This project develops a new mathematical framework for deep learning based on the interpretation of certain deep neural networks as nonlinear time-dependent differential equations. This interpretation offers a new way to analyze the successes and failures of deep learning. Advances in deep learning will be made by adapting the wide array of tools available for related optimal control problems, including numerical optimization, partial and ordinary differential equations, inverse problems theory, and parallel processing, to the deep learning problem. Using these currently untapped resources provides rigorous new ways to design and train very deep neural networks that generalize well.

Funding: US National Science Foundation CAREER award DMS 1751636

We develop new mathematical approaches, algorithms and systems for identifying pathogen strains using whole genome multilocus sequence typing (wgMLST) data in mixed samples. Strain identification is an underdetermined inverse problem when the samples are not cultured, i.e., contain multiple strains at unknown proportions. Therefore, we encode prior knowledge on both the strains and their proportion in a Bayesian framework. Our approaches use mixed-integer optimization to identify the most likely strains and Monte-Carlo sampling to quantify the uncertainty associated with those estimates. Motivating applications of our algorithms include the identification of P. falciparum (malaria) and E.coli (foodborne illness) strains to improve visibility and monitoring for public health officials.

Funding: Center for Disease Control and Prevention

This project develops efficient numerical methods for solving big data parameter estimation problems that involve partial differential equations (PDEs). Parameter estimation problems of this kind impose key challenges in many scientific disciplines, e.g., in medical imaging, geophysical imaging, and deep learning. The inverse problem can be formulated as an optimization problem with constraints that are given by the PDEs. The unknowns are parameters of the PDEs (e.g., physical properties of the object to be measured). The objective is to minimize the misfit between PDE simulations and measured data plus some regularization term. The project aims at deriving efficient algorithms and massively parallel solvers for solving big data PDE parameter estimation problems. Its main thrusts are reduced-order modeling techniques, stochastic optimization methods, and parallel and distributed optimization schemes.

Funding: US National Science Foundation award DMS 1522599