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VAE-KRnet and its Applications to Variational Bayes

Communications in Computational Physics 2022年第4期31卷 1049-1082页

作者： Xiaoliang Wan Shuangqing Wei Department of Mathematics Center for Computation and TechnologyLouisiana State UniversityBaton Rouge 70803USA Division of Electrical&Computer Engineering Louisiana State UniversityBaton Rouge 70803USA

In this work,we have proposed a generative model,called VAE-KRnet,for density estimation or approximation,which combines the canonical variational autoencoder(VAE)with our recently developed flow-based generativemodel,called KRnet.VAE is used as a dimension reduction technique to capture the latent space,and KRnet is used to model the distribution of the latent variable.Using a linear model between the data and the latent variable,we show that VAE-KRnet can be more effective and robust than the canonical VAE.VAE-KRnet can be used as a density model to approximate either data distribution or an arbitrary probability density function(PDF)known up to a constant.VAE-KRnet is flexible in terms of dimensionality.When the number of dimensions is relatively small,KRnet can effectively approximate the distribution in terms of the original random variable.For high-dimensional cases,we may use VAE-KRnet to incorporate dimension reduction.One important application of VAE-KRnet is the variational Bayes for the approximation of the posterior distribution.The variational Bayes approaches are usually based on the minimization of the Kullback-Leibler(KL)divergence between the model and the posterior.For highdimensional distributions,it is very challenging to construct an accurate densitymodel due to the curse of dimensionality,where extra assumptions are often introduced for efficiency.For instance,the classical mean-field approach assumes mutual independence between dimensions,which often yields an underestimated variance due to oversimplification.To alleviate this issue,we include into the loss the maximization of the mutual information between the latent random variable and the original random variable,which helps keep more information from the region of low density such that the estimation of variance is improved.Numerical experiments have been presented to demonstrate the effectiveness of our model.

关键词： Deep learning variational Bayes uncertainty quantification Bayesian inverse problems generative modeling

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AN ACCELERATION STRATEGY FOR RANDOMIZE-THEN-OPTIMIZE SAMPLING VIA DEEP NEURAL NETWORKS

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Journal of Computational Mathematics 2021年第6期39卷 848-864页

作者： Liang Yan Tao Zhou School of Mathematics Southeast UniversityNanjing210096China Nanjing Center for Applied Mathematics Nanjing211135China LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

Randomize-then-optimize (RTO) is widely used for sampling from posterior distributions in Bayesian inverse problems. However, RTO can be computationally intensive forcomplexity problems due to repetitive evaluations of the expensive forward model and itsgradient. In this work, we present a novel goal-oriented deep neural networks (DNN) surrogate approach to substantially reduce the computation burden of RTO. In particular,we propose to drawn the training points for the DNN-surrogate from a local approximatedposterior distribution – yielding a flexible and efficient sampling algorithm that convergesto the direct RTO approach. We present a bayesian inverse problem governed by ellipticPDEs to demonstrate the computational accuracy and efficiency of our DNN-RTO approach, which shows that DNN-RTO can significantly outperform the traditional RTO.

关键词： Bayesian inverse problems Deep neural network Markov chain Monte Carlo

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