  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.
