Is there a deep learning method for 3D labels?
Since the probability are summing up to zero, so you can simply treat it as Multi-class problem and use a network with Softmax at the end.
Last layer and compile -

model.add(keras.layers.Dense( 3, activation=&quot;softmax&quot;))
model.compile( optimizer='adam, loss=&quot;categorical_crossentropy&quot;, metrics='accuracy')

Metrics - Accuracy is not appropriate. Define a custom metrics based on the interpretation of 3 probabilities
The labels will be as per the probability-
e.g. This is for MNIST 10 digits -

Digit 1 - [0.05, 0.55, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05]


Prediction - [0.064, 0.356, 0.059, 0.069, 0.068, 0.050, 0.044, 0.122, 0.064, 0.101]

Code for MNIST - Colab link