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I've been asked to solve a problem for a project. I'm working on Python or R. I need to approximate a function with multiplayer perceptron (neural network).

The function is: $y= 2\text{cos}(x)+4$ on the interval $[0,10]$.

After reading a lot about perceptron and neural network for the approximation of functions, I found a code that helped me a lot and my program is based on this code. Here is the code (with comments) I use to approximate my function followed by questions and examples :

# Create vector of random data between 0 and 10, lenght 50
x <- sort(10*runif(50))

# We generate data from our fonction (that we need to approximate)
y <- 2*cos(x)+4 

# We generate vector of 100 valors from 0 to 10 that we will use to draw the basic function and the one approximate with neural network (multilayer perceptron)
x1 <- seq(0, 10, by=0.1)

# Basic function (to draw the curve of the basic function that we need to approximate)
f <- 2*cos(x1)+4

# Charging library nnet (to apply neural network to our model)
library(nnet)

# We apply neural network to our generated data (random points of the function we want to approximate)
# We use one hidden layer, 6 neurons and 40 iteration
nn <- nnet(x, y, size=6, maxit=40, linout=TRUE)

# We display the 50 random generated points of our function
plot(x, y)

# Original function curve (2*cos+4)
lines(x1, f)

# Approximate function curve, generated by prediction on the results of the neural network
lines(x1, predict(nn, data.frame(x=x1)), col="green") 

Few questions

1) In the example I found, we generate 50 points (vector x), but why 50 ? If I use 20 or 1000 results can vary.

2) How to fix the number of neurons to use ? I just try with a lot of different numbers but I don’t have rule.

3) How to fix the number of iteration the neural network use (maxit=40 here) ? Should I use other parameters of nnet ? I'm a bit lost

4) I have 1 hidden layer as I see that most of approximate function case use only one but I dont really know why.

5) When I create vector « x1 » I dont know how to fix the number of element I want (here 100 but what if I choose 20 or 1000)

6) When I run my code, results are changing, sometimes the approximation is quite precise and sometimes It’s just not good from a certain points of the curve and I dont really know why.

I'm also interested by this method on python but it seems more complicated to implement

Example 1 : a good approximation (not bad)

green line : approximated function with neural network (multilayer perceptron)

dark line: our function (2*cos(x)+4)

dots are generated points from the function (y) that we use to build the model 1st approximation (not bad)

Example 2 : a bad approximation (with the same code) enter image description here

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