Reliability analysis and risk assessment for complex physical and
engineering systems governed by partial differential equations (PDEs)
are computationally intensive, especially when high-dimensional random
parameters are involved. Since standard numerical schemes for solving
these complex PDEs are expensive, traditional Monte Carlo methods
which require repeatedly solving PDEs are infeasible. Alternative
approaches which are typically the surrogate based methods suffer from
the so-called ``curse of dimensionality'', which limits their
application to problems with high-dimensional parameters. The purpose
of this mini-symposium is to bring researchers from different fields
to discuss the recent machine learning methods for such problems,
focusing on both novel machine learning surrogates and alternative
Monte Carlo methods.
14:00
Generative Model based on Winslow Mapping for Sampling unnormalised Distribution
Xiang Zhou | City University of Hong Kong | Hong Kong
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Author:
Xiang Zhou | City University of Hong Kong | Hong Kong
As an alternative to MCMC, the generative model transforms a simple distribution to a complicated target distribution and becomes increasingly popular due to the advance of deep neural network (such as the Knothe-Rosenblatt mapping in NICE and NVP works) to represent the high dim map. The principle is to minimise the discrepancy (such as KL divergence in variational inference or the Wasserstein distance in Monge-Ampere flow) between the target density and the transformed density. Variational structure is critically beneficial in this context for the implementation by machine learning techniques. In this work, our generative model is based on the harmonic map which is the foundation of many traditional moving mesh methods in adaptive numerical method for PDE. Our approach fully enjoys the variational formulation associated with the so-called Winslow’s energy. With a simple choice of the monitor function in the Winslow’s energy, the induced density is already close to the target density in many cases and thus serves a good initial guess for the further refining in the same framework of variational inference. Furthermore, the functional derivative to iteratively refine the monitor function also enjoys a variational formulation and thus by learning the map and gradient simultaneously, we have a robust and efficient descent algorithm. Numerical examples are included in the talk to demonstrate our ideas.
14:30
- CANCELED - Deep density estimation for Fokker-Planck equations using flow-based generative model
Kejun Tang | ShanghaiTech University | China
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Author:
Kejun Tang | ShanghaiTech University | China
In this work, we proposed a general flow-based generative model based onKnothe-Rosenblatt rearrangement, and we study the performance and applications of flow-based generative models subject to an invertible mapping. We apply this generative model to solve the Fokker-Planck equation. In particular, an adaptive deep learning based algorithm is proposed for solving the Fokker-Planck equation. Unlike the standard deep learning based method for solving partial differential equations, this approach can provide an exact sampling instead of uniform sampling on the domain. Numerical experiments demonstrate the efficiency of our method.
15:00
Proper orthogonal decomposition method for multiscale elliptic PDEs with random coefficients
Dingjiong Ma | University of Hong Kong | Hong Kong
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Authors:
Dingjiong Ma | University of Hong Kong | Hong Kong
Zhiwen Zhang | University of Hong Kong | Hong Kong
In this talk, we develop an efficient multiscale reduced basis method to solve elliptic PDEs with multiscale and random coefficients in a multi-query setting. Our method consists of offline and online stages. In the offline stage, a small number of reduced multiscale basis functions are constructed within each coarse grid block using the proper orthogonal decomposition (POD) method. Moreover, local tensor spaces are defined to approximate the solution space of the multiscale random PDEs. In the online stage, a weak formulation is derived and discretized using the Galerkin method to compute the solution. Since the multiscale reduced basis functions can efficiently approximate the high-dimensional solution space, our method is very efficient in solving multiscale elliptic PDEs with random coefficients. Convergence analysis of the proposed method is presented, which shows the dependence of the numerical error on the number of snapshots and the truncation threshold in the POD method. Finally, numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale problems with or without scale separation in the physical space.
15:30
- CANCELED - Domain decomposed uncertainty analysis based on RealNVP
Qifeng Liao | ShanghaiTech University | China
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Author:
Qifeng Liao | ShanghaiTech University | China
The domain decomposition uncertainty quantification method (DDUQ)
(SIAM J. SCI. COMPUT (37) pp. A103-A133) provides a decomposed strategy
to conduct uncertainty analysis for complex engineering systems governed by PDEs.
In DDUQ, uncertainty analysis on each local component is independently conducted in
an ``offline" phase, and global uncertainty analysis results are assembled using precomputed
local information in an ``online" phase through importance sampling.
The performance of DDUQ relies on the coupling surrogates and probability density estimation
during the importance sampling procedure.
Since coupling surrogates can give high-dimensional interface parameters,
we in this work develop a RealNVP based interface coupling strategy, which dramatically
improve the efficiency of DDUQ.