Approximate Function With Neural Network
One of the most straightforward tasks in machine learning is approximating a given mathematical function. Mainly because it’s very easy to know and measure what’s right and what’s wrong: for any given point we can simply calculate the correct answer. Thus the dataset is very easy to obtain.
In Statistics and Machine Learning, this task is called “regression” (as opposed to “classification”) because we’re trying to estimate an underlying distribution (or a particular point of this distribution) instead a classifying a given point.
In this post I’m going to describe how to set up a Multilayer Perceptron for regression on any arbitrary mathematical function.
Function y
We’ll start with this combined sinusoid:
y(x) = sin(2π * x) + sin(5π * x)
with x = 1:0.002:1
First, we implement the above function in Python:


Which we then use to generate our dataset:


Depending on which activation function we use for our neurons, we also need to normalize the data between 1 and 1 (e.g. for sigmoid).
We store the y_max
value so we can restore the original values later (e.g. for plotting).


Then we split our dataset into a training and a testing set. Typical splits range anywhere between ^{70}⁄_{30} and ^{90}⁄_{10}, so we’re using a ^{80}⁄_{20} split here:


We could also manually do this, but Scikitlearn already provides a nice function that splits and shuffles the dataset.
Let’s have a look at what our data currently looks like:


The blue dots are the training data for our network, the yellow crosses are the test data  the network should predict these later!
Since we have 1D input data for our network, we need to reshape it:


After all this setup, we can move on to the heart of our application.
We use the MLPRegressor
function from Scikitlearn to set up our MLP.
I cannot explain all its parameters here, so go have a look at its documentation.
In the example below we are using just a single hidden layer with 30 neurons.


Great, we have our predictions! But what to do with them? First we can visualize them, of course:


In the above example we can see that the network has barely detected the first (main) sinus function. The network simply does not have enough inherent complexity to model the already complex sinus function  let alone two of them together!
But we can also look at the predictions in a more abstract way, by calculating the mean squared error (MSE):


Essentially this will just go over all predictions, calculate their distance from the correct answer, square it and sum them all up.
Since the prediction performance is really terrible and our network is really tiny, we can try adding a few more layers and neurons:


That looks much better already! Go ahead and play around with MLPRegressor’s parameters and see how well you can fit the function.
Function z
The same can be done for a twodimensional function like the following:
z(x,y) = exp( (x² + y²) / 0.1)
We begin again by implementing our function in Python:


So far so good. To create the dataset this time, we need a list of (x,y) coordinates and calculate the corresponding z value:


Depending on the function you might want to normalize the output values at this point. For the above function this is not required, since the value of the exponential function for negative inputs is between 0 and 1 anyway.
Again, split the dataset into training and testing data:


And depict it as a 3D plot:


Here we need to do some data structure crafting to get the points in the right shape for plotting. First we split the list of (x,y) points into one array containing the indices of the x1 axis (called x1_vals
) and another array containing the indices of the x2 axis (called x2_vals
).
Then we can use the indices along with the function output values to create a 3D scatter plot.
Due to the equidistant spacing of the data points this nicely looks like a surface.
The same is done for the test data.
The training and prediction part is the same as before:


And plot the predictions again:


Even with only a single hidden layer containing 20 neurons the network performs quite well on this task (compared to the previous one). This because the underlying shape is much simpler (symmetric exponential function), therefore a less complex network is required to adequately fit it.
Go ahead and see how small you can make the network before loosing too much fidelity!
References
 Full Source Code func_y.py, func_z.py
 Numpy
 Scikitlearn
 Matplotlib pyplot
 Python 3