  Reduced order models based on the transport of a lower dimensional manifold
representation of the thermochemical state, such as Principal Component (PC)
transport and Machine Learning (ML) techniques, have been developed to reduce
the computational cost associated with the Direct Numerical Simulations (DNS)
of reactive flows. Both PC transport and ML normally require an abundance of
data to exhibit sufficient predictive accuracy, which might not be available
due to the prohibitive cost of DNS or experimental data acquisition. To
alleviate such difficulties, similar data from an existing dataset or domain
(source domain) can be used to train ML models, potentially resulting in
adequate predictions in the domain of interest (target domain). This study
presents a novel probabilistic transfer learning (TL) framework to enhance the
trust in ML models in correctly predicting the thermochemical state in a lower
dimensional manifold and a sparse data setting. The framework uses Bayesian
neural networks, and autoencoders, to reduce the dimensionality of the state
space and diffuse the knowledge from the source to the target domain. The new
framework is applied to one-dimensional freely-propagating flame solutions
under different data sparsity scenarios. The results reveal that there is an
optimal amount of knowledge to be transferred, which depends on the amount of
data available in the target domain and the similarity between the domains. TL
can reduce the reconstruction error by one order of magnitude for cases with
large sparsity. The new framework required 10 times less data for the target
domain to reproduce the same error as in the abundant data scenario.
Furthermore, comparisons with a state-of-the-art deterministic TL strategy show
that the probabilistic method can require four times less data to achieve the
same reconstruction error.
