Nonlocality in Continuum Mechanics, Population Dynamics, and Neural Networks

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Nonlocal systems have been used to predict cracks and damage in dynamic fracture, to denoise images, and to provide more accurate models in binary fluids and in phase separation. This project concerns theories employed to model discontinuous, singular, or irregular behavior encountered in such applications. Currently, they are applied to study damage-resistant cover glass such as the one produced for cellphones, composite-based aircrafts, blades for wind turbines, oil and gas extraction, and in the nuclear energy industry, as in fracturing of concrete in extreme situations and storage of spent nuclear fuel. The project also aims to develop mathematical tools to advance the understanding of nonlocal models, with the goal of answering questions such as how much damage a specified force will induce in a material, what is the make-up of a population under given growth, decay, and diffusion rules, and how can efficiency and efficacy of neural networks performance be improved. The project provides opportunities for research training of graduate and undergraduate students. The investigators will conduct a mathematical and computational investigation of nonlocal models that arise in dynamic fracture in plates, image processing, and phase transitions. They will create neural networks models with continuous depth, which involve nonlocality, by considering a cumulative effect of layers in the hidden states. To provide a theoretical and methodological foundation for the study of nonlocal models within the realm of applied mathematics, the researchers will adapt existing techniques from the local theory to the nonlocal framework, as well as develop new methods. Applied mathematicians from academia and research laboratories will provide expertise in modeling and computational aspects.This project is jointly funded by the DMS Applied Mathematics Program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.