平方 (x^2) 近似的神经网络 [英] Neural network for square (x^2) approximation

问题描述

我是 TensorFlow 和数据科学的新手.我做了一个简单的模块，应该弄清楚输入和输出数字之间的关系.在这种情况下，x 和 x 平方.Python中的代码:

I'm new to TensorFlow and Data Science. I made a simple module that should figure out the relationship between input and output numbers. In this case, x and x squared. The code in Python:

import numpy as np
import tensorflow as tf

# TensorFlow only log error messages.
tf.logging.set_verbosity(tf.logging.ERROR)

features = np.array([-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
                    9, 10], dtype = float)
labels = np.array([100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 64,
                    81, 100], dtype = float)

model = tf.keras.Sequential([
    tf.keras.layers.Dense(units = 1, input_shape = [1])
])

model.compile(loss = "mean_squared_error", optimizer = tf.keras.optimizers.Adam(0.0001))
model.fit(features, labels, epochs = 50000, verbose = False)
print(model.predict([4, 11, 20]))

我尝试了不同数量的单元，并添加更多层，甚至使用relu激活函数，但结果总是错误.它适用于 x 和 2x 等其他关系.这里有什么问题?

I tried a different number of units, and adding more layers, and even using the relu activation function, but the results were always wrong. It works with other relationships like x and 2x. What is the problem here?

推荐答案

您犯了两个非常基本的错误:

You are making two very basic mistakes:

• 您的超简单模型(具有单个单元的单层网络)根本不符合神经网络的要求，更不用说深度学习"模型了(因为您的问题已被标记)
• 同样，您的数据集(只有 20 个样本)也非常小

当然可以理解，如果神经网络要解决像 x*x 这样简单"的问题，就需要具有一定的复杂性；它们真正闪耀的地方是在接受大型训练数据集时.

It is certainly understood that neural networks need to be of some complexity if they are to solve problems even as "simple" as x*x; and where they really shine is when fed with large training datasets.

尝试解决此类函数近似的方法不仅仅是列出(少数可能的)输入，然后将其连同所需的输出一起输入模型；请记住，NN 是通过示例而不是符号推理来学习的.而且例子越多越好.在类似情况下，我们通常会生成大量示例，然后将其提供给模型进行训练.

The methodology when trying to solve such function approximations is not to just list the (few possible) inputs and then fed to the model, along with the desired outputs; remember, NNs learn through examples, and not through symbolic reasoning. And the more examples the better. What we usually do in similar cases is to generate a large number of examples, which we subsequently feed to the model for training.

话虽如此，这里是一个相当简单的 Keras 3 层神经网络演示，用于逼近函数 x*x，使用 [-50, 50]:

Having said that, here is a rather simple demonstration of a 3-layer neural network in Keras for approximating the function x*x, using as input 10,000 random numbers generated in [-50, 50]:

import numpy as np
import keras
from keras.models import Sequential
from keras.layers import Dense
from keras.optimizers import Adam
from keras import regularizers
import matplotlib.pyplot as plt

model = Sequential()
model.add(Dense(8, activation='relu', kernel_regularizer=regularizers.l2(0.001), input_shape = (1,)))
model.add(Dense(8, activation='relu', kernel_regularizer=regularizers.l2(0.001)))
model.add(Dense(1))

model.compile(optimizer=Adam(),loss='mse')

# generate 10,000 random numbers in [-50, 50], along with their squares
x = np.random.random((10000,1))*100-50
y = x**2

# fit the model, keeping 2,000 samples as validation set
hist = model.fit(x,y,validation_split=0.2,
             epochs= 15000,
             batch_size=256)

# check some predictions:
print(model.predict([4, -4, 11, 20, 8, -5]))
# result:
[[ 16.633354]
 [ 15.031291]
 [121.26833 ]
 [397.78638 ]
 [ 65.70035 ]
 [ 27.040245]]

嗯，还不错！请记住，NN 是函数逼近器:我们不应该期望它们完全重现函数关系，也不要知道"4 和-4 应该是相同的.

Well, not that bad! Remember that NNs are function approximators: we should expect them neither to exactly reproduce the functional relationship nor to "know" that the results for 4 and -4 should be identical.

让我们在 [-50,50] 中生成一些新的随机数据(请记住，出于所有实际目的，这些是模型的未见数据)并绘制它们，连同原始图片，以获得更一般的图片:

Let's generate some new random data in [-50,50] (remember, for all practical purposes, these are unseen data for the model) and plot them, along with the original ones, to get a more general picture:

plt.figure(figsize=(14,5))
plt.subplot(1,2,1)
p = np.random.random((1000,1))*100-50 # new random data in [-50, 50]
plt.plot(p,model.predict(p), '.')
plt.xlabel('x')
plt.ylabel('prediction')
plt.title('Predictions on NEW data in [-50,50]')

plt.subplot(1,2,2)
plt.xlabel('x')
plt.ylabel('y')
plt.plot(x,y,'.')
plt.title('Original data')

结果:

嗯，可以说它确实看起来像是一个很好的近似......

Well, it arguably does look like a good approximation indeed...

您还可以查看此线程以获得正弦近似值.

You could also take a look at this thread for a sine approximation.

最后要记住的是，尽管即使使用相对简单的模型，我们也确实得到了不错的近似值，但我们应该不期望的是外推，即[-50, 50] 之外的良好表现；有关详细信息，请参阅我在 深度学习不擅长拟合训练范围之外的简单非线性函数吗?

The last thing to keep in mind is that, although we did get a decent approximation even with our relatively simple model, what we should not expect is extrapolation, i.e. good performance outside [-50, 50]; for details, see my answer in Is deep learning bad at fitting simple non linear functions outside training scope?

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