In 1988, Barak Pearlmutter proposed a method for learning state space trajectories in recurrent neural networks (RNNs). The paper describes procedures for computing the gradient of an error function E with respect to the weights and time constants of a continuous RNN. These gradients are essential for gradient descent optimization, enabling the network to minimize E. The approach involves forward and backward simulations to compute these gradients, allowing the network to learn temporal patterns. The paper presents simulations where networks are trained to move through limit cycles, demonstrating the effectiveness of the method. It also discusses elaborations such as mutable time delays and teacher forcing, and provides a complexity analysis. The proposed RNNs are particularly suited for temporally continuous domains like signal processing, control, and speech. The paper introduces a forward/backward technique to compute gradients by simulating the network forward and backward in time. It derives differential equations for the gradients and shows how they can be used to update weights and time constants. The method is validated through simulations on tasks like the XOR problem and following circular and figure-eight trajectories. The paper also discusses embellishments such as time delays, avoiding the backward pass by using a shooting method, and an online learning variation inspired by Williams and Zipser. Teacher forcing is introduced to improve learning efficiency by using the training signal to guide the network's states. The analysis highlights the computational power and complexity of the networks, noting that they can represent complex trajectories and have potential applications in signal processing, control, and speech recognition. The paper concludes with future work, including generalization to novel inputs and applications in identification and control. It also explores the relationship to other work, comparing its approach to Pineda's and Jordan's methods, and discusses the stability and potential for hybrid learning algorithms.In 1988, Barak Pearlmutter proposed a method for learning state space trajectories in recurrent neural networks (RNNs). The paper describes procedures for computing the gradient of an error function E with respect to the weights and time constants of a continuous RNN. These gradients are essential for gradient descent optimization, enabling the network to minimize E. The approach involves forward and backward simulations to compute these gradients, allowing the network to learn temporal patterns. The paper presents simulations where networks are trained to move through limit cycles, demonstrating the effectiveness of the method. It also discusses elaborations such as mutable time delays and teacher forcing, and provides a complexity analysis. The proposed RNNs are particularly suited for temporally continuous domains like signal processing, control, and speech. The paper introduces a forward/backward technique to compute gradients by simulating the network forward and backward in time. It derives differential equations for the gradients and shows how they can be used to update weights and time constants. The method is validated through simulations on tasks like the XOR problem and following circular and figure-eight trajectories. The paper also discusses embellishments such as time delays, avoiding the backward pass by using a shooting method, and an online learning variation inspired by Williams and Zipser. Teacher forcing is introduced to improve learning efficiency by using the training signal to guide the network's states. The analysis highlights the computational power and complexity of the networks, noting that they can represent complex trajectories and have potential applications in signal processing, control, and speech recognition. The paper concludes with future work, including generalization to novel inputs and applications in identification and control. It also explores the relationship to other work, comparing its approach to Pineda's and Jordan's methods, and discusses the stability and potential for hybrid learning algorithms.