The training of backpropagation networks involves adjusting the weights between the computing nodes in the artificial neural network to minimize the errors between the network's predictions and the known outputs in the training set. This least-squares minimization problem is conventionally solved by an iterative fixed-step technique, using gradient descent, which occasionally exhibits instabilities and converges slowly. The authors show that training of the backpropagation network can be expressed as a problem of solving coupled ordinary differential equations for the weights as a (continuous) function of time. These differential equations are usually mathematically stiff. The use of a stiff differential equation solver ensures quick convergence to the nearest least-squares minimum. Training proceeds at a rapidly accelerating rate as the accuracy of the predictions increases, in contrast with gradient descent and conjugate gradient methods. The number of presentations required for accurate training is reduced by up to several orders of magnitude over the conventional method.