How many hidden layers are typically required in a MLP-based neural network to capture a discontinuous relationship?

In a multilayer perceptron (MLP) based neural network, the ability to model complex relationships, including discontinuous ones, largely depends on the architecture, particularly the number of hidden layers. For capturing discontinuous relationships effectively, having at least two hidden layers is beneficial.

The first hidden layer can start learning basic features from the inputs, while the second hidden layer can combine these features more effectively, allowing the model to understand and represent more complex patterns, including discontinuities. This hierarchical feature learning is crucial for adequately mapping the input to the desired output when the relationship isn't continuous.

While one hidden layer can suffice for simpler tasks or relationships that are mostly continuous, it often lacks the capacity to fully capture the intricacies involved in more complex, discontinuous functions. Hence, two hidden layers provide additional flexibility and depth, enhancing the network's ability to generalize and perform well on challenging tasks.

For more intricate mappings that involve severe discontinuities or high variability, deeper architectures (three or more hidden layers) might be employed, but the fundamental requirement for capturing discontinuities typically arises with at least two hidden layers. Thus, the emphasis on two hidden layers as generally required aligns well with the theoretical foundations of neural networks.