报告一：Solving Differential Equations via Machine Learning

报告人：George Em Karniadakis教授

时间：12月21日（周四）上午10:00-11:00

地点：王克桢楼348会议室

主持人：张东晓 教授

 

报告内容摘要：

In the last 30 years I have pursued the numerical solution of partial differential equations (PDEs) for diverse application in mechanics and soft matter using spectral and spectral elements methods for diverse applications, starting from deterministic PDEs in complex geometries, to stochastic PDEs for uncertainty quantification, and to fractional PDEs that describe non-local behavior in disordered media and viscoelastic materials. More recently, I have been working on solving PDEs in a fundamentally different way. I will present a new paradigm in solving linear and nonlinear PDEs from noisy measurements without the use of the classical numerical discretization. Instead, we infer the solution of PDEs from noisy data, which can represent measurements of variable fidelity. The key idea is to encode the structure of the PDE into prior distributions and train Bayesian nonparametric (Gaussian process, GP) regression models on available noisy data. The resulting posterior distributions can be used to predict the PDE solution with quantified uncertainty, efficiently identify extrema via Bayesian optimization, and acquire new data via active learning. Moreover, I will present how we can use this new framework to learn PDEs from noisy measurements of the solution and the forcing terms. Alternatively, we can solve PDEs using optimized deep neural networks (dNN) by encoding the PDE directly into the network. I will present both approaches and compare the relative performance of GPs and dNNs.

 

报告人简介：

George Karniadakis received his S.M. and Ph.D. from Massachusetts Institute of Technology. He was appointed Lecturer in the Department of Mechanical Engineering at MIT in 1987 and subsequently he joined the Center for Turbulence Research at Stanford / Nasa Ames. He joined Princeton University as Assistant Professor in the Department of Mechanical and Aerospace Engineering and as Associate Faculty in the Program of Applied and Computational Mathematics. He was a Visiting Professor at Caltech in 1993 in the Aeronautics Department and joined Brown University as Associate Professor of Applied Mathematics in the Center for Fluid Mechanics in 1994. After becoming a full professor in 1996, he continues to be a Visiting Professor and Senior Lecturer of Ocean/Mechanical Engineering at MIT. He is a Fellow of the Society for Industrial and Applied Mathematics (SIAM, 2010-), Fellow of the American Physical Society (APS, 2004-), Fellow of the American Society of Mechanical Engineers (ASME, 2003-) and Associate Fellow of the American Institute of Aeronautics and Astronautics (AIAA, 2006-). He received the Ralf E Kleinman award from SIAM (2015), the J. Tinsley Oden Medal (2013), and the CFD award (2007) by the US Association in Computational Mechanics. His h-index is 83 and he has been cited over 35,500 times. He recently received the Alexander von Humboldt Award. He is an associate editor for SIAM J. Sci. Comp., SIAM Reviews, SIAM J. Uncertainty Quantification, J. Comput. Physics, M3AS, and Calcolo.

 

---

报告二： Learning subsurface flow equations via LASSO

报告人：常海滨 副研究员

时间：12月21日（周四）上午11:00-12:00

地点：王克桢楼348会议室

主持人：张东晓 教授

 

报告内容摘要：

The governing equations of physical problems are traditionally derived by resorting to conservation laws or physical principles. However, there are still some complex problems, for which these first principle derivations cannot be implemented. As the increasing of the data acquisition and storage ability, data driven methods have grained more and more attention. In recently years, several publications have discussed how to learn the dynamical systems and partial differential equations using data driven methods. Following their ideas, in this talk, we will show how to learn the subsurface flow equations via Lasso (the least absolute shrinkage and selection operator). The learning of single phase groundwater flow equation and contaminant flow equation are demonstrated. Considering that the subsurface flow model parameters are always heterogeneous, we investigated and established a procedure for learning the partial differential equations with heterogeneous model parameters. Computation of the derivative is required for implementing the learning procedure and we discussed how to compute the derivative from noisy data.