One potential challenge when training a deep neural network is the problem of vanishing or exploding gradients. This occurs when the gradients become extremely small or large, hindering the network's ability to learn effectively. To mitigate this, techniques like gradient clipping or weight initialization methods can be used. Another challenge is overfitting, where the network memorizes the training data instead of learning general patterns. Regularization techniques such as dropout or L1/L2 regularization can help combat this issue.

A common pitfall is the issue of dataset bias. If the training dataset is not representative of the real-world data, the network may not perform well on unseen examples. Bias can arise due to various factors such as imbalanced classes, over- or under-sampling, or sampling from a non-random distribution. It's important to carefully curate and preprocess the data to minimize bias. Another challenge is selecting the right architecture and hyperparameters for the network. This requires experimentation and fine-tuning, as there is no one-size-fits-all solution. Additionally, lack of interpretability and explainability of deep neural networks can be a challenge, especially in fields where transparency is crucial, like healthcare or finance.