Quantifying the uncertainty of quantities of interest (QoIs) from physical systems is a primary objective in model validation. However, achieving this goal entails balancing the need for computational efficiency with the requirement for numerical accuracy. To address this trade-off, we propose a novel bi-fidelity formulation of variational auto-encoders (BF-VAE) designed to estimate the uncertainty associated with a QoI from low-fidelity (LF) and high-fidelity (HF) samples of the QoI. This model allows for the approximation of the statistics of the HF QoI by leveraging information derived from its LF counterpart. Specifically, we design a bi-fidelity auto-regressive model in the latent space that is integrated within the VAE's probabilistic encoder-decoder structure. An effective algorithm is proposed to maximize the variational lower bound of the HF log-likelihood in the presence of limited HF data, resulting in the synthesis of HF realizations with a reduced computational cost. Additionally, we introduce the concept of the bi-fidelity information bottleneck (BF-IB) to provide an information-theoretic interpretation of the proposed BF-VAE model. Our numerical results demonstrate that BF-VAE leads to considerably improved accuracy, as compared to a VAE trained using only HF data when limited HF data is available.

翻译：物理系统中感兴趣量（QoIs）的不确定性量化是模型验证的主要目标。然而，实现这一目标需要在计算效率与数值精度之间寻求平衡。为解决这一权衡问题，我们提出了一种新颖的双保真变分自编码器（BF-VAE）公式，旨在通过QoI的低保真（LF）与低保真（HF）样本估计其相关不确定性。该模型通过利用低保真对应信息近似低保真QoI的统计特性。具体而言，我们在潜在空间中设计了一种集成于VAE概率编码器-解码器结构内的双保真自回归模型。针对低保真数据有限的情况，提出了一种最大化低保真对数似然变分下界的有效算法，从而以较低计算成本合成低保真实例。此外，我们引入双保真信息瓶颈（BF-IB）概念，为所提BF-VAE模型提供信息论解释。数值结果表明，在低保真数据有限的情况下，与仅使用低保真数据训练的VAE相比，BF-VAE显著提升了精度。