Less is more: a new machine-learning methodology for spatiotemporal systems

Machine learning provides a way to use only portions of the variables of a spatiotemporal system to predict its subsequent evolution and consequently avoids the curse of dimensionality. The learning machines employed for this purpose, in essence, are time-delayed recurrent neural networks with multiple input neurons and multiple output neurons. We show in this paper that such kinds of learning machines have a poor generalization ability to variables that have not been trained with. We then present a one-dimensional time-delayed recurrent neural network for the same aim of model-free prediction. It can be trained on different spatial variables in the training stage but initiated by the time series of only one spatial variable, and consequently possess an excellent generalization ability to new variables that have not been trained on. This network presents a new methodology to achieve fine-grained predictions from a learning machine trained on coarse-grained data, and thus provides a new strategy for certain applications such as weather forecasting. Numerical verifications are performed on the Kuramoto coupled oscillators and the Barrio- Varea-Aragon-Maini model.