改进的变分自编码器实现更精准的贝叶斯反问题不确定性量化。
Enhanced uncertainty quantification variational autoencoders for the solution of Bayesian inverse problems
- 设计新损失函数提升变分自编码器在贝叶斯反问题中的训练效果。
- 理论证明在前向映射为仿射时，隐变量收敛于参数后验分布。
- 在拉普拉斯方程上验证，精度与泛化性优于已有方法。
神经网络是实时求解确定性和贝叶斯反问题的强大工具，其中变分自编码器可通过观测数据实现实时贝叶斯参数估计及其分布推断，支持不确定性量化。本文在现有研究基础上，提出一种新型损失函数以训练用于贝叶斯反问题的变分自编码器。当前向映射为仿射时，理论上证明了变分自编码器的隐状态收敛于模型参数的后验分布。通过数值实验验证该理论结果，并在精度和泛化能力方面对比了所提方法与文献中已有方法。最后，在拉普拉斯方程上测试该方法，与原始变分自编码器及马尔可夫链蒙特卡洛方法进行比较。
原文摘要 · Abstract (English)
Among other uses, neural networks are a powerful tool for solving deterministic and Bayesian inverse problems in real-time, where variational autoencoders, a specialized type of neural network, enable the Bayesian estimation of model parameters and their distribution from observational data allowing real-time inverse uncertainty quantification. In this work, we build upon existing research [Goh, H. et al., Proceedings of Machine Learning Research, 2022] by proposing a novel loss function to train variational autoencoders for Bayesian inverse problems. When the forward map is affine, we provide a theoretical proof of the convergence of the latent states of variational autoencoders to the posterior distribution of the model parameters. We validate this theoretical result through numerical tests and we compare the proposed variational autoencoder with the existing one in the literature both in terms of accuracy and generalization properties. Finally, we test the proposed variational autoencoder on a Laplace equation, with comparison to the original one and Markov Chains Monte Carlo.
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