Applied Mathematics

Research Grants

PI: Professor George Karniadakis

Learning and Meta-Learning of Partial Differential Equations via Physics-Informed Neural Networks: Theory, Algorithms, and Applications sponsored by Air Force Office of Scientific Research Multidisciplinary Research Program of the University Research Initiative (AFOSR MURI)

Abstract:  Despite the significant progress over the last 50 years in simulating multiphysics problems using numerical discretization of partial differential equations (PDEs), we still cannot incorporate seamlessly noisy data into existing algorithms, mesh-generation is complex, and we cannot tackle high-dimensional problems governed by parametrized PDEs. Moreover, solving inverse problems is often prohibitively expensive and requires different formulations and new computer codes. We are overcoming these obstacles by introducing physics-informed learning, integrating seamlessly data and mathematical models, and implementing them using physics-informed neural networks (PINNs) and other new physics- informed networks (PINs). We are blending knowledge on existing methods, e.g. domain decomposition and uncertainty quantification, with new concepts in deep neural networks and more general networks and regressions. Inversely, we are employing synergistically the lessons learned from PINNs/PINs to enhance the performance of existing numerical methods, e.g., for low-dimensional modeling and high-dimensional PDEs.

PI: Professor Bjorn Sandstede

RTG: Mathematics of Information, Data, and Applications of Science

Abstract:  The big-data revolution has transformed many areas of engineering, industry, mathematics, and science. For instance, data generated through genome-wide association studies, observations from telescopes, and computational simulations on petaflop computers provide tremendous opportunities to better understand the world around us. These massive datasets also pose many challenges, and there has been an explosion of research in recent years about how to manipulate, store, and extract meaningful information from ever-larger amounts of data. Mathematical analysis, algorithms, and insights have been and continue to be a crucial component of this research effort. This RTG project focuses on the mathematical foundations and applications of data science and catalyze research collaborations that combine different mathematical perspectives to address emerging challenges and opportunities. The mathematical challenges of data science also presents a unique and transformative opportunity to develop more systematic and integrated training for the next generation; this project broadens and enhances the scope and quality of the educational and research training provided to graduate students and postdoctoral fellows and involve more undergraduate students, particularly students from historically underrepresented groups, in courses and research experiences in applied mathematics, increasing the workforce trained in data science.

The project focuses on research and training in the mathematical foundations of data science and its applications. The research projects have strong interdisciplinary flavor, combining fundamental stochastic, statistical, combinatorial, dynamical, and computational aspects with concrete applications. Projects involve collaborations with domain scientists from other disciplines, including astrophysics, biology, engineering, and neuroscience. Topics include applying machine learning and Bayesian statistics tools to deriving, analyzing, and simulating partial differential equations; designing optimal closed-loop experiments using statistical inference; advancing techniques in discrete optimization; developing combinatorial models in neuroscience; understanding random projections of high-dimensional measures; and constructing dimension reduction techniques that preserve relevant structure of large data sets. The educational activities focus on vertically integrated training of undergraduates, graduate students, and postdoctoral fellows. Training activities include a first-year seminar focused on the interface of data and social justice, enhanced undergraduate and graduate curricula, summer research experiences for undergraduates, graduate students, and postdoctoral fellows, and working groups for advanced graduate students and postdoctoral fellows. The broader impacts include the recruitment, retention, and training of a diverse cohort of applied mathematicians trained in data science. In addition, the research planned in genome-wide association studies, design of closed-loop neuroscience experiments, single-cell data alignment, image restoration, simulation of Hubble data, self-assembly, inference of dynamical brain data, and data compression and reduction aims to have impact in applications.

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.

PI:  Kavita Ramanan

Title:  High-dimensional Stochastic Dynamics on Diverse Network Topologies

The overall goal of this research program develops a foundational theory for a general class of interacting particle systems and multi-agent (cooperative and non-cooperative) games on realistic but tractable network topologies, including those with heterogeneous (dense and sparse) connectivity structures, with possibly non-Markovian dynamics that may be either diffusive or of jump type. The project aims to forge a new field of inquiry into a systematic study of the effect of network topology on not only equilibrium behavior and phase transitions and large deviations behavior, but also transient phenomena such as oscillatory behavior, which although of great importance in applications, are currently poorly understood at a rigorous mathematical level.

The project also entails the analysis and development of neuromorphic learning algorithms, which are special examples of processes with local interactions. A third thrust of the program includes the development of rough paths analysis for both interacting diffusions on graphs and constrained diffusions, and the use of the signature from rough paths analysis to study stochastic optimization and prediction under model uncertainty.

The project will also involve collaborations with scientists that have synergistic research interests at the Flatiron Institute and IBM’s Thomas J. Watson Research Center.