Adaptable reservoir computing: A paradigm for model-free data-driven prediction of critical transitions in nonlinear dynamical systems

A problem in nonlinear and complex dynamical systems with broad applications is forecasting the occurrence of a critical transition based solely on data without knowledge about the system equations. When such a transition leads to system collapse, as often is the case, all the available data are from the pre-critical regime where the system still functions normally, making the prediction problem challenging. In recent years, a machine-learning based approach tailored to solving this difficult prediction problem, adaptable reservoir computing, has been articulated. This Perspective introduces the basics of this machine-learning scheme and describes representative results. The general setting is that the system dynamics live on a normal attractor with oscillatory dynamics at the present time and, as a bifurcation parameter changes into the future, a critical transition can occur after which the system switches to a completely different attractor, signifying system collapse. To predict a critical transition, it is essential that the reservoir computer not only learns the dynamical "climate" of the system of interest at some specific parameter value but, more importantly, discovers how the system dynamics changes with the bifurcation parameter. It is demonstrated that this capability can be endowed into the machine through a training process with time series from a small number of distinct, pre-critical parameter values, thereby enabling accurate and reliable prediction of the catastrophic critical transition. Three applications are presented: predicting crisis, forecasting amplitude death, and creating digital twins of nonlinear dynamical systems. Limitations and future perspectives are discussed.