Question

Define the following with individual proper diagram:

a. Why Exact Interpolation is important in RBF implementation?

b. How Bayesian Neural Network work is more feasible in ANN. Explain with proper example

Answer 1

ans .1
The radial basis function (RBF) network has its foundation in the conventional approximation theory.
It has the capability of universal approximation.
The RBF network is a popular alternative to the well-known multilayer perceptron
since it has a simpler structure and a much faster way of the training process.
Many aspects associated with the RBF network, such as network structure, universal approximation capability,
radial basis functions, RBF network learning, structure optimization, normalized RBF networks, application to dynamic system modeling,
and nonlinear complex-valued signal processing and many more

RBF network learning can be formulated as the minimization of the MSE function

1?=???=1‖‖??−??⃗??‖‖2=1?‖‖?−???‖‖2?,(3.1)where ?=[?1,?2,…,??], ⃗?

? is the target output for the
? th sample in the training set, and
‖⋅‖2? is the Frobenius norm defined as
‖?‖2?=tr(???).
Basically the Selection of the RBF centers is most critical to RBF network implementation.
Radial Basis Functions (RBF) interpolation is primarily used for interpolation of scattered data in higher dimensions.
If t-varying data or vector data are to be interpolated, the proposed method offers a significant speed-up as well.

Ans.2
Bayesian inference is a well-established statistical technique used to solve a wide range of inverse problems.
For the great majority of practical problems, it is not possible to formulate the posterior distribution analytically
and the most practical manner to solve the problem is by using sampling techniques.
Metropolis-Hastings algorithm that belongs to the class of Markov Chain Monte Carlo (MCMC) is very suitable to sample the posterior distribution because it is not necessary to know the normalization constant that arises from the Bayes theorem.

The main purpose of this Bayesian inference is to replace the flow simulator by proxy models
generated by an artificial neural network (ANN) to make feasible the application of the sampling algorithm in the history matching.
An iterative procedure combining MCMC sampling and ANN training is proposed.

Hope this will help,
Regards
Chegg.