|The Schedule of the Student Form|
|Day1. 2021.4.24 (Saturday)|
|09:00-10:00||Prof. Perdikaris||Learning the Solution Operator of Parametric Partial Differential Equations with Physics-informed DeepONets|
|10:10-10:40||Keunsu Kim||Introduction to Persistent Homology|
|10:50-11:20||Yantong Xie||A Structure Preserving Numerical Scheme for Fokker-Planck Equations of Neuron Networks: Numrical Analysis and Exploration|
|13:40-14:10||Hyomin Shin||Introduction to Variational Gaussian Process|
|14:20-14:50||Zeyu Jin||High Order Numerical Homogenization for Dissipative Ordinary Differential Equations|
|15:00-15:30||Eunsuh Kim||Solving Forward and Inverse Problems of Partial Differential Equations with Physics Informed Neural Network|
|Day2. 2021.4.25 (Sunday)|
|09:00-10:00||Prof. Zhou||Introduction to Uncertainty Quantification|
|10:10-10:40||Jeahan Jung||Introduction to Bayesian Deep Learning|
|10:50-11:20||Yixiao Lu||A Two-fluid Model for Plasma with Prandtl Number Correction|
Time: 9:00 am, April 24
Title: Learning the Solution Operator of Parametric Partial Differential Equations with Physics-informed DeepONets
Abstract : Deep operator networks (DeepONets) are receiving increased attention thanks to their demonstrated capability to approximate nonlinear operators between infinite-dimensional Banach spaces. However, despite their remarkable early promise, they typically require large training data-sets consisting of paired input-output observations which may be expensive to obtain, while their predictions may not be consistent with the underlying physical principles that generated the observed data. In this work, we propose a novel model class coined as physics-informed DeepONets, which introduces an effective regularization mechanism for biasing the outputs of DeepOnet models towards ensuring physical consistency. This is accomplished by leveraging automatic differentiation to impose the underlying physical laws via soft penalty constraints during model training. We demonstrate that this simple, yet remarkably effective extension can not only yield a significant improvement in the predictive accuracy of DeepOnets, but also greatly reduce the need for large training data-sets. To this end, a remarkable observation is that physics-informed DeepONets are capable of solving parametric partial differential equations (PDEs) without any paired input-output observations, except for a set of given initial or boundary conditions. We illustrate the effectiveness of the proposed framework through a series of comprehensive numerical studies across various types of PDEs. Strikingly, a trained physics informed DeepOnet model can predict the solution of O(1000) time-dependent PDEs in a fraction of a second — up to three orders of magnitude faster compared to a conventional PDE solver.
Time: 9:00 am, April 25
Title: Introduction to Uncertainty Quantification
Abstract : Uncertainty quantification (UQ) has been a hot research topic recently. UQ has a variety of applications, including hydrology, fluid mechanics, data assimilation, and weather forecasting. Among others, high order numerical methods have become one of the most important tools for UQ; and the relevant computational techniques and their mathematical theory have attracted great attention in recent years. This talk will present a brief introduction to some recent developments on high order algorithms for both forward UQ and inverse UQ.
Tao Zhou is currently an Associate Professor in Chinese Academy of Sciences. Before joining CAS, he was a postdoc fellow in EPFL in Switzerland during 2011-2012. Dr. Zhou’s research interests include Uncertainty Quantification (UQ), Parallel-in-Time Algorithms, Spectral Methods and Stochastic Optimal Control. He has published more than 50 papers in top international journals such as SIAM Review, SINUM and JCP. He was a recipient of the NSFC Career Award for Excellent Young Scholars (2018) and CSIAM Excellent Young Scholar Prize (2016). Dr. Zhou serves as Associate Editors for many international journals, such as SIAM Journal on Scientific Computing (SISC) and Communications in Computational Physics (CiCP). He also serves as the Associate Editor-in-Chief of International Journal for UQ. Since 2018, he has been the Chief Scientist of Science Challenge Project on UQ supported by State Administration of Science, Technology and Industry for National Defense.