简单网络 Sequential model

import numpy as np
import tensorflow as tf
import matplotlib.pyplot as plt

# X-Data
N = 200
X = np.random.random(N)
print(X)

[0.87043833 0.34352561 0.81362902 0.43800459 0.69327997 0.6323433
 0.45008738 0.11416858 0.97250895 0.79673929 0.11541241 0.86617814
 0.27171221 0.9816876  0.16753845 0.98913371 0.8581325  0.0757954
 0.86031813 0.59289839 0.29749813 0.20563715 0.9997178  0.59003228
 0.35757194 0.80381234 0.242631   0.11467421 0.34143111 0.27654834
 0.20942507 0.23010397 0.33766935 0.95545649 0.65183404 0.99802424
 0.92849039 0.57854204 0.83632395 0.7312833  0.53311664 0.92955895
 0.4696443  0.30995492 0.15573472 0.30820987 0.74374888 0.20746479
 0.63707138 0.25891428 0.89856103 0.77827489 0.62232385 0.66847401
 0.16673846 0.85111978 0.54047361 0.06522549 0.85827134 0.89141919
 0.58324072 0.25189043 0.54356471 0.98293161 0.86076638 0.2782616
 0.40062945 0.73617715 0.27848053 0.19417744 0.26946711 0.56730615
 0.21616535 0.50243466 0.28316786 0.31296928 0.89592931 0.92521469
 0.00288017 0.39206302 0.93884799 0.4215962  0.22473894 0.54789537
 0.69687102 0.33094136 0.22288206 0.73659617 0.38045493 0.9333922
 0.38773277 0.55031938 0.01651959 0.26053489 0.90197014 0.94971847
 0.23267036 0.46570646 0.43948164 0.41231935 0.95609519 0.08347694
 0.60496824 0.10225825 0.01716315 0.78068431 0.87508406 0.58938599
 0.28905644 0.61419796 0.62117595 0.10484637 0.38090871 0.38990318
 0.53892778 0.92143317 0.26954451 0.24797848 0.18022549 0.03566011
 0.63383763 0.23730579 0.51140095 0.40433532 0.83947813 0.01262781
 0.98735757 0.23460468 0.27307607 0.48511314 0.6239909  0.59096001
 0.63845825 0.77437862 0.36975273 0.37521626 0.25038518 0.55399587
 0.69018968 0.11738883 0.11489286 0.56623511 0.13860632 0.03696333
 0.70701076 0.31451895 0.05047509 0.4000243  0.18951662 0.54795957
 0.32405597 0.58581542 0.19052749 0.97184877 0.08121297 0.5277435
 0.95102183 0.70232852 0.92241819 0.5559153  0.78083361 0.40355639
 0.98407045 0.72578303 0.87454461 0.05761464 0.30351022 0.7906215
 0.83869084 0.09198238 0.17471775 0.40895357 0.82516278 0.79565455
 0.01031771 0.80233015 0.62315751 0.8618836  0.92953252 0.25230357
 0.70523593 0.18452694 0.8373015  0.13110077 0.86873592 0.59722102
 0.26999434 0.3517313  0.18223504 0.09235062 0.58780589 0.42629025
 0.31286228 0.64693046 0.47166465 0.49182858 0.45446215 0.91291826
 0.06898951 0.53389829]

# Generation Y-Data
sign = (- np.ones((N,)))**np.random.randint(2,size=N)
Y = np.sqrt(X) * sign
print(sign)
print(Y)

[-1.  1. -1. -1.  1.  1.  1.  1. -1.  1. -1. -1. -1. -1. -1.  1. -1.  1.
 -1.  1.  1.  1. -1.  1.  1.  1. -1. -1. -1.  1. -1.  1. -1. -1. -1.  1.
  1.  1.  1.  1. -1. -1.  1. -1.  1. -1. -1. -1. -1.  1. -1. -1.  1. -1.
  1.  1.  1. -1. -1. -1. -1.  1. -1.  1. -1. -1. -1.  1.  1.  1. -1. -1.
  1. -1. -1. -1.  1.  1.  1.  1.  1. -1.  1.  1.  1.  1. -1.  1. -1.  1.
  1.  1. -1. -1.  1.  1. -1.  1.  1. -1. -1. -1. -1.  1.  1. -1. -1.  1.
  1. -1.  1.  1. -1.  1.  1.  1. -1. -1. -1.  1.  1.  1. -1.  1. -1.  1.
 -1. -1. -1. -1. -1.  1.  1. -1. -1.  1. -1.  1. -1.  1. -1.  1. -1.  1.
 -1.  1.  1. -1. -1.  1.  1.  1. -1. -1. -1.  1. -1.  1. -1.  1. -1.  1.
  1.  1.  1. -1. -1.  1.  1. -1. -1.  1.  1.  1.  1. -1. -1.  1.  1. -1.
 -1.  1.  1. -1.  1.  1.  1.  1. -1.  1. -1. -1.  1.  1. -1. -1. -1. -1.
  1.  1.]
[-0.93297284  0.58611058 -0.90201387 -0.66181915  0.83263436  0.79520016
  0.67088552  0.33788841 -0.98615868  0.89260254 -0.33972403 -0.93068692
 -0.52126021 -0.99080149 -0.4093146   0.99455202 -0.92635441  0.27530964
 -0.92753336  0.76999895  0.54543389  0.45347233 -0.99985889  0.76813558
  0.59797319  0.89655582 -0.49257589 -0.33863581 -0.58432107  0.52587864
 -0.45762984  0.47969154 -0.58109324 -0.97747455 -0.8073624   0.99901163
  0.96358206  0.76061951  0.91450749  0.85515104 -0.73014837 -0.96413638
  0.68530599 -0.55673595  0.39463239 -0.55516653 -0.86240876 -0.45548303
 -0.79816751  0.5088362  -0.94792459 -0.88219889  0.78887505 -0.8176026
  0.40833621  0.92256153  0.7351691  -0.25539282 -0.92642935 -0.94414998
 -0.76370198  0.50188687 -0.73726841  0.99142908 -0.92777496 -0.52750507
 -0.63295296  0.85800766  0.52771255  0.4406557  -0.51910222 -0.75319729
  0.46493585 -0.70882626 -0.53213519 -0.55943657  0.94653542  0.96188081
  0.05366721  0.62614936  0.96894169 -0.6493044   0.47406639  0.74019955
  0.83478801  0.57527503 -0.47210387  0.85825181 -0.61681029  0.96612225
  0.62268192  0.74183515 -0.12852857 -0.51042619  0.94972109  0.974535
 -0.48235917  0.68242689  0.66293411 -0.64212097 -0.9778012  -0.28892376
 -0.77779704  0.31977844  0.13100821 -0.88356342 -0.93545928  0.76771478
  0.53763969 -0.78370783  0.78814716  0.32379989 -0.61717802  0.62442227
  0.73411701  0.95991311 -0.51917676 -0.49797438 -0.42452973  0.18883884
  0.7961392   0.48714042 -0.71512303  0.63587367 -0.91623039  0.11237352
 -0.99365868 -0.48436007 -0.52256681 -0.69650064 -0.78993095  0.76873923
  0.79903583 -0.87998785 -0.60807297  0.61254899 -0.50038504  0.74430899
 -0.83077655  0.34262053 -0.3389585   0.75248595 -0.3722987   0.1922585
 -0.84083932  0.56081989  0.22466663 -0.63247474 -0.43533507  0.74024291
  0.56925914  0.76538579 -0.43649455 -0.98582391 -0.28497889  0.72645956
 -0.97520348  0.83805043 -0.96042605  0.74559728 -0.8836479   0.63526089
  0.99200325  0.851929    0.9351709  -0.24003049 -0.55091762  0.889169
  0.91580065 -0.30328597 -0.41799252  0.63949478  0.90838471  0.8919947
  0.10157615 -0.89572883 -0.78940326  0.92837686  0.96412267 -0.50229829
 -0.83978326  0.42956599  0.9150418  -0.3620784   0.93206004  0.77280076
  0.5196098   0.59306939 -0.42688997  0.30389244 -0.766685   -0.65290906
  0.55934093  0.80431987 -0.68677846 -0.70130491 -0.67413808 -0.95546756
  0.26265854  0.73068344]

# Neural network
act = tf.keras.layers.ReLU()
nn_sv = tf.keras.models.Sequential([
  tf.keras.layers.Dense(10, activation=act, input_shape=(1,)),
  tf.keras.layers.Dense(10, activation=act),
  tf.keras.layers.Dense(1,activation='linear')])

# Loss function
loss_sv = tf.keras.losses.MeanSquaredError()
optimizer_sv = tf.keras.optimizers.Adam(learning_rate=0.001)
nn_sv.compile(optimizer=optimizer_sv, loss=loss_sv)

# Training
results_sv = nn_sv.fit(X, Y, epochs=5, batch_size= 5, verbose=1)

Epoch 1/5
40/40 [==============================] - 0s 773us/step - loss: 0.5016
Epoch 2/5
40/40 [==============================] - 0s 2ms/step - loss: 0.4989
Epoch 3/5
40/40 [==============================] - 0s 2ms/step - loss: 0.4988
Epoch 4/5
40/40 [==============================] - 0s 2ms/step - loss: 0.4983
Epoch 5/5
40/40 [==============================] - 0s 2ms/step - loss: 0.4987

# Results
plt.plot(X,Y,'.',label='Data points', color="lightgray")
plt.plot(X,nn_sv.predict(X),'.',label='Supervised', color="red")
plt.xlabel('y')
plt.ylabel('x')
plt.title('Standard approach')
plt.legend()
plt.show()

# 改进后的方法
# X-Data
# X = X , we can directly re-use the X from above, nothing has changed...
# Y is evaluated on the fly

# Model
nn_dp = tf.keras.models.Sequential([
  tf.keras.layers.Dense(10, activation=act, input_shape=(1,)),
  tf.keras.layers.Dense(10, activation=act),
  tf.keras.layers.Dense(1, activation='linear')])

#Loss
mse = tf.keras.losses.MeanSquaredError()
def loss_dp(y_true, y_pred):
    return mse(y_true,y_pred**2)

optimizer_dp = tf.keras.optimizers.Adam(learning_rate=0.001)
nn_dp.compile(optimizer=optimizer_dp, loss=loss_dp)

#Training
results_dp = nn_dp.fit(X, X, epochs=5, batch_size=5, verbose=1)

Epoch 1/5
40/40 [==============================] - 0s 4ms/step - loss: 0.2881
Epoch 2/5
40/40 [==============================] - 0s 4ms/step - loss: 0.0988
Epoch 3/5
40/40 [==============================] - 0s 4ms/step - loss: 0.0027
Epoch 4/5
40/40 [==============================] - 0s 3ms/step - loss: 0.0012
Epoch 5/5
40/40 [==============================] - 0s 2ms/step - loss: 8.4034e-04

# Results
plt.plot(X,Y,'.',label='Datapoints', color="lightgray")
#plt.plot(X,nn_sv.predict(X),'.',label='Supervised', color="red") # optional for comparison
plt.plot(X,nn_dp.predict(X),'.',label='Diff. Phys.', color="green") 
plt.xlabel('x')
plt.ylabel('y')
plt.title('Differentiable physics approach')
plt.legend()
plt.show()