Showing papers on "Basis function published in 2021"

A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs

Abstract: Conventional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) may incur in severe limitations when dealing with nonlinear time-dependent parametrized PDEs, as these are strongly anchored to the assumption of modal linear superimposition they are based on. For problems featuring coherent structures that propagate over time such as transport, wave, or convection-dominated phenomena, the RB method may yield inefficient reduced order models (ROMs) when very high levels of accuracy are required. To overcome this limitation, in this work, we propose a new nonlinear approach to set ROMs by exploiting deep learning (DL) algorithms. In the resulting nonlinear ROM, which we refer to as DL-ROM, both the nonlinear trial manifold (corresponding to the set of basis functions in a linear ROM) as well as the nonlinear reduced dynamics (corresponding to the projection stage in a linear ROM) are learned in a non-intrusive way by relying on DL algorithms; the latter are trained on a set of full order model (FOM) solutions obtained for different parameter values. We show how to construct a DL-ROM for both linear and nonlinear time-dependent parametrized PDEs. Moreover, we assess its accuracy and efficiency on different parametrized PDE problems. Numerical results indicate that DL-ROMs whose dimension is equal to the intrinsic dimensionality of the PDE solutions manifold are able to efficiently approximate the solution of parametrized PDEs, especially in cases for which a huge number of POD modes would have been necessary to achieve the same degree of accuracy.

A Novel Combination of Whale Optimization Algorithm and Support Vector Machine with Different Kernel Functions for Prediction of Blasting-Induced Fly-Rock in Quarry Mines

Abstract: This study proposed a novel data-driven model for estimating distance of fly-rock in bench blasting in open-pit mines using a robust combination of the whale optimization algorithm (WOA), support vector machine (SVM) and kernel functions Four kernel functions were investigated for embedding in the SVM model, including linear (L), radius basis function (RBF), polynomial (P), and hyperbolic tangent (HT) functions Then, the WOA was applied to optimize the kernel-based SVM models, namely WOA–SVM–L, WOA–SVM–P, WOA–SVM–RBF, and WOA–SVM–HT A variety of conventional data-driven models were also developed for predicting fly-rock distance, including adaptive neuro-fuzzy inference system (ANFIS), gradient boosting machine (GBM), random forest (RF), classification and regression tree (CART), and artificial neural network (ANN) The blasting parameters and maximum fly-rock distance, as well as their relationship, were carefully investigated for this aim The predictive results of the models were evaluated through two performance indices: root-mean-squared error (RMSE) and correlation coefficient (R2) These indices indicated that the linear function-based WOA–SVM model (ie, WOA–SVM–L) seems to be not fit for predicting fly-rock with the largest error (ie, RMSE = 9080 and R2 = 0937) In contrast, the WOA–SVM–RBF model yielded the highest accuracy in predicting the distance of fly-rock (ie, RMSE = 5241, R2 = 0977) Meanwhile, the WOA–SVM–P and WOA–SVM–HT models provided lower performances than those of the WOA–SVM–RBF model, but they are acceptable The conventional models (ie, ANFIS, GBM, RF, CART, and ANN) are pretty well (ie, RMSE in the range of 5804 to 6567; R2 in the range of 0965 to 0973); however, their performance is lower than those of the WOA–SVM–RBF model as well Based on these results, the WOA–SVM model was proposed as a useful data-driven model for predicting fly-rock with high reliability in practical engineering

Numerical solutions of time-fractional Klein-Gordon equations by clique polynomials

Abstract: This work adopts to the time-fractional Klein–Gordon equation (FKGE) in the Caputo sense. We present a new technique using the clique polynomial as basis function for the operational matrices to obtain solution of time-FKGE. The key advantage of this technique is converting the time-FKGE to algebraic equations, which can be simply solved the problem under study. For the approximation of a bivariate function using the clique polynomial, an error bound is given. Numerical results derived using the proposed technique are compared with the exact solution. The results show that the proposed technique is very user friendly for solving the time-FKGE and accurate.

An autoencoder‑based reduced‑order model for eigenvalue problems with application to neutron diffusion

Abstract: Using an autoencoder for dimensionality reduction, this paper presents a novel projection-based reduced-order model for eigenvalue problems Reduced-order modelling relies on finding suitable basis functions which define a low-dimensional space in which a high-dimensional system is approximated Proper orthogonal decomposition (POD) and singular value decomposition (SVD) are often used for this purpose and yield an optimal linear subspace Autoencoders provide a nonlinear alternative to POD/SVD, that may capture, more efficiently, features or patterns in the high-fidelity model results Reduced-order models based on an autoencoder and a novel hybrid SVD-autoencoder are developed These methods are compared with the standard POD-Galerkin approach and are applied to two test cases taken from the field of nuclear reactor physics

Debiased machine learning of conditional average treatment effects and other causal functions

Abstract: This paper provides estimation and inference methods for the best linear predictor (approximation) of a structural function, such as conditional average structural and treatment effects, and structural derivatives, based on modern machine learning (ML) tools. We represent this structural function as a conditional expectation of an unbiased signal that depends on a nuisance parameter, which we estimate by modern machine learning techniques. We first adjust the signal to make it insensitive (Neyman-orthogonal) with respect to the first-stage regularization bias. We then project the signal onto a set of basis functions, growing with sample size, which gives us the best linear predictor of the structural function. We derive a complete set of results for estimation and simultaneous inference on all parameters of the best linear predictor, conducting inference by Gaussian bootstrap. When the structural function is smooth and the basis is sufficiently rich, our estimation and inference result automatically targets this function. When basis functions are group indicators, the best linear predictor reduces to group average treatment/structural effect, and our inference automatically targets these parameters. We demonstrate our method by estimating uniform confidence bands for the average price elasticity of gasoline demand conditional on income.

Meshless upwind local radial basis function-finite difference technique to simulate the time- fractional distributed-order advection–diffusion equation

Abstract: The main objective in this paper is to propose an efficient numerical formulation for solving the time-fractional distributed-order advection–diffusion equation. First, the distributed-order term has been approximated by the Gauss quadrature rule. In the next, a finite difference approach is applied to approximate the temporal variable with convergence order $$\mathcal{O}(\tau ^{2-\alpha })$$ as $$0<\alpha <1$$ . Finally, to discrete the spacial dimension, an upwind local radial basis function-finite difference idea has been employed. In the numerical investigation, the effect of the advection coefficient has been studied in terms of accuracy and stability of the proposed difference scheme. At the end, two examples are studied to approve the impact and ability of the numerical procedure.

A Semi-Algebraic Optimization Approach to Data-Driven Control of Continuous-Time Nonlinear Systems

Abstract: This letter considers the problem of designing state feedback data-driven controllers for nonlinear continuous-time systems. Specifically, we consider a scenario where the unknown dynamics can be parametrized in terms of known basis functions and the available measurements are corrupted by unknown-but-bounded noise. The goal is to use this noisy experimental data to directly design a rational state-feedback control law guaranteed to stabilize all plants compatible with the available information. The main result of this letter shows that, by using Rantzer’s Dual Lyapunov approach, combined with elements from convex analysis, the problem can be recast as an optimization over positive polynomials, which can be relaxed to a semi-definite program through the use of Sum-of-Squares and semi-algebraic optimization arguments. Three academic examples are considered to illustrate the effectiveness of the proposed method.

Local radial basis function–finite-difference method to simulate some models in the nonlinear wave phenomena: regularized long-wave and extended Fisher–Kolmogorov equations

Abstract: In this investigation, we concentrate on solving the regularized long-wave (RLW) and extended Fisher–Kolmogorov (EFK) equations in one-, two-, and three-dimensional cases by a local meshless method called radial basis function (RBF)–finite-difference (FD) method. This method at each stencil approximates differential operators such as finite-difference method. In each stencil, it is necessary to solve a small-sized linear system with conditionally positive definite coefficient matrix. This method is relatively efficient and has low computational cost. How to choose the shape parameter is a fundamental subject in this method, since it has a palpable effect on coefficient matrix. We will employ the optimal shape parameter which results from algorithm of Sarra (Appl Math Comput 218:9853–9865, 2012). Then, we compare the approximate solutions acquired by RBF–FD method with theoretical solution and also with results obtained from other methods. The numerical results show that the RBF–FD method is suitable and robust for solving the RLW and EFK equations.

Basis Function Matrix-Based Flexible Coefficient Autoregressive Models: A Framework for Time Series and Nonlinear System Modeling

Abstract: We propose, in this paper, a framework for time series and nonlinear system modeling, called the basis function matrix-based flexible coefficient autoregressive (BFM-FCAR) model. It has very flexible nonlinear structure. We show that many famous nonlinear time series models can be derived under this framework by choosing the proper basis function matrices. Some probabilistic properties (the conditions of geometrical ergodicity) of the BFM-FCAR model are investigated. Taking advantage of the model structure, we present an efficient parameter estimation algorithm for the proposed framework by using the variable projection method. Finally, we show how new models are generated from the proposed framework.

The impact of LRBF-FD on the solutions of the nonlinear regularized long wave equation

Abstract: This paper develops a method for the numerical solution of the nonlinear regularized long wave equation This method discretizes the unknown solution in two main schemes The time discretization is accomplished by means of an implicit method based on the $$\theta $$ -weighted and finite difference methods, while the spatial discretization is described with the help of the finite difference scheme derived from the local radial basis function method The advantage of the local collocation method is based only the discretization nodes located in each sub-domain, requiring to be considered when obtaining the approximate solution at every node It also tackles the ill-conditioning problem derived from global collocation method Besides, the stability analysis of the proposed method is analyzed and the accuracy of it is examined with $$L_{\infty }$$ and $$L_2$$ norm errors At the end, the results obtained by the proposed method are compared with the methods given in previous works and it indicates an improvement in comparison with previous works

A local radial basis function-finite difference (RBF-FD) method for solving 1D and 2D coupled Schrödinger-Boussinesq (SBq) equations

Abstract: In this study, one-dimensional (1D) and two-dimensional (2D) coupled Schrodinger-Boussinesq (SBq) equations are examined numerically. A local meshless method based on radial basis function-finite difference (RBF-FD) method for spatial approximation is devised. We use polyharmonic splines as radial basis function along with augmented polynomials. By using polyharmonic splines we avoid to choose optimal shape parameter which requires special algorithms in meshless methods. For temporal discretization, low-storage ten-stage fourth-order explicit strong stability preserving Runge Kutta method is used which gives more flexibility on temporal step width. L ∞ and L 2 error norms are calculated to show accuracy of the proposed method. Further, conserved quantities are monitoried during numerical simulations to see how good the proposed method preserves them. Stability of the proposed method is dicussed numerically. Some codes are developed in Julia programming language to achieve more speed up in numerical simulations. Obtained results and their comparison with some studies such as wavelet, difference schemes and Fourier spectral methods available in literature verify the efficiency and reliability of the proposed method.

Adaptive Consensus of Non-Strict Feedback Switched Multi-Agent Systems With Input Saturations

Abstract: This paper considers the leader-following consensus for a class of nonlinear switched multi-agent systems (MASs) with non-strict feedback forms and input saturations under unknown switching mechanisms. First, in virtue of Gaussian error functions, the saturation nonlinearities are represented by asymmetric saturation models. Second, neural networks are utilized to approximate some unknown packaged functions, and the structural property of Gaussian basis functions is introduced to handle the non-strict feedback terms. Third, by using the backstepping process, a common Lyapunov function is constructed for all the subsystems of the followers. At last, we propose an adaptive consensus protocol, under which the tracking error under arbitrary switching converges to a small neighborhood of the origin. The effectiveness of the proposed protocol is illustrated by a simulation example.

The role of feature space in atomistic learning

Abstract: Eficient, physically-inspired descriptors of the structure and composition of molecules and materials play a key role in the application of machine-learning techniques to atomistic simulations. The proliferation of approaches, as well as the fact that each choice of features can lead to very different behavior depending on how they are used, e.g. by introducing non-linear kernels and non-Euclidean metrics to manipulate them, makes it difficult to objectively compare different methods, and to address fundamental questions on how one feature space is related to another. In this work we introduce a framework to compare different sets of descriptors, and different ways of transforming them by means of metrics and kernels, in terms of the structure of the feature space that they induce. We define diagnostic tools to determine whether alternative feature spaces contain equivalent amounts of information, and whether the common information is substantially distorted when going from one feature space to another. We compare, in particular, representations that are built in terms of n-body correlations of the atom density, quantitatively assessing the information loss associated with the use of low-order features. We also investigate the impact of different choices of basis functions and hyperparameters of the widely used SOAP and Behler-Parrinello features, and investigate how the use of non-linear kernels, and of a Wasserstein-type metric, change the structure of the feature space in comparison to a simpler linear feature space.

Heterogeneous Hypergraph Embedding for Graph Classification

Abstract: Recently, graph neural networks have been widely used for network embedding because of their prominent performance in pairwise relationship learning. In the real world, a more natural and common situation is the coexistence of pairwise relationships and complex non-pairwise relationships, which is, however, rarely studied. In light of this, we propose a graph neural network-based representation learning framework for heterogeneous hypergraphs, an extension of conventional graphs, which can well characterize multiple non-pairwise relations. Our framework first projects the heterogeneous hypergraph into a series of snapshots and then we take the Wavelet basis to perform localized hypergraph convolution. Since the Wavelet basis is usually much sparser than the Fourier basis, we develop an efficient polynomial approximation to the basis to replace the time-consuming Laplacian decomposition. Extensive evaluations have been conducted and the experimental results show the superiority of our method. In addition to the standard tasks of network embedding evaluation such as node classification, we also apply our method to the task of spammers detection and the superior performance of our framework shows that relationships beyond pairwise are also advantageous in the spammer detection. To make our experiment repeatable, source codes and related datasets are available at https://xiangguosun.mystrikingly.com

Tuned hybrid nonuniform subdivision surfaces with optimal convergence rates

Abstract: This paper presents an enhanced version of our previous work, hybrid non-uniform subdivision surfaces [19], to achieve optimal convergence rates in isogeometric analysis. We introduce a parameter $\lambda$ ($\frac{1}{4}<\lambda<1$) to control the rate of shrinkage of irregular regions, so the method is called tuned hybrid non-uniform subdivision (tHNUS). Our previous work corresponds to the case when $\lambda=\frac{1}{2}$. While introducing $\lambda$ in hybrid subdivision significantly complicates the theoretical proof of $G^1$ continuity around extraordinary vertices, reducing $\lambda$ can recover the optimal convergence rates when tuned hybrid subdivision functions are used as a basis in isogeometric analysis. From the geometric point of view, the tHNUS retains comparable shape quality as [19] under non-uniform parameterization. Its basis functions are refinable and the geometric mapping stays invariant during refinement. Moreover, we prove that a tuned hybrid subdivision surface is globally $G^1$-continuous. From the analysis point of view, tHNUS basis functions form a non-negative partition of unity, are globally linearly independent, and their spline spaces are nested. We numerically demonstrate that tHNUS basis functions can achieve optimal convergence rates for the Poisson's problem with non-uniform parameterization around extraordinary vertices.

A Study on Basis Functions of the Parameterized Level Set Method for Topology Optimization of Continuums

Abstract: In recent years, the parameterized level set method (PLSM), which rests on radial basis functions in most early work, has gained growing attention in structural optimization. However, little work has been carried out to investigate the effect of the basis functions in the parameterized level set method. This paper examines the basis functions of the parameterized level set method, including radial basis functions, B-spline functions, and shape functions in the finite element method (FEM) for topology optimization of continuums. The effects of different basis functions in the PLSM are examined by analyzing and comparing the required storage, convergence speed, computational efficiency, and optimization results, with the benchmark minimum compliance problems subject to a volume constraint. The linear basis functions show relatively satisfactory overall performance. Besides, several schemes to boost computational efficiency are proposed. The study on examples with unstructured 2D and 3D meshes can also be considered as a tentative investigation of prospective possible commercial applications of this method.

Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems With Unknown Dynamics

Abstract: Nonlinear dynamical systems can be made easier to control by lifting them into the space of observable functions, where their evolution is described by the linear Koopman operator. This letter describes how the Koopman operator can be used to generate approximate linear, bilinear, and nonlinear model realizations from data, and argues in favor of bilinear realizations for characterizing systems with unknown dynamics. Necessary and sufficient conditions for a dynamical system to have a valid linear or bilinear realization over a given set of observable functions are presented and used to show that every control-affine system admits an infinite-dimensional bilinear realization, but does not necessarily admit a linear one. Therefore, approximate bilinear realizations constructed from generic sets of basis functions tend to improve as the number of basis functions increases, whereas approximate linear realizations may not. To demonstrate the advantages of bilinear Koopman realizations for control, a linear, bilinear, and nonlinear Koopman model realization of a simulated robot arm is constructed from data. In a trajectory following task, the bilinear realization exceeds the prediction accuracy of the linear realization and the computational efficiency of the nonlinear realization when incorporated into a model predictive control framework.