Data-Driven Model Reduction Strategies for Dynamical Systems

Many physically occurring phenomena are nonlinear in nature and can be understood through dynamical systems theory which describes how the state of the particular system evolves in time. However, it is generally cumbersome to analyze these processes in depth because of the nonlinearities in the mathematical model or large sets of equations. Model reduction strategies are employed for such nonlinear processes to reduce the model dimensionality and approximate the full model dynamics. In this study, we focus on data driven model reduction strategies for various biological systems where only observable data is available and illustrate their efficacy.

Our first work focused on treatment of circadian misalignment by proposing a data-driven operational phase-coordinate framework to identify a reduced model framework and employing optimal control to derive treatment inputs. It is to be noted though that conventional data-driven techniques usually have various disadvantages particularly their sensitivity to noise. Thus, we shift our focus towards neural networks as an alternative data-driven strategy for robust reduced order model identification. The second work structures a neural network based on a proposed isostable coordinate-based model reduction framework for fixed point systems which is trained on sets of observable data to predict linear unknown coefficients for the reduced model.

The notion of reduced model based neural networks is extended in the following work for systems with oscillatory dynamics where a phase isostable coordinate-based model reduction scheme is employed to build the neural network. The neural network is trained to learn the coefficients of the Fourier series expansion of the reduced model’s unknown terms. Finally, we propose a data-driven framework based on neural networks in the last work that learns the eigenspace representation of a nonlinear system for model reduction. The eigenspace is constrained such that the resulting system representation can outperform traditional model reduction strategies like DMD that solely rely on least square minimization.

The results of this study illustrate clearly that data-driven model reduction strategies can be employed successfully to approximate the full model dynamics of the underlying processes.