Neural Networks Part 2: Learning Pi

Thousands of years ago, probably around the invention of the wheel, and before the time of Solomon, humans must have been measuring various objects… calculating some distances, and wondering,  is there a better way to measure how far around the outside of a wheel is compared to its diameter? Perhaps they tried for various sizes, small pebbles, small wheels… large wheels… to try and see a pattern that makes it not necessary to measure off a very long string every time to get a circumference?

Thankfully this is an easy function to figure out, for each measured radius and circumference around the circle, circumference is 3.14159… times the diameter, and the function here is linear.

 

Let’s make a neural network to recreate this “aha” moment and discovery of pi using machine learning. First let’s get a simple input and expected output array to run with a neural network:

from keras.models import Sequential
from keras.layers import Dense
import numpy as np
import math

model = Sequential()
model.add(Dense(1, activation='softmax',input_dim=1))
train_x = np.array(range(250))
train_y = np.array([x*math.pi for x in train_x])
print(train_y)

As you may know, machine learning runs well with a whole bunch of data. Here we give an exact value of what robot-brain would measure if it were physically reading measurements for these various round objects with diameter 0 through 249. You will see…

[   0.            3.14159265    6.28318531    9.42477796   12.56637061
   15.70796327   18.84955592   21.99114858   25.13274123   28.27433388
   31.41592654   34.55751919   37.69911184   40.8407045    43.98229715
   47.1238898    50.26548246   53.40707511   56.54866776   59.69026042
   62.83185307   65.97344573   69.11503838   72.25663103   75.39822369
   78.53981634   81.68140899   84.82300165   87.9645943    91.10618695
   94.24777961   97.38937226  100.53096491  103.67255757  106.81415022
  109.95574288  113.09733553  116.23892818  119.38052084  122.52211349
  ...

Now let’s give a very very simple robot-brain the inputs [0,1,2,3…] and try to get it to find the relation to the expected outputs…

model.compile(loss='mse',optimizer='adam', metrics=['accuracy'])
model.fit(train_x, train_y, epochs=700, batch_size=100)

model.save("pi.hdf")

for diameter in train_x:
	prediction = model.predict( np.array([diameter]))
	print( '%s:%s, computer thinks pi is %s' % ( diameter, prediction, prediction/diameter ))

This generates a model for the problem, saves it to file (this step is completely optional) and gives the prediction for each x diameter.

Run this code, and… oops, there is a problem. It is giving a “1 circumference” result for each value.

...
Epoch 693/700
250/250 [==============================] - 0s 16us/step - loss: 203603.4469 - acc: 0.0000e+00
Epoch 694/700
250/250 [==============================] - 0s 11us/step - loss: 203603.4562 - acc: 0.0000e+00
Epoch 695/700
250/250 [==============================] - 0s 9us/step - loss: 203603.4438 - acc: 0.0000e+00
Epoch 696/700
250/250 [==============================] - 0s 8us/step - loss: 203603.4469 - acc: 0.0000e+00
Epoch 697/700
250/250 [==============================] - 0s 8us/step - loss: 203603.4500 - acc: 0.0000e+00
Epoch 698/700
250/250 [==============================] - 0s 12us/step - loss: 203603.4562 - acc: 0.0000e+00
Epoch 699/700
250/250 [==============================] - 0s 10us/step - loss: 203603.4469 - acc: 0.0000e+00
Epoch 700/700
250/250 [==============================] - 0s 10us/step - loss: 203603.4469 - acc: 0.0000e+00
./nnpi.py:19: RuntimeWarning: divide by zero encountered in true_divide
  print( '%s:%s, computer thinks pi is %s' % ( diameter, prediction, prediction/diameter ))
0:[[ 1.]], computer thinks pi is [[ inf]]
1:[[ 1.]], computer thinks pi is [[ 1.]]
2:[[ 1.]], computer thinks pi is [[ 0.5]]
3:[[ 1.]], computer thinks pi is [[ 0.33333334]]
4:[[ 1.]], computer thinks pi is [[ 0.25]]
5:[[ 1.]], computer thinks pi is [[ 0.2]]
6:[[ 1.]], computer thinks pi is [[ 0.16666667]]
7:[[ 1.]], computer thinks pi is [[ 0.14285715]]
8:[[ 1.]], computer thinks pi is [[ 0.125]]
9:[[ 1.]], computer thinks pi is [[ 0.11111111]]
10:[[ 1.]], computer thinks pi is [[ 0.1]]
11:[[ 1.]], computer thinks pi is [[ 0.09090909]]
12:[[ 1.]], computer thinks pi is [[ 0.08333334]]
13:[[ 1.]], computer thinks pi is [[ 0.07692308]]
14:[[ 1.]], computer thinks pi is [[ 0.07142857]]
...

It turns out the softmax function is for multiple inputs coming in… in fact the docs say it should throw an error for just one input??

Let’s try a linear activation for this function and see what it does…


from keras.models import Sequential
from keras.layers import Dense
import numpy as np
import math

model = Sequential()
model.add(Dense(1, activation='linear',input_dim=1))
train_x = np.array(range(250))
train_y = np.array([x*math.pi for x in train_x])
print(train_y)

model.compile(loss='mse',optimizer='adam', metrics=['accuracy'])
model.fit(train_x, train_y, epochs=700, batch_size=100)

model.save("pi.hdf")

for diameter in train_x:
prediction = model.predict( np.array([diameter]))
print( '%s:%s, computer thinks pi is %s' % ( diameter, prediction, prediction/diameter ))

And the output is:

...
250/250 [==============================] - 0s 10us/step - loss: 7385.4433 - acc: 0.0000e+00
Epoch 688/700
250/250 [==============================] - 0s 12us/step - loss: 7346.4989 - acc: 0.0000e+00
Epoch 689/700
250/250 [==============================] - 0s 10us/step - loss: 7307.1777 - acc: 0.0000e+00
Epoch 690/700
250/250 [==============================] - 0s 11us/step - loss: 7270.0424 - acc: 0.0000e+00
Epoch 691/700
250/250 [==============================] - 0s 8us/step - loss: 7230.2705 - acc: 0.0000e+00
Epoch 692/700
250/250 [==============================] - 0s 8us/step - loss: 7191.8240 - acc: 0.0000e+00
Epoch 693/700
250/250 [==============================] - 0s 7us/step - loss: 7152.7945 - acc: 0.0000e+00
Epoch 694/700
250/250 [==============================] - 0s 7us/step - loss: 7114.3938 - acc: 0.0000e+00
Epoch 695/700
250/250 [==============================] - 0s 8us/step - loss: 7076.5807 - acc: 0.0000e+00
Epoch 696/700
250/250 [==============================] - 0s 8us/step - loss: 7036.8849 - acc: 0.0000e+00
Epoch 697/700
250/250 [==============================] - 0s 7us/step - loss: 6998.8652 - acc: 0.0000e+00
Epoch 698/700
250/250 [==============================] - 0s 7us/step - loss: 6961.6289 - acc: 0.0000e+00
Epoch 699/700
250/250 [==============================] - 0s 18us/step - loss: 6922.9076 - acc: 0.0000e+00
Epoch 700/700
250/250 [==============================] - 0s 15us/step - loss: 6886.2189 - acc: 0.0000e+00
./nnpi.py:19: RuntimeWarning: divide by zero encountered in true_divide
print( '%s:%s, computer thinks pi is %s' % ( diameter, prediction, prediction/diameter ))
0:[[ 1.60629654]], computer thinks pi is [[ inf]]
1:[[ 4.16281796]], computer thinks pi is [[ 4.16281796]]
2:[[ 6.71933937]], computer thinks pi is [[ 3.35966969]]
3:[[ 9.27586079]], computer thinks pi is [[ 3.09195352]]
4:[[ 11.8323822]], computer thinks pi is [[ 2.95809555]]
5:[[ 14.38890362]], computer thinks pi is [[ 2.87778068]]
6:[[ 16.94542503]], computer thinks pi is [[ 2.82423759]]
7:[[ 19.5019455]], computer thinks pi is [[ 2.78599215]]
8:[[ 22.05846786]], computer thinks pi is [[ 2.75730848]]
9:[[ 24.61499023]], computer thinks pi is [[ 2.73499894]]
10:[[ 27.1715107]], computer thinks pi is [[ 2.71715117]]
11:[[ 29.72803116]], computer thinks pi is [[ 2.70254827]]
12:[[ 32.28455353]], computer thinks pi is [[ 2.69037938]]
13:[[ 34.8410759]], computer thinks pi is [[ 2.6800828]]
14:[[ 37.39759445]], computer thinks pi is [[ 2.67125678]]
15:[[ 39.95411682]], computer thinks pi is [[ 2.66360784]]
16:[[ 42.51063919]], computer thinks pi is [[ 2.65691495]]
17:[[ 45.06716156]], computer thinks pi is [[ 2.65100956]]
18:[[ 47.62368393]], computer thinks pi is [[ 2.6457603]]
19:[[ 50.18020248]], computer thinks pi is [[ 2.64106321]]
20:[[ 52.73672485]], computer thinks pi is [[ 2.63683629]]
21:[[ 55.29324722]], computer thinks pi is [[ 2.63301182]]
22:[[ 57.84976578]], computer thinks pi is [[ 2.62953472]]
23:[[ 60.40628815]], computer thinks pi is [[ 2.62636042]]
24:[[ 62.96281052]], computer thinks pi is [[ 2.62345052]]
...

Now this is getting somewhere! Since the loss is getting smaller and smaller, give it more epochs training time – let’s try “epochs=7000″…
With this change you should get….

...
Epoch 6989/7000
250/250 [==============================] - 0s 11us/step - loss: 8.6996e-10 - acc: 0.0040
Epoch 6990/7000
250/250 [==============================] - 0s 9us/step - loss: 1.1100e-09 - acc: 0.0040
Epoch 6991/7000
250/250 [==============================] - 0s 10us/step - loss: 2.0296e-09 - acc: 0.0040
Epoch 6992/7000
250/250 [==============================] - 0s 10us/step - loss: 1.3331e-09 - acc: 0.0040
Epoch 6993/7000
250/250 [==============================] - 0s 9us/step - loss: 1.9024e-09 - acc: 0.0040
Epoch 6994/7000
250/250 [==============================] - 0s 8us/step - loss: 2.9781e-09 - acc: 0.0040
Epoch 6995/7000
250/250 [==============================] - 0s 8us/step - loss: 1.8108e-09 - acc: 0.0040
Epoch 6996/7000
250/250 [==============================] - 0s 8us/step - loss: 8.5055e-10 - acc: 0.0040
Epoch 6997/7000
250/250 [==============================] - 0s 8us/step - loss: 8.8337e-10 - acc: 0.0040
Epoch 6998/7000
250/250 [==============================] - 0s 8us/step - loss: 6.0212e-10 - acc: 0.0040
Epoch 6999/7000
250/250 [==============================] - 0s 8us/step - loss: 5.9753e-10 - acc: 0.0040
Epoch 7000/7000
250/250 [==============================] - 0s 8us/step - loss: 8.7943e-10 - acc: 0.0040
./nnpi.py:19: RuntimeWarning: divide by zero encountered in true_divide
  print( '%s:%s, computer thinks pi is %s' % ( diameter, prediction, prediction/diameter ))
0:[[  1.26110535e-05]], computer thinks pi is [[ inf]]
1:[[ 3.14160514]], computer thinks pi is [[ 3.14160514]]
2:[[ 6.2831974]], computer thinks pi is [[ 3.1415987]]
3:[[ 9.42479038]], computer thinks pi is [[ 3.14159679]]
4:[[ 12.56638241]], computer thinks pi is [[ 3.1415956]]
5:[[ 15.70797443]], computer thinks pi is [[ 3.14159489]]
6:[[ 18.84956932]], computer thinks pi is [[ 3.14159489]]
7:[[ 21.99116135]], computer thinks pi is [[ 3.14159441]]
8:[[ 25.13275337]], computer thinks pi is [[ 3.14159417]]
9:[[ 28.2743454]], computer thinks pi is [[ 3.14159393]]
10:[[ 31.41593742]], computer thinks pi is [[ 3.14159369]]
11:[[ 34.55752945]], computer thinks pi is [[ 3.14159369]]
12:[[ 37.69912338]], computer thinks pi is [[ 3.14159369]]
13:[[ 40.8407135]], computer thinks pi is [[ 3.14159346]]
14:[[ 43.98230743]], computer thinks pi is [[ 3.14159346]]
...

Congratulations! this concludes the most ridiculous (ridiculously simple) neural network! If you read the generated hdf file in hdfview, you can see the bias added should be super close to zero, and the matrix just has one multiplication, by approximately pi:

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