal step size in each iteration. Here, we propose an efficient algorithm for obtaining the step size by first reformulating the objective function as a fourth order polynomial in terms of the step size, where the polynomial coefficients are required to be calculated only once per iteration. An optimal step size search procedure using the Newton’s method is developed with the step size is efficiently obtained for each iteration. Simulation results in a simulated room environment and a real environment show that the proposed algorithm converges faster than the existing methods with a lower number of iterations and a lower computational complexity. In addition, the proposed algorithm can separate the speech signals and reduce the background noise simultaneously.

Predicting PM2.5 Concentrations at a Regional Background Station Using Second Order Self-Organizing Fuzzy Neural Network

This study aims to develop a second order self-organizing fuzzy neural network (SOFNN) to predict the hourly concentrations of fine particulate matter (PM2.5) for the next 24 h at a regional background station called Shangdianzi (SDZ) in China from 14 to 23 January 2010. The structure of the SOFNN was automatically adjusted according to the sensitivity analysis (SA) of model output and the parameter-learning phase was performed applying a second order gradient (SOG) algorithm. Principal component analysis (PCA) was employed to select the dominating factors for PM2.5 concentrations as the input variables for the SOFNN. It was found that the dominating variables (relative humidity (RH), pressure (Pre), aerosol optical depth (AOD), wind speed (WS) and wind direction (WD)) extracted by PCA agreed well with the characteristics of PM2.5 at SDZ where the PM2.5 concentrations were heavily affected by meteorological parameters and were closely related to AOD. The forecasting results showed that the proposed SOG-SASOFNN performed better than other models with higher coefficient of determination (R2) during both training phase and test phase (0.89 and 0.84, respectively) in predicting PM2.5 concentrations at SDZ. In conclusion, the developed SOG-SASOFNN provided satisfying results for modeling the hourly distribution of PM2.5 at SDZ during the studied period.

An innovative fractional order LMS based on variable initial value and gradient order

This article presents a novel fractional order LMS (FOLMS) algorithm, which involves a variable gradient order scheme. The fractional order gradient descent method is revisited firstly. A variable initial value scheme is proposed to attenuate the non–locality of fractional order calculus and to ensure the convergence of the proposed FOLMS algorithm. Furthermore, it is noticed that a contradiction between rapidity and accuracy always appears together with the advancement of FOLMS algorithm; namely, a larger value of the gradient order can not only give a faster convergence speed, but also correspond to a larger estimation error. For the purpose of removing the contradiction between rapidity and accuracy, a variable gradient order scheme is designed for the FOLMS algorithm. Based on a sufficient large number of independent runs, the efficiency and superiority of the proposed algorithm are demonstrated in numerical examples finally.

Regularity for free interface variational problems in a general class of gradients

We present a way to study a wide class of optimal design problems with a perimeter penalization. More precisely, we address existence and regularity properties of saddle points of energies of the form $$ (u,A) \quad \mapsto \quad \int_\Omega 2fu \; \text{d}x \; - \int_{\Omega \cap A} \sigma_1 \mathscr A u \cdot \mathscr A u \; \text{d}x \; - \int_{\Omega \setminus A} \sigma_2\mathscr A u\cdot \mathscr A u \; \text{d}x \; + \; \text{Per}(A;\overline \Omega),$$ where $\Omega$ is a bounded Lipschitz domain, $A\subset \mathbb R^N$ is a Borel set, $u:\Omega \subset \mathbb R^N \to \mathbb R^d$, $\mathscr A$ is an operator of gradient form, and $\sigma_1, \sigma_2$ are two not necessarily well-ordered symmetric tensors. The class of operators of gradient form includes scalar- and vector-valued gradients, symmetrized gradients, and higher order gradients. Therefore, our results may be applied to a wide range of problems in elasticity, conductivity or plasticity models. In this context and under mild assumptions on $f$, we show for a solution $(w,A)$, that the topological boundary of $A \cap \Omega$ is locally a $\rm{C}^1$-hypersurface up to a closed set of zero $\mathscr H^{N-1}$-measure.

Enhanced micropolar model for wave propagation in ordered granular materials

The vibrational properties of a face-centered cubic granular crystal of monodisperse particles are predicted using a discrete model as well as two micropolar models, first the classical Cosserat and second an enhanced Cosserat-type model, that properly takes into account all degrees of freedom at the contacts between the particles. The continuum models are derived from the discrete model via a micro–macro transition of the discrete relative displacements and particle rotations to the respective continuum field variables. Next, only the long wavelength approximations of the models are compared and, considering the discrete model as reference, the Cosserat model shows inconsistent predictions of the bulk wave dispersion relations. This can be explained by an insufficient modeling of sliding mode of particle interactions in the Cosserat model. An enhanced micropolar model is proposed including only one new elastic tensor from the more complete second--order gradient micropolar theory. This enhanced micropolar model then involves the minimum number of elastic constants to consistently predict the dispersion relations in the long wavelength limit.

Effect of passivation on higher order gradient plasticity models for non-proportional loading: energetic and dissipative gradient components

In this work, a new class of thermodynamic-based higher order gradient plasticity theory is proposed and applied to the stretch-surface passivation problem for investigating the material behaviour under the non-proportional loading condition. This paper incorporates the thermal and mechanical responses of microsystems. It addresses phenomena such as size and boundary effects and in particular microscale heat transfer in fast-transient processes. The stored energy of cold work is considered in the development of the recoverable counterpart of the free energy. The main distinction in this formulation is the presence of the dissipative higher order microstress quantity that is known to give rise to the stress jump phenomenon, which causes a controversial dispute in the field of strain gradient plasticity theory with respect to whether it is physically acceptable or not. The finite element solution for the stretch-surface passivation problem is developed and validated by comparing with three sets of small-scale experiments. Based on the validated finite element solution, the stress jump phenomenon under the stretch-surface passivation condition is investigated with the effects of the various material parameters. The evolution of the free energy and dissipation potentials is investigated at elevated temperatures. The two-dimensional simulation is also given to examine the gradient and grain boundary effect, the mesh sensitivity of the two-dimensional model and the stress jump phenomenon. Finally, some significant conclusions are presented.

Influence of elastic strain gradient on the upper limit of flexocoupling strength, spatially modulated phases, and soft phonon dispersion in ferroics

Within the framework of Landau-Ginzburg-Devonshire (LGD) theory we studied the role of the flexocoupling between the order parameter and elastic strain gradients in the stability of a spatially-modulated phase (SMP) in ferroics with commensurate and incommensurate long-range ordered phases under the presence of squired elastic strain gradient. The squired elastic strain gradient is required for the free energy stability to arbitrary strain gradients. Obtained analytical expressions showed that the fundamental upper limit for the magnitude of the static bulk flexoelectric effect strength, established by Yudin and Tagantsev under the absence of squired elastic strain gradient and higher order gradients terms, should be substituted by the temperature-dependent condition on the flexoelectric coupling strength under the presence of the gradient terms. Moreover, we established that the SMP appears and becomes stable in commensurate ferroics if the flexocoupling constant exceeds the critical value, defined by the reduced temperature, strain and order parameter gradients constants, striction and expansion coefficients in the LGD functional. We calculated the soft phonon dispersion in ferroics with commensurate and incommensurate long-range ordered phases allowing for the squired elastic strain gradient, as well as static and dynamic flexocoupling. Appeared that the dispersion for the optic mode is slightly sensitive to the flexocoupling, while the dispersion of acoustic mode strongly depends on the coupling strength. Obtained results demonstrate that the non-trivial differences in the dispersion of optic and acoustic modes appear under the change of flexocoupling constant.