使用神经网络和 ReLU 逼近正弦函数

Question

I am trying to approximate a sine function with a neural network (Keras).

Yes, I read the related posts :)

• Link 1

• Link 2

• Link 3

Using four hidden neurons with sigmoid and an output layer with linear activation works fine.

But there are also settings that provide results that seem strange to me.

Since I am just started to work with I am interested in what and why things happen, but I could not figure that out so far.

# -*- coding: utf-8 -*-

import numpy as np
np.random.seed(7)

from keras.models import Sequential
from keras.layers import Dense
import pylab as pl
from sklearn.preprocessing import MinMaxScaler

X = np.linspace(0.0 , 2.0 * np.pi, 10000).reshape(-1, 1)
Y = np.sin(X)

x_scaler = MinMaxScaler()
#y_scaler = MinMaxScaler(feature_range=(-1.0, 1.0))
y_scaler = MinMaxScaler()

X = x_scaler.fit_transform(X)
Y = y_scaler.fit_transform(Y)

model = Sequential()
model.add(Dense(4, input_dim=X.shape[1], kernel_initializer='uniform', activation='relu'))
# model.add(Dense(4, input_dim=X.shape[1], kernel_initializer='uniform', activation='sigmoid'))
# model.add(Dense(4, input_dim=X.shape[1], kernel_initializer='uniform', activation='tanh'))
model.add(Dense(1, kernel_initializer='uniform', activation='linear'))

model.compile(loss='mse', optimizer='adam', metrics=['mae'])

model.fit(X, Y, epochs=500, batch_size=32, verbose=2)

res = model.predict(X, batch_size=32)

res_rscl = y_scaler.inverse_transform(res)

Y_rscl = y_scaler.inverse_transform(Y)

pl.subplot(211)
pl.plot(res_rscl, label='ann')
pl.plot(Y_rscl, label='train')
pl.xlabel('#')
pl.ylabel('value [arb.]')
pl.legend()
pl.subplot(212)
pl.plot(Y_rscl - res_rscl, label='diff')
pl.legend()
pl.show()

This is the result for four hidden neurons (ReLU) and linear output activation.

Why does the result take the shape of the ReLU?

Does this have something to do with the output normalization?

Answer 1

Two things here:

1. Your network is really shallow and small. Having only 4 neurons with relu makes a case when a couple of this neurons are completely saturated highly possible. This is probably why your network result looks like that. Try he_normal or he_uniform as initializer to overcome that.
2. In my opinion your network is too small for this task. I would definitely increase both depth and width of your network by intdoucing more neurons and layers to your network. In case of sigmoid which has a similiar shape to a sin function this might work fine - but in case of relu you really need a bigger network.

Answer 2

Try adding more hidden layers, each with more hidden units. I used this code:

model = Sequential()
model.add(Dense(50, input_dim=X.shape[1], activation='relu'))
model.add(Dense(50, input_dim=X.shape[1], activation='relu'))
model.add(Dense(1, activation='linear'))

and got these results: