Neural networks are highly non-linear functions often parametrized by a staggering number of weights. Miniaturizing these networks and implementing them in hardware is a direction of research that is fueled by a practical need, and at the same time connects to interesting mathematical problems. For example, by quantizing, or replacing the weights of a neural network with quantized (e.g., binary) counterparts, massive savings in cost, computation time, memory, and power consumption can be attained. Of course, one wishes to attain these savings while preserving the action of the function on domains of interest. 
We discuss connections to problems in discrepancy theory, present data-driven and computationally efficient stochastic methods for quantizing the weights of already trained neural networks and we prove that our methods have favorable error guarantees under a variety of assumptions.  

https://sites.uci.edu/inverse/

The phase field crystal modeling framework describes materials at atomic space scales on diffusive time scales. It has been used to study grain growth, fracture, crystallization, and other phenomena. In this talk I will describe some recent work with collaborators developing a thermodynamically consistent phase field crystal model that includes heat transport and lattice expansion and contraction. We use the theory of non-equilibrium thermodynamics, a formalism developed by Alt and Pawlow, and Onager's principle to give consistent laws of entropy production, and mass and energy conservation. I will show some preliminary numerical simulation results involving heat transport during solidification, and I will discuss some ideas on developing entropy and energy stable numerical methods.

In this talk we discuss inverse problems of determining time-dependent coefficients appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines time-dependent coefficients of the wave equation uniquely in a space-time cylinder. We shall discuss two problems: (1) Recovery of a potential appearing in the wave equation, when the Dirichlet and Neumann values are measured on opposite parts of the lateral boundary of the space-time cylinder. (2) Recovery of both a damping coefficient and a potential appearing in the wave equation, when the Dirichlet values are measured on the whole lateral boundary and the Neumann data is collected on roughly half of the boundary. This talk is based on joint works with Teemu Saksala (NC State University) and Lili Yan (University of Minnesota).