Interpretable Dimension Reduction Procedures for Dynamical Systems

Modeling of complex and high-dimensional physical systems is becoming increasingly relevant not only in academic research, but in practical applications as well. Many high-dimensional simulations, however, require immense computational expense, often taking days to complete a single simulation. In many cases, physical insight into the behavior of these systems can guide the discovery of dimensionally-reduced models, allowing for drastic decrease in computational expense, though different classes of physical systems require different forms of dimensionality reduction. 

One class of systems are those governed by the minimization of functionals, which is largely the case in solid mechanics. In this case, it is important to take advantage of any small parameter included in the representative physics in order to achieve dimension reduction. To illustrate the process, this thesis first presents a derivation of a dimensionally-reduced ribbon model for strips of pre-curved liquid crystalline elastomers (LCEs). LCEs are smart-active materials that are highly programmable due to control over the so-called nematic director during the manufacturing process. Long, thin LCE strips have been shown to display a large breadth of mechanical deformations that can be activated when exposed to various external stimuli (e.g. heat, light, electric field). The 3D modeling of these strips is extensive, but by appropriately scaling relevant parameters and taking advantage of the long, thin geometry, we can derive an effective 1D model. Finite element simulations show that the 1D model is capable of picking up a wide range of behaviors observed in 3D simulations and laboratory experiments. 

Another class of physical systems that lend themselves to dimension reduction are those that produce mass amounts of discrete data. If the data obtained are theorized to exist on a low-dimensional manifold, then we can exploit that structure to drastically speed up computation. In the second part of this thesis, I have developed a general procedure for developing surrogate models for dynamical systems that generate large data sets. I begin by implementing a modified diffusion maps algorithm to discover the low-dimensional structure, then develop a means of integrating over discrete point cloud data representative of a manifold in order to create a surrogate model. As a test case, we demonstrate the efficacy of this method on data representing trajectories in the chemical kinetics of cyclotrimethylenetrinitramine (also known as RDX) decomposition. Comparing trajectories from the full physics system and the surrogate model shows a minimal loss in accuracy for the low-dimensional results.

Ultimately, I present in this thesis an effective guide to interpretable dimension reduction in two classes of physical systems. The work with LCEs shows the efficacy of exploiting small parameters in systems governed by the minimization of functionals, while the work on the chemical kinetics of RDX implements my general framework for the development of data-driven surrogate models.