Data-driven inference of low-order isostable-coordinate-based dynamical models using neural networks

The isostable coordinate system, which encodes for level sets of the slowest decaying eigenmodes of the Koopman operator, provides an effective framework with which to represent the dynamics of a general nonlinear system using a low-order basis. When the underlying model equations are known, transformation to an isostable-coordinate-based reduced order model is relatively straightforward. In a data-driven setting, where only time series measurements of observables are available, robust, accurate, and general strategies for inference of these reduced order models have yet to be developed. To this end, in this work we reframe the reduced order isostable coordinate dynamics of a general nonlinear dynamical system in the basin of attraction of a stable fixed point in terms of the composition of a set of known nonlinear functions and unknown linear functions. This framing allows for the use of an artificial neural network to identify the weights of the unknown linear functions without any need of prior estimation of the isostable coordinates. Once learning is completed, these weights can be extracted to yield a nonlinear reduced order model that is independent of the artificial neural network. The proposed technique is illustrated in a collection of models including one that considers the dynamics of a synaptically coupled population of tonically firing conductance-based neurons.