Method for reconstructing nonlinear modes with adaptive structure from multidimensional data

We present a detailed description of a new approach for the extraction of principal nonlinear dynamical modes (NDMs) from high-dimensional data. The method of NDMs allows the joint reconstruction of hidden scalar time series underlying the observational variability together with a transformation mapping these time series to the physical space. Special Bayesian prior restrictions on the solution properties provide an efficient recognition of spatial patterns evolving in time and characterized by clearly separated time scales. In particular, we focus on adaptive properties of the NDMs and demonstrate for model examples of different complexities that, depending on the data properties, the obtained NDMs may have either substantially nonlinear or linear structures. It is shown that even linear NDMs give us more information about the internal system dynamics than the traditional empirical orthogonal function decomposition. The performance of the method is demonstrated on two examples. First, this approach is successfully tested on a low-dimensional problem to decode a chaotic signal from nonlinearly entangled time series with noise. Then, it is applied to the analysis of 250-year preindustrial control run of the INMCM4.0 global climate model. There, a set of principal modes of different nonlinearities is found capturing the internal model variability on the time scales from annual to multidecadal.