System Theory and Sparse Learning for Data Driven Modeling

The main focus of this talk will be addressing how to produce parsimonious dynamical models using large volumes of computational and/or experimental data. The major challenge in nonlinear approximation theory and its applications is the high dimensionality of the system. The performance of most of the conventional modeling algorithms decreases drastically as the dimension of the system increases. Also, the ease with which a mathematical model can be used in various post-processing computations such as controller design may determine the suitability of an approximation method for a particular problem at hand. In summary, many different factors, such as intended use of the model, problem dimensionality, quality of the measurement data, offline or online learning etc., can result in ad-hoc decisions leading to an inappropriate model architecture. All these issues make system modeling a challenging task and motivate us to seek more general and universal modeling methods that can be applied to a wide class of structurally different systems. This talk will discuss the role of dynamical system theory to efficiently deal with the issues of nonlinearity and high dimensioned output vector. The basic idea is to split the identification process into two steps: (1) linear system identification followed by the (2) nonlinear system identification process. Dynamic system theory is utilized to automatically identify time-varying finite dimensional subspace over which the system dynamics evolves. Automated construction of time varying input-output models facilitates the identification of the hidden state dynamics from the data, enabling the analyst to uncover the linear components of the invariant manifolds that are observed by the high dimensioned input-output data. Identification of the time varying modes captures the detectable invariant sets of the dynamical system and provides an effective means of data compression. The identified linear model is further refined to learn the nonlinear dynamics. The appropriate nonlinear functional structure is identified from an over-complete dictionary of basis-functions using sparse approximation tools. Thus, these new nonlinear system approximation algorithms not only have the ability to capture underlying physics, but they also have the benefit of naturally accommodating model reduction algorithms such as proper orthogonal decomposition (POD) and principal component analysis (PCA). The talk will highlight examples arising from various academic and engineering problems to assess the reliability and limitations of the newly established sparse approximation schemes.