  Surrogate modeling and uncertainty quantification tasks for PDE systems are
most often considered as supervised learning problems where input and output
data pairs are used for training. The construction of such emulators is by
definition a small data problem which poses challenges to deep learning
approaches that have been developed to operate in the big data regime. Even in
cases where such models have been shown to have good predictive capability in
high dimensions, they fail to address constraints in the data implied by the
PDE model. This paper provides a methodology that incorporates the governing
equations of the physical model in the loss/likelihood functions. The resulting
physics-constrained, deep learning models are trained without any labeled data
(e.g. employing only input data) and provide comparable predictive responses
with data-driven models while obeying the constraints of the problem at hand.
This work employs a convolutional encoder-decoder neural network approach as
well as a conditional flow-based generative model for the solution of PDEs,
surrogate model construction, and uncertainty quantification tasks. The
methodology is posed as a minimization problem of the reverse Kullback-Leibler
(KL) divergence between the model predictive density and the reference
conditional density, where the later is defined as the Boltzmann-Gibbs
distribution at a given inverse temperature with the underlying potential
relating to the PDE system of interest. The generalization capability of these
models to out-of-distribution input is considered. Quantification and
interpretation of the predictive uncertainty is provided for a number of
problems.
