  In this article, we present a data-driven method for parametric models with
noisy observation data. Gaussian process regression based reduced order
modeling (GPR-based ROM) can realize fast online predictions without using
equations in the offline stage. However, GPR-based ROM does not perform well
for complex systems since POD projection are naturally linear. Conditional
variational autoencoder (CVAE) can address this issue via nonlinear neural
networks but it has more model complexity, which poses challenges for training
and tuning hyperparameters. To this end, we propose a framework of CVAE with
Gaussian process regression recognition (CVAE-GPRR). The proposed method
consists of a recognition model and a likelihood model. In the recognition
model, we first extract low-dimensional features from data by POD to filter the
redundant information with high frequency. And then a non-parametric model GPR
is used to learn the map from parameters to POD latent variables, which can
also alleviate the impact of noise. CVAE-GPRR can achieve the similar accuracy
to CVAE but with fewer parameters. In the likelihood model, neural networks are
used to reconstruct data. Besides the samples of POD latent variables and input
parameters, physical variables are also added as the inputs to make predictions
in the whole physical space. This can not be achieved by either GPR-based ROM
or CVAE. Moreover, the numerical results show that CVAE-GPRR may alleviate the
overfitting issue in CVAE.
