UC Santa Barbara

The underlying theme that characterizes my research arc has been that of phenomenological modeling of dynamical systems and mathematical abstraction of networks. By abstracting complicated networks (such as neural networks trained to learn dynamic system behavior), we can build much lower parameter and geometrically interpretable models that capture the same properties of the more expensive and harder to understand deep neural networks. Additionally, an abstracted vision of biochemical networks allows us to discover new mechanisms within cellular transcriptional networks to better understand how bacteria can adapt to and perform computation in challenging and changing environments. In Chapter 2 we analyze the learned dictionaries from the deep network based Koopman system identification algorithm, deepDMD. We explore the theoretical basis for the strong performance of this algorithm. We do so by discovering dictionaries to satisfy a property of subspace approximation, which we define as uniform finite approximate closure. We also discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions can be used to achieve the same accuracy and dimensional scaling as the deep-learning-based deepDMD algorithm. This mixed dictionary achieves similar accuracy and dimensional scaling to deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. In Chapter 3 we connect the concepts of Koopman system identification and Koopman state feedback control. For control affine systems, we demonstrate examples of state feedback controllers to illustrate that, when control is the objective, the Koopman model becomes subject to the control design process. This implies that certain scenarios merit data-driven control strategies different from system identification strategies that prioritize best fit. We also present a controller design framework to stabilize systems modeled with a data-driven Koopman model. Since Koopman models propose a linear state-space representation of the true dynamics, they often imply that a proportional control law applied to the states of the Koopman model can stabilize the origin. We compute error bounds on the minimal nonlinear control action needed to stabilize a system that has already been assigned a Koopman model with stabilizing proportional control. In Chapter 4, we shift gears to model cell sensory transcription networks, the intracellular computation structure that regulates and drives cellular activity. Here we explore the computation (and classification) potential of single-celled organisms. We present a model for identification and response to environmental changes in resilient bacteria. This model combines two known motifs in transcription networks: dense overlapping regulons (DORs) and single input modules (SIMs). Combined in a hybrid network motif we have processes for decision making, control actuation and feedback. In Chapter 4, we model this hybrid network motif (which we call the DOR2SIM motif) with a superposition of modular nonlinear functions to describe protein signaling in the network and basic mass action kinetics to describe the other chemical reactions in this process. We also introduce a notion of classification for dynamic systems to explain bacterial decision making in terms of classification. Given this definition, we provide sufficient conditions for models of the DOR2SIM motif to classify. These conditions suggest that relatively low monomer degradation rates as well as low expression of source node genes (the inputs to the DORs) at equilibrium enables classification.