# VAE-KRnet and its applications to variational Bayes

arXiv: https://arxiv.org/abs/2006.16431 · HTML: https://mlsanity.com/paper/2006.16431/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jun 2020.

## Built on by (most cited first, 11 of 11)

- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elements](https://mlsanity.com/paper/2110.14121/) (2021)
- [Numerical Computation of Partial Differential Equations by Hidden-Layer Concatenated Extreme Learning Machine](https://mlsanity.com/paper/2204.11375/) (2022)
- [Bounded KRnet and its applications to density estimation and approximation](https://mlsanity.com/paper/2305.09063/) (2023)
- [Weak Generative Sampler to Efficiently Sample Invariant Distribution of Stochastic Differential Equation](https://mlsanity.com/paper/2405.19256/) (2024)
- [Estimating Committor Functions via Deep Adaptive Sampling on Rare Transition Paths](https://mlsanity.com/paper/2501.15522/) (2025)
- [Pathway to O(√d) Complexity bound under Wasserstein metric of flow-based models](https://mlsanity.com/paper/2512.06702/) (2025)
- [Moving Sampling Physics-informed Neural Networks induced by Moving Mesh PDE](https://mlsanity.com/paper/2311.16167/) (2023)
- [Dimension-reduced KRnet maps for high-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2303.00573/) (2023)
- [Error Analysis of Physics-Informed Neural Networks for Approximating Dynamic PDEs of Second Order in Time](https://mlsanity.com/paper/2303.12245/) (2023)
- [Error Analysis and Numerical Algorithm for PDE Approximation with Hidden-Layer Concatenated Physics Informed Neural Networks](https://mlsanity.com/paper/2406.06350/) (2024)

## Abstract (arXiv)

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 generative model, 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 high-dimensional distributions, it is very challenging to construct an accurate density model 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.