Title: Smart Sampling Algorithm for Surrogate Model Development

It helps to convert a high-fidelity simulation model into a computationally inexpensive surrogate model that captures its essential features with prescribed numerical accuracy

Surrogate modelling1

1. Polynomial Surface Response Models (PRSM)
2. Radial Basis Functions
3. Support Vector Regression
4. Artificial Neural Netwroks

Sampling method

• Non-adaptive: grid-based, pattern/geometry-based, stochastic and quasi-random
• Adaptive:

Problem Statement

描述了一个实验或者复杂得难以计算的单元、过程或者系统。为了解决这个问题，我们选择替代来代替原有的模型。总的来说，问题如下：

• 
• 寻找替代模型：
• 终止条件：最大采样数量或者替代精度

Motivation

Samplling methods

• uniform(US)
• random(RS)
• Statistical
• Central composite design
• quasi-random low-discrepancy

Problem

• How many sample points should we use
• How should we generate tem
• How do they affect the quality of surrogate approximation
• Which sampling method gives the best approximation for given form of S(x)

Key Concepts

Normalize:
这篇文章与Cozad的文章不同之处在于，Cozad的文章主要在于寻找最优的代理模型，而这篇文章的关注点：对于给定的代理模型，怎么选取点是最佳的

Crowding Distance Metric (CDM):
This is the spartial charateristic

Departure Function:
indicates the surrogate model derived from I sample points(i=1,2,…,I)
indicates the surrogate derived from all points except
This is the quality charateristic:

Optimal Points Placement

• select points which is far from the existing points
• select points which has a great influence on S(x)

Define a NLP problem:2

Smart Sampling Algorithm

对于估计模型S(x)至少需要采样个数为P：

• Set K = P+1 <
• Generate K sampling method, shows US is the best,
• compute
• Using () to generate
• If ，stop. Otherwise, proceed next
• 
• calculate
• SolveNLP，添加p=p+1，set

example

# The example
x=c(0, 0.35, 0.47, 0.55, 0.69, 1)
y=(6*x-2)^2*sin(12*x-4)

# S(x)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4
inputData<-data.frame("x"=x,"y"=y)
fit4<-lm(y~x+I(x^2)+I(x^3)+I(x^4),data=inputData)
summary(fit4)

# coefficients
coef4<-fit4$coefficients

# define a function to calculate CDM

CDMCalc<-function(x){
  CDMValue<-lapply(x,function(x_i){
    xi<-rep(x_i,length(x))
    return(sum((xi-x)^2))
  })
  return(unlist(CDMValue))
}

# Calulate CDM
CDMValue<-CDMCalc(x)
orderCDM<-order(CDMValue,decreasing = TRUE)

inputNum<-c(2,3,4,5,6)
inputData6<-inputData[inputNum,]
fit1<-lm(y~x+I(x^2)+I(x^3)+I(x^4),data=inputData6)
coef1<-fit1$coefficients
summary(fit1)

# # Symbolic Computation in R，using rsympy package
# library(Ryacas)
# t<-Sym("t")
# 
# tmatrix<-List(List(1,t,t**2,t**3,t**4))
# coef1<-List(coef1)
# dataMatrix<-tmatrix*coef1