(753d) Data Driven Parameter Reduction

Traditional model reduction techniques build parsimonious representations of a system by (a) neglecting states expected to have minor effects on the system behavior or (b) by discovering linear or nonlinear combinations of the states that influence behavior. Traditionally, the reduction can be achieved by analyzing the mathematical equations modeling the system. Time-scale analysis underpins one of the well-known reduction methods, exploiting a disparity of scales in chemical kinetics (e.g. Quasi-Steady State (QSS) approximation).

In addition to its states, however, a system is also influenced by several parameters. In many multiparameter models, there exist a set of so-called effective parameters, which are those combinations of the model parameters that dictate the system behavior (such as the Reynolds number in fluid mechanics). The number of effective parameters may be much lower than the number of model parameters in the system original description. If the mathematical model of the system is available, one can traditionally perform (a) sensitivity analysis to discriminate the importance of every single parameter and (b) dimensional analysis to extract the effective parameters.

Here, we present a data-driven approach to find the effective parameters of a system, and thus reduce the dimensionality of its parameter space. This data-driven approach does not require access to the model’s mathematical description, and can even work if only partial (but rich enough) observations of the system are available. Specifically, we modify a manifold learning technique, namely Diffusion Maps, through the use of an output-informed kernel; this allows us to find nonlinear directions in parameter space corresponding to a reduced, effective parameter set. In the same step, we discover parameter directions in which the system behavior does not change. This framework will be illustrated through prototype examples, such as singular and regular perturbations, and then through effective parameter identification in coupled oscillator networks.