To solve the XOR problem using a Deep Neural Network (DNN) in Python, you can use libraries like TensorFlow or PyTorch. I'll provide an example using TensorFlow:

```python
import tensorflow as tf
import numpy as np

# Define the XOR dataset
x_data = np.array([[0, 0], [0, 1], [1, 0], [1, 1],], dtype=np.float32)
y_data = np.array([[0], [1], [1], [0],], dtype=np.float32)

# Define the DNN model
model = tf.keras.models.Sequential([
    tf.keras.layers.Input(shape=(2,)),
    tf.keras.layers.Dense(8, activation='relu'),
    tf.keras.layers.Dense(1, activation='sigmoid')
])

# Compile the model
model.compile(optimizer='adam', loss='binary_crossentropy', metrics=['accuracy'])

# Train the model
model.fit(x_data, y_data, epochs=10000, verbose=0)

# Evaluate the model
loss, accuracy = model.evaluate(x_data, y_data)
print(f'Loss: {loss}, Accuracy: {accuracy}')

# Make predictions
predictions = model.predict(x_data)
print("Predictions:")
print(predictions)
```

In this example, we first define the XOR dataset, create a DNN model with an input layer, a hidden layer with ReLU activation, and an output layer with sigmoid activation. We compile the model using binary cross-entropy loss and the Adam optimizer. Then, we train the model for a large number of epochs and evaluate its performance. Finally, we make predictions using the trained model.

Make sure you have TensorFlow installed before running this code. You can install it with `pip install tensorflow`.

(0, array([0.6921862], dtype=float32), array([-0.04412946], dtype=float32))
(20, array([0.21779542], dtype=float32), array([0.23556986], dtype=float32))
(40, array([0.1230993], dtype=float32), array([0.28736547], dtype=float32))
(60, array([0.10452971], dtype=float32), array([0.29752243], dtype=float32))
(80, array([0.10088827], dtype=float32), array([0.29951417], dtype=float32))
(100, array([0.10017419], dtype=float32), array([0.29990473], dtype=float32))
(120, array([0.10003416], dtype=float32), array([0.29998133], dtype=float32))
(140, array([0.10000671], dtype=float32), array([0.29999635], dtype=float32))
(160, array([0.10000133], dtype=float32), array([0.2999993], dtype=float32))
(180, array([0.10000026], dtype=float32), array([0.29999986], dtype=float32))
(200, array([0.10000005], dtype=float32), array([0.29999998], dtype=float32))

WARNING:tensorflow:From /usr/local/lib/python2.7/dist-packages/tensorflow/python/framework/op_def_library.py:263: colocate_with (from tensorflow.python.framework.ops) is deprecated and will be removed in a future version.
Instructions for updating:
Colocations handled automatically by placer.
WARNING:tensorflow:From /usr/local/lib/python2.7/dist-packages/tensorflow/python/util/tf_should_use.py:193: initialize_all_variables (from tensorflow.python.ops.variables) is deprecated and will be removed after 2017-03-02.
Instructions for updating:
Use `tf.global_variables_initializer` instead.