Learning Partial Differential Equation (PDE) and Beyond

Duke University, Durham NC

Investigators

Abstract

Partial differential equations (PDE) are powerful tools in modeling physical, biological and social worlds. They are used to understand, predict, simulate and infer complex phenomena and quantities across disciplines of science and engineering, from heat transfer, wave propagation, and biological systems to weather forecast, climate change, and economic and social behavior. In the past, most of the equations are derived from basic physical laws and assumptions. With the advancement of technologies, abundant data are available from measurements and observations in many complex situations where the underlying model is not yet available or not accurate enough. Whether one can learn a PDE model directly from observed data becomes a both interesting and important question. In this project, a new perspective of PDE learning based on observed solution data, from theory and methodology to efficient algorithms and applications, will be developed. Mathematical theories and computational tools developed in this project will be useful for researchers and practitioners in many disciplines that have to deal with computational modeling of systems using differential equations. The developed mathematical tools and computational algorithms will be disseminated broadly for advancing scientific and technological progress. Integration with education at different levels will be designed to provide training for young computational mathematicians. A data driven approach to PDE learning can provide a useful tool to discover new or more accurate quantitative models for complex systems and dynamics not only in traditional physical sciences but also in biological, social and other disciplines where basic principles and laws are not available but data are abundant. In return, the learned PDE model will provide more understandings and insights of the underlying problem as well as computational tools. For any data driven, data intensive and data-enabled approach, it is a fundamental task to study and understand the complexity of the underlying problem, model, or data and using this understanding to design effective representation, dimension reduction and fast algorithm that are problem specific. This project is aimed to address a few important questions and challenges which include (1) understanding and characterization of the data, e.g., how much data are available and how much data are needed, (2) data selection and data processing to deal with numerical errors and noise, and (3) development of efficient and accurate algorithms based on different formulations. Systematic experiments will be designed to verify the developed theories and test proposed methodologies and algorithms. The insights obtained in this project will also provide new perspectives on mathematics and physics based 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.