Our days are filled with instances of reasoning about the physics of the world, from simple tasks such as stacking dishes in a way that keeps them stable, to life-and-death decisions such as not crossing the street because we presume an oncoming car would hit us if we did. Yet the process we use to make inferences about physical events is not well understood. Here I argue that these interactions are based on a rich, approximately accurate simulation of physical events, but we must account for uncertainty about the current properties of objects in the world. In this thesis I investigate the structure of this simulation process and how it relates to other facets of cognition, including (1) demonstrating that the principles underlying interactions with the world are based on accurate physics, even if our explanations of those same principles are idiosyncratic and erroneous, (2) mapping out the types of uncertainty that this process accounts for, and demonstrating that the simulations themselves are therefore stochastic, and (3) explaining how physical predictions are updated over time due to changing evidence from evolving simulations. This provides a framework for understanding how people form and update representations of both the current and future state of the world based on rich, structured, probabilistic reasoning.

Networks are ubiquitous in natural and engineered systems, from critical infrastructures (like power grids and water distribution systems) to contact networks in epidemiological models. Managing these systems requires a broad array of tools to monitor the configuration and state of the network, identify optimal operating points, and design controls. This thesis examines a collection of topics broadly related to this theme.

In Part 1, we consider problems related to control and optimization in network systems. Chapter 1 studies the problem of safety-critical control in networks of grid-forming inverters. Coupling a physically-meaningful Lyapunov-like function with an optimization approach to identifying forward-invariant sets, we propose a method to certify that a post-fault trajectory achieves frequency synchronization while respecting safety constraints. In Chapter 2, we consider the network resource allocation problem of optimally distributing resources to mitigate the spread of an epidemic. We propose a convex optimization framework for minimizing the basic reproduction number for general compartmental epidemiological models. Chapter 3 addresses optimal control in infinitesimally contracting systems. We provide new convergence criteria for a common indirect optimal control algorithm, and we establish the uniqueness of the optimal control in the limits of large contraction rates and short time horizons.

In Part 2, we use tools from statistical inference and machine learning to solve estimation problems in network systems. Chapter 4 examines the problem of inferring routing topologies from endpoint data in communication networks. Extending a technique called network tomography to use higher-order statistics, and using Mobius inversion to disentangle the interactions between different network paths that are reflected in these statistics, we are able to estimate routing matrices without the cooperation of intermediate routers. Finally, in Chapter 5, we use machine learning to predict edge flows. We propose an implicit neural network that incorporates two fundamental physical principles to estimate flows on unlabeled edges, and we provide a contraction mapping to evaluate the model and backpropagate loss gradients.