  We propose a novel composite framework to find unknown fields in the context
of inverse problems for partial differential equations (PDEs). We blend the
high expressibility of deep neural networks as universal function estimators
with the accuracy and reliability of existing numerical algorithms for partial
differential equations as custom layers in semantic autoencoders. Our design
brings together techniques of computational mathematics, machine learning and
pattern recognition under one umbrella to incorporate domain-specific knowledge
and physical constraints to discover the underlying hidden fields. The network
is explicitly aware of the governing physics through a hard-coded PDE solver
layer in contrast to most existing methods that incorporate the governing
equations in the loss function or rely on trainable convolutional layers to
discover proper discretizations from data. This subsequently focuses the
computational load to only the discovery of the hidden fields and therefore is
more data efficient. We call this architecture Blended inverse-PDE networks
(hereby dubbed BiPDE networks) and demonstrate its applicability for recovering
the variable diffusion coefficient in Poisson problems in one and two spatial
dimensions, as well as the diffusion coefficient in the time-dependent and
nonlinear Burgers' equation in one dimension. We also show that this approach
is robust to noise.
