Nonlinear Partial Differential Equations and Applications

This project develops methods for the qualitative and quantitative study of nonlinear first- and second-order partial differential equations (PDE) set in finite and/or infinite dimensional state spaces arising in science and engineering, and some of their applications. The emphasis is on (i) the theory of mean field games (MFG) and the techniques associated to them including the well-posedeness of nonlinear PDE in infinite dimensional spaces and applications to mean field limits, large deviations of random matrices, filtering, information acquisition, deep learning and economics; and (ii) viscosity solutions techniques to study traffic models and goal-based stochastic control of portfolio selection with time inconsistency and mental accounting. The project aims at creating a multifaceted platform for theoretical advances across the areas of mean field games, PDE set in infinite dimensional spaces, large deviations of random matrices, mean field limits of interacting particle systems with singular interactions, deep learning, filtering and control of partial information, robust control, time-inconsistent stochastic optimization, and the modeling for applications in asset pricing, traffic management, information acquisition, personalized goal-setting optimization and mental accounting. The analysis of the proposed applications will first require a deeper theoretical understanding that will prompt the formulation of new mathematical questions and their study. It will also generate substantial synergies among distinct areas, bringing together expertise from different areas of analysis and probability. The project cultivates various collaborations with both established and early-career researchers, including graduate students. The project is concerned with the development pf the mathematical methodologies and innovation and the modeling required to study many applications of current interest in several diverse fields like filtering and control of partial information, assessment of risk and model ambiguity in economics, information acquisition, asset allocation with time inconsistency and mental accounting, random matrices, traffic modeling, and, finally, deep learning. Most of the applications considered here are described using either deterministic and stochastic control in finite and infinite spaces (mean field games of control) or mean field games. These are very active areas of research with many open mathematical problems and questions. Some of the mathematical difficulties that this project focus on are the development of the methodology to study the well-posedness of solutions to PDE set in spaces of probability measures and to systems of backward-forward equations, the understanding of mean field limits for models with singular dynamics, the study of large deviations of random matrices, and the rigorous mathematical description of some of the algorithms of deep learning. 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.