  Many-query problems, arising from uncertainty quantification, Bayesian
inversion, Bayesian optimal experimental design, and optimization under
uncertainty-require numerous evaluations of a parameter-to-output map. These
evaluations become prohibitive if this parametric map is high-dimensional and
involves expensive solution of partial differential equations (PDEs). To tackle
this challenge, we propose to construct surrogates for high-dimensional
PDE-governed parametric maps in the form of projected neural networks that
parsimoniously capture the geometry and intrinsic low-dimensionality of these
maps. Specifically, we compute Jacobians of these PDE-based maps, and project
the high-dimensional parameters onto a low-dimensional derivative-informed
active subspace; we also project the possibly high-dimensional outputs onto
their principal subspace. This exploits the fact that many high-dimensional
PDE-governed parametric maps can be well-approximated in low-dimensional
parameter and output subspace. We use the projection basis vectors in the
active subspace as well as the principal output subspace to construct the
weights for the first and last layers of the neural network, respectively. This
frees us to train the weights in only the low-dimensional layers of the neural
network. The architecture of the resulting neural network captures to first
order, the low-dimensional structure and geometry of the parametric map. We
demonstrate that the proposed projected neural network achieves greater
generalization accuracy than a full neural network, especially in the limited
training data regime afforded by expensive PDE-based parametric maps. Moreover,
we show that the number of degrees of freedom of the inner layers of the
projected network is independent of the parameter and output dimensions, and
high accuracy can be achieved with weight dimension independent of the
discretization dimension.
