  Many real-world applications demand accurate and fast predictions, as well as
reliable uncertainty estimates. However, quantifying uncertainty on
high-dimensional predictions is still a severely under-invested problem,
especially when input-output relationships are non-linear. To handle this
problem, the present work introduces an innovative approach that combines
autoencoder deep neural networks with the probabilistic regression capabilities
of Gaussian processes. The autoencoder provides a low-dimensional
representation of the solution space, while the Gaussian process is a Bayesian
method that provides a probabilistic mapping between the low-dimensional inputs
and outputs. We validate the proposed framework for its application to
surrogate modeling of non-linear finite element simulations. Our findings
highlight that the proposed framework is computationally efficient as well as
accurate in predicting non-linear deformations of solid bodies subjected to
external forces, all the while providing insightful uncertainty assessments.
