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PDE-Net: Learning PDEs from Data
Zichao Long · Yiping Lu · Xianzhong Ma · Bin Dong

Fri Jul 13 09:15 AM -- 12:00 PM (PDT) @ Hall B #34

Partial differential equations (PDEs) play a prominent role in many disciplines of science and engineering. PDEs are commonly derived based on empirical observations. However, with the rapid development of sensors, computational power, and data storage in the past decade, huge quantities of data can be easily collected and efficiently stored. Such vast quantity of data offers new opportunities for data-driven discovery of physical laws. Inspired by the latest development of neural network designs in deep learning, we propose a new feed-forward deep network, called PDE-Net, to fulfill two objectives at the same time: to accurately predict dynamics of complex systems and to uncover the underlying hidden PDE models. Comparing with existing approaches, our approach has the most flexibility by learning both differential operators and the nonlinear response function of the underlying PDE model. A special feature of the proposed PDE-Net is that all filters are properly constrained, which enables us to easily identify the governing PDE models while still maintaining the expressive and predictive power of the network. These constrains are carefully designed by fully exploiting the relation between the orders of differential operators and the orders of sum rules of filters (an important concept originated from wavelet theory). Numerical experiments show that the PDE-Net has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.

Author Information

Zichao Long (Peking University)
Yiping Lu (Peking University)
Xianzhong Ma (Peking University)
Bin Dong (Peking University)

I received my B.S. from Peking University in 2003, M.Sc from National University of Singapore in 2005 and Ph.D from University of California Los Angeles (UCLA) in 2009. Then I spent 2 years in University of California San Diego (UCSD) as a visiting assistant professor. I was a tenure-track assistant professor at University of Arizona since 2011 and joined Peking University as an associate professor in 2014. I received the Qiu Shi Outstanding Young Scholar award in 2014 and the award of the Project of Thousand Youth Talents of China in 2015. My research interest is in mathematical modeling and computations in imaging and data science, which includes (but not limited to) biological and medical imaging and image analysis , image guided diagnosis and treatment of disease, (semi-)supervised learning. A special feature of my research is blending different branches in mathematics which includes: bridging wavelet frame theory, variational techniques and nonlinear PDEs; combining sparse approximation and partial differential equations with machine and deep learning. We are working on projects aiming at addressing new and fascinating connections among these subjects, which not only leads to new understandings of the subjects themselves, but also gives rise to new and effective mathematical and computational tools for imaging/data science.

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