In a neural network with tensor input X, it seems there are times when it will never learn... Why?
In short


You are waaaaay undertraining. Increase the number of times you show the network your data. I am guessing training may take longer than you expect because typically networks train best with 0-mean data, which yours is not.
ReLU seems to cause problems with such a shallow network. Try increasing depth or using elu activation instead.
I confirmed having an activation in the last layer doesn't cause huge problems, but it is still a good idea of getting in the habit of knowing when you should and should not have an activation in the last layer.


Undertraining

I handled this by increasing the number of training examples by 10,000 times (you could increase the number of epochs instead but this results in better printing):

x_train=np.array([[1,2], [3,4], [5,6]] * 10000)
y_train=np.array([f(x) for x in x_train])[:, np.newaxis]


Problems with ReLU

The problems with ReLU can be handled in one of two ways, increase the number of layers when using ReLU, or use a different activation such as elu. Both trained just fine for me:

model = km.Sequential()
model.add(kl.Dense(units=3, activation='relu', input_shape=(in_dim,)))
model.add(kl.Dense(units=3, activation='relu'))
model.add(kl.Dense(units=out_dim))
model.compile(
  loss=kloss.mean_squared_error, optimizer=ko.Adam(lr=0.1)
)


or

model = km.Sequential()
model.add(kl.Dense(units=3, activation='elu', input_shape=(in_dim,)))
model.add(kl.Dense(units=out_dim))
model.compile(
  loss=kloss.mean_squared_error, optimizer=ko.Adam(lr=0.1)
)


Full working code

Below shows the code with elu, you can swap the block out for the ReLU version (shown above) instead and it prints very similar values.

import numpy as np
import keras.models as km
import keras.layers as kl
import keras.optimizers as ko
import keras.losses as kloss

# This will cause no learning
np.random.seed(1692585618)

def f(x):
    a = x[0]* 3.141 + x[1]
    return a;

# Create a sample dataset 
# Input is (*, 2)
# Output is (*, )
x_train=np.array([[1,2], [3,4], [5,6]] * 10000)
y_train=np.array([f(x) for x in x_train])[:, np.newaxis]

# These are required by the shape of x_train and y_train
in_dim = x_train.shape[1]
out_dim = 1

model = km.Sequential()
model.add(kl.Dense(units=3, activation='elu', input_shape=(in_dim,)))
model.add(kl.Dense(units=out_dim))
model.compile(
  loss=kloss.mean_squared_error, optimizer=ko.Adam(lr=0.1)
)

model.fit(x_train, y_train, epochs=3, verbose=True)
print('predicted: {}'.format(model.predict(x_train)[:3, 0]))
print('actual   : {}'.format(y_train[:3, 0]))


prints

Epoch 1/3
30000/30000 [==============================] - 1s 29us/step - loss: 3.0553
Epoch 2/3
30000/30000 [==============================] - 1s 20us/step - loss: 5.0199e-06
Epoch 3/3
30000/30000 [==============================] - 1s 20us/step - loss: 5.4414e-06
predicted: [ 5.1426897 13.420667  21.707573 ]
actual   : [ 5.141 13.423 21.705]