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Artificial Neural Networks
Published in Paresh Chra Deka, A Primer on Machine Learning Applications in Civil Engineering, 2019
Radial basis function (RBF) networks correspond to a particular class of function approximators which can be trained by using a set of samples. First proposed by Broomhead and Lowe (1988) and Moody and Darken (1988), an RBF consists of three layers: input, hidden, and output. Compared to the multilayer perceptron (MLP), an RBF has only one hidden layer. The hidden units provide a set of functions that constitute an arbitrary basis for the input parameters. These units are also known as radial centers and are represented by vectors c1, c2, …ch. The transformation from an input space to a hidden unit space is nonlinear while transformation from hidden units to output space is linear. The hidden layer produces a significant non-zero response only when the input falls within a small localized region of the input space. The output of the network is a linear combination of RBF of inputs and neuron parameters. They have many applications mainly in the field of function approximation, time series prediction, classification, and system control. An RBF performs classification by measuring the input’s similarity to examples from the training set. Each RBF neuron stores a prototype which is one of the examples from the training set. To classify a new input, each neuron computes Euclidean distance between the input and its prototype. If the input is closer to A class prototype than B class prototype, then it classified as A. Figure 2.4 shows the structure of the radial basis function neural network (RBFNN).
Artificial Neural Networks
Published in Praveen Kumar, Jay Alameda, Peter Bajcsy, Mike Folk, Momcilo Markus, Hydroinformatics: Data Integrative Approaches in Computation, Analysis, and Modeling, 2005
Typical selections for RBF are thin-plate-spline Φ(r) = r2log(r), multiquadratic Φ(r) = (r2 + β2)1/2, inverse multiquadratic Φ(r) = [(r2 + β2)1/2]-1, and Gaussian Φ(r) = exp(-r2/β2). Symbol r represents the Euclidean distance between a center μ_ and the data points x_, and β is a real variable. The MRAN activation function (RBF) is Gaussian and is given by: ϕk(x_)=exp(−‖x_−μ_k‖2σk2),
3D geological modelling for the design of complex underground works
Published in Daniele Peila, Giulia Viggiani, Tarcisio Celestino, Tunnels and Underground Cities: Engineering and Innovation meet Archaeology, Architecture and Art, 2020
F. Giovacchini, M. Vendramini, L. Soldo, M. Merlo, D. Marchisio, G. Ricci, A. Eusebio
The used interpolant is based on FastRBF™. Radial Basis Functions describe the predicted value at a point X as a function that depends on the distance from the point where the value is known. RBFs are a group of global interpolation methods where the interpolant is dependent on all data points. Fast RBF interpolant uses advanced algorithms to solve the combination of Radial Basis Functions and evaluate the estimates quickly. The interpolation of surfaces depends on the distance from the points where values are known.
Using a hybrid artificial intelligence method for estimating the compressive strength of recycled aggregate self-compacting concrete
Published in European Journal of Environmental and Civil Engineering, 2022
Gholamreza Pazouki, Arash Pourghorban
As shown in Figure 2 the RBFNN is a simple neural network that is configured by three main layers and each layer consists of a number(s) of the neuron. The first layer is the inputs layer. This simple layer achieves data from the environment and transmits this data to the next layer (hidden layer) via non-linear mapping. The second layer is the hidden layer, the calculation is performed in this layer, each node of this layer includes its unique radial basis function (the radial basis is a function, whose value is related to the distance of inputs from a central point). There are various types of radial basis function such as Gaussian, Inverse multi-quadratic, Hardly multi-quadratic, and Sigmoid functions. One of the most well-known and prevalent types of RBF is Gaussian (Golafshani & Pazouki, 2018). The last layer is the output layer which presents the model results. Moreover, in this model the information transmission from hidden layer to the output layer is via linear mapping.
Numerical solution of steady MHD duct flow in a square annulus duct under strong transverse magnetic field
Published in International Journal of Ambient Energy, 2020
M. Prasanna Jeyanthi, S. Ganesh
One of the important mesh-free numerical methods in solving partial differential equations is radial basis function method. Compared to traditional methods, RBF can give accurate results and has flexibility in the computational domain. This method can be extended to the multidimensional problem. The classical FDM can be generalised as RBF-FD method. As in FDM, the partial derivative of the kth order at the arbitrary node xj can be written as the linear combination of its (n) neighbourhood nodes. For example, in 1D problem at xj, derivative of the rth order can be approximated as where xi is the nodes on the neighbourhood. Some of the radial basis functions are Multiquadric, Inverse Multiquadric, inverse Quadrics and Gaussians. Among all radial basis functions, Hardy’s Multiquadric gives accurate solutions. In this article, Multiquadric Radial Basis Functions is used to solve the given governing equations.
Metamodel-based optimization of stochastic computer models for engineering design under uncertain objective function
Published in IISE Transactions, 2019
Guilin Li, Matthias Hwai-yong Tan, Szu Hui Ng
Frequently, computer codes that simulate an engineering system are very time-consuming to run. Consequently, a practically appealing approach is to approximate the computer model by a more computationally efficient metamodel. A metamodel-based optimization strategy requires one to first identify a metamodel form, then design an experiment to collect data by running the expensive computer code, and finally fit and optimize the metamodel (Barton and Meckesheimer, 2006). Metamodels can be built using many types of regression models with a variety of prediction power. For example, linear regression models are easy to built and have been widely used. Although simple, this type of model lacks the ability to model complicated surfaces. By using more sophisticated methods such as Gaussian process models or Radial Basis Function (RBF) models, one can achieve better prediction. RBF models using compactly supported RBFs can be fitted efficiently to very large data sets. RBF models are also applicable to problems with high-dimensional design variable spaces, as generally few restrictions are imposed on the location of sample points. In this section, we focus on using the RBF metamodel to predict and optimize the stochastic computer model under a fixed objective function.