Recurrent neural network architectures can have useful computational properties, with complex temporal dynamics. However, evaluation of recurrent dynamic architectures requires solution of systems of differential equations, and the number of evaluations required to determine their response to a given input can vary with the input, or can be indeterminate altogether in the case of oscillations or instability. In feed-forward networks, by contrast, only a single pass through the network is needed to determine the response to a given input. Modern machine-learning systems are designed to operate efficiently on feed-forward architectures. We hypothesised that two-layer feedforward architectures with simple, deterministic dynamics could approximate the responses of single-layer recurrent network architectures. By identifying the fixed-point responses of a given recurrent network, we trained two-layer networks to directly approximate the fixed-point response to a given input. These feed-forward networks then embodied useful computations, including competitive interactions, information transformations and noise rejection. Our approach was able to find useful approximations to recurrent networks, which can then be evaluated in linear and deterministic time complexity.