This paper addresses the challenges of training Recurrent Neural Networks (RNNs), specifically the vanishing and exploding gradient problems. The authors analyze these issues from analytical, geometric, and dynamical systems perspectives to better understand their underlying causes. They propose two solutions: gradient norm clipping to handle exploding gradients and a soft constraint to address vanishing gradients. The proposed methods are validated through experiments. The vanishing gradient problem occurs when long-term components of the gradient decay exponentially to zero, making it difficult for the model to learn temporal dependencies. The exploding gradient problem occurs when these components grow exponentially, leading to unstable training. The authors show that the behavior of these gradients depends on the eigenvalues of the recurrent weight matrix. If the largest eigenvalue is less than 1, gradients vanish; if it is greater than 1, gradients explode. The paper also draws parallels between RNNs and dynamical systems, noting that bifurcations in the system can lead to sudden changes in the gradient behavior. The authors propose a regularization term that forces the error signal to maintain its norm as it propagates backward in time, helping to prevent vanishing gradients. The authors validate their solutions through experiments on synthetic and real-world tasks. They show that their proposed methods significantly improve performance on tasks requiring long-term memory, such as the temporal order problem and other pathological tasks. The methods also perform well on natural tasks like polyphonic music prediction and language modeling. The paper concludes that the proposed solutions provide a practical way to address the vanishing and exploding gradient problems in RNNs, improving their ability to learn from long sequences of data.This paper addresses the challenges of training Recurrent Neural Networks (RNNs), specifically the vanishing and exploding gradient problems. The authors analyze these issues from analytical, geometric, and dynamical systems perspectives to better understand their underlying causes. They propose two solutions: gradient norm clipping to handle exploding gradients and a soft constraint to address vanishing gradients. The proposed methods are validated through experiments. The vanishing gradient problem occurs when long-term components of the gradient decay exponentially to zero, making it difficult for the model to learn temporal dependencies. The exploding gradient problem occurs when these components grow exponentially, leading to unstable training. The authors show that the behavior of these gradients depends on the eigenvalues of the recurrent weight matrix. If the largest eigenvalue is less than 1, gradients vanish; if it is greater than 1, gradients explode. The paper also draws parallels between RNNs and dynamical systems, noting that bifurcations in the system can lead to sudden changes in the gradient behavior. The authors propose a regularization term that forces the error signal to maintain its norm as it propagates backward in time, helping to prevent vanishing gradients. The authors validate their solutions through experiments on synthetic and real-world tasks. They show that their proposed methods significantly improve performance on tasks requiring long-term memory, such as the temporal order problem and other pathological tasks. The methods also perform well on natural tasks like polyphonic music prediction and language modeling. The paper concludes that the proposed solutions provide a practical way to address the vanishing and exploding gradient problems in RNNs, improving their ability to learn from long sequences of data.