During the past two decades, tremendous efforts have been made to understand and 

quantify the uncertainty in myriad  applications such as fluid and solid mechanics, 

electromagnetics, materials, experimental design, climate modeling, etc,  where there 

exist two fundamental hurdles to deal with: the slow convergence rate of Monte Carlo 

(MC) method and the curse of dimensionality, in addition to the large simulation cost 

for one realization when complex systems are considered. In this project, we have 

endeavored to capture rare events in infinite dimensional systems by integrating ideas 

from large deviation theory, computational mathematics and deep learning. 

We developed the first minimum action method for dynamical systems with time delays, 

which finds the most probable transition path from one state to another one. Dynamical 

systems with time delays find many applications in control problems. 

To capture rare events in problems modeled by partial differential equations, we 

successfully coupled reduced-order model with deep generative model to achieve  

an effective importance sampling estimator, which lays a foundation to apply the 

cross-entropy method together with the multi-fidelity strategy.  

To employ the Fokker-Planck approach to study rare events, we have developed 

KRnet, which is a normalizing flow model that can be used as an approximator for 

high-dimensional probability density functions (PDFs). We have successfully used KRnet to 

approximate high-dimensional Fokker-Planck equations, which cannot be afforded by tradition 

numerical methods such as the finite element method.

Using KRnet, we further developed deep adaptive sampling techniques that help reduce 

the statistical errors of neural network approximation of partial differential equations, 

which can significantly improve the accuracy of the surrogate model of a parametric partial 

differential equation such that sampling a complex system becomes much cheaper.  

Overall, with the help of deep learning techniques, we have developed promising algorithms 

to merge PDF approximation and sample generation, which is beyond traditional approaches. 

These algorithms will be helpful for the quantification of uncertainty in complex systems.

Last Modified: 11/30/2023
Modified by: Xiaoliang Wan