Keras model to fit polynomial

Question

I generated some data from a 4th degree polynomial and wanted to create a regression model in Keras to fit this polynomial. The problem is that predictions after fitting seem to be basically linear. Since this is my first time working with neural nets I assume I made a very trivial and stupid mistake.

Here is my code:

model = Sequential()
model.add(Dense(units=200, input_dim=1))
model.add(Activation('relu'))
model.add(Dense(units=45))
model.add(Activation('relu'))
model.add(Dense(units=1))

model.compile(loss='mean_squared_error',
              optimizer='sgd')

model.fit(x_train, y_train, epochs=20, batch_size=50)

loss_and_metrics = model.evaluate(x_test, y_test, batch_size=100)

classes = model.predict(x_test, batch_size=1)

x_train and y_train are numpy arrays containing the first 9900 entries from this file.

I tried different batch_sizes, number of epochs, layer sizes and amounts of training data. Nothing seems to help.

Please point out everything you see that does not make sense!

Answer 1

Neural networks generally won't do a good job in extrapolating polynomial functions. However, if your training and testing data are from the same range, you could achieve quite nice results. I generated some data and used your code:

import numpy as np
x_train=np.random.rand(9000)
y_train=x_train**4+x_train**3-x_train
x_train=x_train.reshape(len(x_train),1)

x_test=np.linspace(0,1,100)
y_test=x_test**4+x_test**3-x_test
x_test=x_test.reshape(len(x_test),1)


model = Sequential()
model.add(Dense(units=200, input_dim=1))
model.add(Activation('relu'))
model.add(Dense(units=45))
model.add(Activation('relu'))
model.add(Dense(units=1))

model.compile(loss='mean_squared_error',
              optimizer='sgd')

model.fit(x_train, y_train, epochs=40, batch_size=50, verbose=1)

loss_and_metrics = model.evaluate(x_test, y_test, batch_size=100)

classes = model.predict(x_test, batch_size=1)

test=x_test.reshape(-1)
plt.plot(test,classes,c='r')
plt.plot(test,y_test,c='b')
plt.show()

Note that I increased epochs to 40 to get more iterations and more accurate results. I also set verbose=1 to be able to see how the loss behaves. The loss is indeed decreasing down to 7.4564e-04, and below is the result that I got. The red line is the prediction of the network, and the blue line is the correct value. You can see that they are quite close to each other.