Non-polynomial Functions Research Articles

Solving Volterra-Fredholm integral equations by non-polynomial spline functions

It depends on our information, non-polynomial spline functions have not been applied for solving Volterra- Fredholm integral equations of the second kind yet. In this paper, we want to use such functions for finding approximation solutions of Volterra-Fredholm integral equations. In our approach, the coefficients of the non-polynomial spline were found by solving a system of linear equations. Then, these functions were utilized to reduce the fredholm integral equations to the solution of algebraic equations. Analysis of convergences investigated. Finally, three examples were presented to show the effectiveness of the method. This was done with the help of a computer program that used the Python code program version 3.9.

Efficient evaluation of lattice Green’s functions

Abstract We develop and implement efficient algorithms for calculating lattice Green’s functions (LGFs) at any point and argument. This includes several approaches: recurrence relations in lattice coordinates, series at zero and infinity, and finite-precision uniform approximations. The methodology can be applied to any simple lattice, whereas program code is provided for triangular and hypercubic lattices. In particular, the obtained generic recurrence relations are applicable to any lattice with a root-free band dispersion. Except for lattices with a high coordination number, these relations allow LGF to be presented as a linear combination of d non-polynomial functions with polynomial coefficients, where d is the lattice dimension. The non-polynomial functions are solutions of d-order differential equation with polynomial coefficients which allows their series expansion at singularities to be performed. For series at infinity, we estimate the remainder, thus extending its use to the zero value of the argument. The remainder itself provides a good finite-precision estimate for the LGF. Finally, we derive a large-scale approximation that smoothly connects the lattice and the continuum Green’s functions. The provided open-source code allows for arbitrary-precision and symbolic computations of LGF.

Infinite quantum signal processing

Quantum signal processing (QSP) represents a real scalar polynomial of degree d using a product of unitary matrices of size 2×2, parameterized by (d+1) real numbers called the phase factors. This innovative representation of polynomials has a wide range of applications in quantum computation. When the polynomial of interest is obtained by truncating an infinite polynomial series, a natural question is whether the phase factors have a well defined limit as the degree d→∞. While the phase factors are generally not unique, we find that there exists a consistent choice of parameterization so that the limit is well defined in the ℓ1 space. This generalization of QSP, called the infinite quantum signal processing, can be used to represent a large class of non-polynomial functions. Our analysis reveals a surprising connection between the regularity of the target function and the decay properties of the phase factors. Our analysis also inspires a very simple and efficient algorithm to approximately compute the phase factors in the ℓ1 space. The algorithm uses only double precision arithmetic operations, and provably converges when the ℓ1 norm of the Chebyshev coefficients of the target function is upper bounded by a constant that is independent of d. This is also the first numerically stable algorithm for finding phase factors with provable performance guarantees in the limit d→∞.

Numerical simulation and error estimation of the Davey-Stewartson equations with virtual element method

This paper aims to present the virtual element method (VEM) for solving the Davey–Stewartson equations with application in fluid mechanics. The VEM is a recent technology that can be regarded as a generalization of the standard finite element method (FEM) to general meshes without the need to integrate complex nonpolynomial functions on the elements. This method only utilizes degrees of freedom associated with the boundary, hence reducing computational complexity compared to the standard FEM. To obtain a full- discrete scheme we combine a semi-implicit scheme with the VEM for time and space variable discretizations, respectively. Furthermore, we obtain an error bound for the full-discrete scheme. The theoretical analysis demonstrates that the convergence rate in the L2 norm is O(h2+τ). Numerical examples confirm efficiency and applicability of the presented method and validate the theoretical outcomes.

Adaptive Privacy-Preserving Coded Computing with Hierarchical Task Partitioning.

Coded computing is recognized as a promising solution to address the privacy leakage problem and the straggling effect in distributed computing. This technique leverages coding theory to recover computation tasks using results from a subset of workers. In this paper, we propose the adaptive privacy-preserving coded computing (APCC) strategy, designed to be applicable to various types of computation tasks, including polynomial and non-polynomial functions, and to adaptively provide accurate or approximated results. We prove the optimality of APCC in terms of encoding rate, defined as the ratio between the computation loads of tasks before and after encoding, based on the optimal recovery threshold of Lagrange Coded Computing. We demonstrate that APCC guarantees information-theoretical data privacy preservation. Mitigation of the straggling effect in APCC is achieved through hierarchical task partitioning and task cancellation, which further reduces computation delays by enabling straggling workers to return partial results of assigned tasks, compared to conventional coded computing strategies. The hierarchical task partitioning problems are formulated as mixed-integer nonlinear programming (MINLP) problems with the objective of minimizing task completion delay. We propose a low-complexity maximum value descent (MVD) algorithm to optimally solve these problems. The simulation results show that APCC can reduce the task completion delay by a range of 20.3% to 47.5% when compared to other state-of-the-art benchmarks.

Quadrature Based Neural Network Learning of Stochastic Hamiltonian Systems

Hamiltonian Neural Networks (HNNs) provide structure-preserving learning of Hamiltonian systems. In this paper, we extend HNNs to structure-preserving inversion of stochastic Hamiltonian systems (SHSs) from observational data. We propose the quadrature-based models according to the integral form of the SHSs’ solutions, where we denoise the loss-by-moment calculations of the solutions. The integral pattern of the models transforms the source of the essential learning error from the discrepancy between the modified Hamiltonian and the true Hamiltonian in the classical HNN models into that between the integrals and their quadrature approximations. This transforms the challenging task of deriving the relation between the modified and the true Hamiltonians from the (stochastic) Hamilton–Jacobi PDEs, into the one that only requires invoking results from the numerical quadrature theory. Meanwhile, denoising via moments calculations gives a simpler data fitting method than, e.g., via probability density fitting, which may imply better generalization ability in certain circumstances. Numerical experiments validate the proposed learning strategy on several concrete Hamiltonian systems. The experimental results show that both the learned Hamiltonian function and the predicted solution of our quadrature-based model are more accurate than that of the corrected symplectic HNN method on a harmonic oscillator, and the three-point Gaussian quadrature-based model produces higher accuracy in long-time prediction than the Kramers–Moyal method and the numerics-informed likelihood method on the stochastic Kubo oscillator as well as other two stochastic systems with non-polynomial Hamiltonian functions. Moreover, the Hamiltonian learning error εH arising from the Gaussian quadrature-based model is lower than that from Simpson’s quadrature-based model. These demonstrate the superiority of our approach in learning accuracy and long-time prediction ability compared to certain existing methods and exhibit its potential to improve learning accuracy via applying precise quadrature formulae.

Numerical integration in the virtual element method with the scaled boundary cubature scheme

AbstractThe virtual element method (VEM) is a stabilized Galerkin method on meshes that consist of arbitrary (convex and nonconvex) polygonal and polyhedral elements. A crucial ingredient in the implementation of low‐ and high‐order VEM is the numerical integration of monomials and nonpolynomial functions over such elements. In this article, we apply the recently proposed scaled boundary cubature (SBC) scheme to compute the weak form integrals in various virtual element formulations over polygonal and polyhedral meshes. In doing so, we demonstrate the flexibility of the approach and the accuracy that it delivers on a broad suite of boundary‐value problems in 2D and 3D over polytopes with affine faces as well as on elements with curved boundaries. In addition, the use of the SBC scheme is exemplified in an enriched Poisson formulation of the VEM in which weakly singular functions are required to be integrated. This study establishes the SBC method as a simple, accurate and efficient integration scheme for use in the VEM.

Positivity-preserving discontinuous Galerkin scheme for linear hyperbolic equations with characteristics-informed augmentation

This paper presents a positivity-preserving discontinuous Galerkin (DG) scheme for the linear hyperbolic problem with variable coefficients on structured Cartesian domains. The standard DG spaces are augmented with either polynomial or non-polynomial basis functions. The primary purpose of these augmented basis functions is to ensure that the cell average from the unmodulated DG scheme remains positive. We explicitly obtain suitable basis functions by inspecting the method of characteristics on an auxiliary problem. A key result is proved which demonstrates that the unmodulated augmented DG scheme will retain a positive cell average, provided that the inflow, source term, and variable coefficients are positive. A simple scaling limiter can then be leveraged to produce a high-order conservative positivity-preserving DG scheme. Numerical experiments demonstrate the scheme is able to retain high-order accuracy as well as robustness for variable coefficients. To improve efficiency, an inexact augmented basis function can be obtained rather than a analytic non-polynomial solution to the auxiliary problem from the method of characteristics.

Approximate Solution of PHI-Four and Allen–Cahn Equations Using Non-Polynomial Spline Technique

The aim of this work is to use an efficient and accurate numerical technique based on non-polynomial spline for the solution of the PHI-Four and Allen–Cahn equations. A recent discovery suggests that the PHI-Four equation focuses on its implications for particle physics and the behavior of scalar fields in the quantum realm. In materials science, ongoing research involves using the Allen–Cahn equation to understand and predict the evolution of microstructures in various materials as well as in biophysics. It depicts pattern formation in biological systems and the dynamics of spatial organization in tissues. To obtain an approximate solution of both equations, this technique uses forward differences for the time and cubic non-polynomial spline function for spatial descretization. The stability of the suggested technique is addressed using the von Neumann technique. Convergence test is carried out theoretically to show the order of convergence of the scheme. Some numerical tests are carried out to confirm accuracy and efficiency in terms of absolute error LR. Convergence rates for different test problems are also computed numerically. Numerical results and simulations obtained are compared with the existing methods.

Improved methods for the enrichment and analysis of the simplicial vector-valued linear finite elements

The simplicial vector linear finite elements are commonly employed for the numerical solution of the stationary Stokes equations. Nevertheless, they exhibit significant limitations when applied to more complex scenarios. In response to these shortcomings, Bernardi and Raugel introduced an enriched finite element which is a generalization of the conventional simplicial vector linear finite element. It employs polynomials as enrichment functions and its application extends across a broad spectrum of practical engineering computational fields. However, for some types of problems, these enrichment functions are not very efficient. The main goal of this paper is to introduce a comprehensive method for enhancing the simplicial vector linear finite element with non-polynomial enrichment functions. The enriched finite element is defined concerning any simplex and can be considered an extension of the Bernardi and Raugel finite element. A crucial component in this context is the characterization result, formulated regarding the nonvanishing of a specific determinant. This result provides both necessary and sufficient conditions for the existence of families of enriched elements. In conclusion, we present numerical tests that show the efficacy of the suggested enrichment strategy.

Universal approximation properties for an ODENet and a ResNet: Mathematical analysis and numerical experiments

We prove a universal approximation property (UAP) for a class of ODENet and a class of ResNet, which are simplified mathematical models for deep learning systems with skip connections. The UAP can be stated as follows. Let $ n $ and $ m $ be the dimension of input and output data, and assume $ m\leq n $. Then we show that ODENet of width $ n+m $ with any non-polynomial continuous activation function can approximate any continuous function on a compact subset on $ \mathbb{R}^n $. We also show that ResNet has the same property as the depth tends to infinity. Furthermore, we derive the gradient of a loss function explicitly with respect to a certain tuning variable. We use this to construct a learning algorithm for ODENet. To demonstrate the usefulness of this algorithm, we apply it to a regression problem, a binary classification, and a multinomial classification in MNIST.

Stability Analysis of a New Fourth-Order Optimal Iterative Scheme for Nonlinear Equations

In this paper, a new parametric class of optimal fourth-order iterative methods to estimate the solutions of nonlinear equations is presented. After the convergence analysis, a study of the stability of this class is made using the tools of complex discrete dynamics, allowing those elements of the class with lower dependence on initial estimations to be selected in order to find a very stable subfamily. Numerical tests indicate that the stable members perform better on quadratic polynomials than the unstable ones when applied to other non-polynomial functions. Moreover, the performance of the best elements of the family are compared with known methods, showing robust and stable behaviour.

An Efficient Method for Solving Nonlinear Boundary Value Problems Via the Non-Polynomial Tension Spline Function

In this research, a nonlinear system of second-order boundary value problems is solved, by using non-polynomial tension spline function. The suggested method's validity and applicability are proved by a few numerical findings. The method's results show that it is both easy to use and efficient.

Intrusive and non-intrusive uncertainty quantification methodologies for pyrolysis modeling

In this work we discuss and compare multiple uncertainty analysis methodologies for pyrolysis modeling. Using Arrhenius equation as the pyrolysis model and kinetic parameters for PMMA material as uncertain variables, different approaches for stochastic uncertainty quantification and their computational effort are analyzed. First non-intrusive (NI) methods are compared through convergence analysis where the deterministic model can be used as is to ascertain the statistical moments, followed by an intrusive polynomial chaos (IPC) approach which requires reformulation of the deterministic model, but solution of the model only once. The applied NI methods include Monte Carlo based simulations, regression and projection based non-intrusive polynomial chaos (NIPC) methods. The uncertainty analysis and convergence comparison is performed for three materials including the effects of overlapping reactions and dependent parameters. In general for different uncertain parameters and materials, IPC gave accurate statistical information in a single model run compared to NIPC projection requiring runs ranging from tens to hundreds and NIPC regression requiring runs in range of thousands to generate converged statistics. However, in case of only a few uncertain parameters, presence of non-polynomial functions in the model and small-scale simulations, projection based NIPC outshines IPC in terms of simplicity and applicability.

Nonlinear vibration study of a rotor blade based on a novel non-polynomial function

This work investigates the nonlinear free vibration of a turbomachinery blade, which is idealized as a rotating, conical, twisted shell of varying thickness reinforced with graphene platelets. Using the von Karman type of nonlinearity, the dynamic equilibrium equation for the twisted panel is obtained and solved by finite element method. The impact of the centrifugal force caused by rotation is taken into account, but the effect of the Coriolis force is neglected. The functionally graded twisted tapering conical shell has two alternative ways in which graphene platelets (GPLs) are reinforced in the thickness direction. The rule of mixture and the modified Halpin-Tsai model are used to compute the effective material properties of the twisted shell. The strain displacement relation is based on a new non-polynomial shear deformation theory. The parametric analysis performed at the endshows that nonlinear frequency increases when twist angle, rotational speed, tapper ratio, and weight fraction of GPLs increase. It is also observed that mode shapes are severely impacted by twist angle and rotational speed.

Refined beam finite element model for thin-walled multi-cell box girders considering distortion and secondary distortional moment deformation effect

In this paper, a novel one-dimensional refined beam finite element model (FEM) accounting for non-uniform distortional warping and secondary distortional moment deformation effect (SDMDE) is formulated for the distortional analysis of thin-walled multi-cell box girders with cantilevered flanges. The extra distortional angle is introduced to incorporate SDMDE, and the validity of which is proved through rigorous theoretical derivation. The in-plane and out-plane deformations at any point around the edge of transverse section built upon the distortional center are unified in the same form and described in detail. The governing differential equations in terms of two generalized displacements, distortional angle χ and primary distortional angle curvature Θ, are derived based on the principle of minimum potential energy (MPE) and solved exactly so as to provide non-polynomial shape functions for the establishment of element stiffness matrix and nodal load vector. A series of multi-cell rectangular hollow section (RHS) girders are analyzed and particular extensions are given to single-cell and multi-cell bridge girders with cantilevered flanges. Comparisons to the results generated from corresponding ones provided by pioneering work or shell FEM analyses not only validate the correctness and reliability of the present beam element formulations but also highlight the significance of the inclusion of SDMDE when the cantilevered flanges are arranged.

LU decomposition and Toeplitz decomposition of a neural network

Any matrix A has an LU decomposition up to a row or column permutation. Less well-known is the fact that it has a ‘Toeplitz decomposition’ A=T1T2⋯Tr where Ti's are Toeplitz matrices. We will prove that any continuous function f:Rn→Rm has an approximation to arbitrary accuracy by a neural network that maps x∈Rn to L1σ1U1σ2L2σ3U2⋯Lrσ2r−1Urx∈Rm, i.e., where the weight matrices alternate between lower and upper triangular matrices, σi(x)≔σ(x−bi) for some bias vector bi, and the activation σ may be chosen to be essentially any uniformly continuous nonpolynomial function. The same result also holds with Toeplitz matrices, i.e., f≈T1σ1T2σ2⋯σr−1Tr to arbitrary accuracy, and likewise for Hankel matrices. A consequence of our Toeplitz result is a fixed-width universal approximation theorem for convolutional neural networks, which so far have only arbitrary width versions. Since our results apply in particular to the case when f is a general neural network, we may regard them as LU and Toeplitz decompositions of a neural network. The practical implication of our results is that one may vastly reduce the number of weight parameters in a neural network without sacrificing its power of universal approximation. We will present several experiments on real data sets to show that imposing such structures on the weight matrices dramatically reduces the number of training parameters with almost no noticeable effect on test accuracy.

Global output feedback control for uncertain strict-feedback nonlinear systems: A logic-based switching event-triggered approach

This paper addresses the global output-feedback stabilization for a class of uncertain nonlinear systems via a switching event-triggered approach. Typically, the system growth rates are allowed to be arbitrary unknown non-polynomial functions of system output. Note that such nonlinearities cannot be handled by generalizing the design rationales employed in polynomial rate cases. For this reason, an event-triggered mechanism together with a logic-based switching mechanism is proposed to determine not only when to sample but also when to switch the control parameter. With the help of the switching control parameter, an observer-based control scheme is developed to achieve global output-feedback stabilization. Particularly, the control parameter can be adaptively adjusted to a sufficiently large value, so the proposed control scheme has a stronger capability to deal with large uncertainties, inherent nonlinearities, and sampling errors of the system. For more efficient resource saving, an extended investigation is presented by constructing a dynamic event-triggered scheme. Finally, two simulation examples are given to illustrate the effectiveness of the switching event-triggered schemes.

On the Omnipresence of Spurious Local Minima in Certain Neural Network Training Problems

AbstractWe study the loss landscape of training problems for deep artificial neural networks with a one-dimensional real output whose activation functions contain an affine segment and whose hidden layers have width at least two. It is shown that such problems possess a continuum of spurious (i.e., not globally optimal) local minima for all target functions that are not affine. In contrast to previous works, our analysis covers all sampling and parameterization regimes, general differentiable loss functions, arbitrary continuous nonpolynomial activation functions, and both the finite- and infinite-dimensional setting. It is further shown that the appearance of the spurious local minima in the considered training problems is a direct consequence of the universal approximation theorem and that the underlying mechanisms also cause, e.g., $$L^p$$ L p -best approximation problems to be ill-posed in the sense of Hadamard for all networks that do not have a dense image. The latter result also holds without the assumption of local affine linearity and without any conditions on the hidden layers.

A general class of enriched methods for the simplicial linear finite elements

Low-order elements are widely used and preferred for finite element analysis, specifically the three-node triangular and four-node tetrahedral elements, both based on linear polynomials in barycentric coordinates. They are known, however, to under-perform when nearly incompressible materials are involved. The problem may be circumvented by the use of higher degree polynomial elements, but their application become both more complex an computationally expensive. For this reason, non-polynomial enriched finite element methods have been proposed for solving engineering problems. In line with previous researches, the main contribution of this paper is to present a general strategy for enriching the standard simplicial linear finite element by non-polynomial functions. A key role is played by a characterization result, given in terms of the non-vanishing of a certain determinant, which provides necessary and sufficient conditions, on the enrichment functions and functionals, that guarantee the existence of families of such enriched elements. We show that the enriched basis functions admit a closed form representation in terms of enrichment functions and functionals. Finally, we provide concrete examples of admissible enrichment functions and perform some numerical tests.