  Physics-informed neural networks (PINNs) are increasingly employed to
replace/augment traditional numerical methods in solving partial differential
equations (PDEs). While having many attractive features, state-of-the-art PINNs
surrogate a specific realization of a PDE system and hence are
problem-specific. That is, each time the boundary conditions and domain shape
change, the model needs to be re-trained. This limitation prohibits the
application of PINNs in realistic or large-scale engineering problems
especially since the costs and efforts associated with their training are
considerable.
  This paper introduces a transferable framework for solving boundary value
problems (BVPs) via deep neural networks which can be trained once and used
forever for various domains of unseen sizes, shapes, and boundary conditions.
First, we introduce \emph{genomic flow network} (GFNet), a neural network that
can infer the solution of a BVP across arbitrary boundary conditions on a small
square domain called \emph{genome}. Then, we propose \emph{mosaic flow} (MF)
predictor, a novel iterative algorithm that assembles or stitches the GFNet's
inferences to obtain the solution of BVPs on unseen, large domains while
preserving the spatial regularity of the solution. We demonstrate that our
framework can estimate the solution of Laplace and Navier-Stokes equations in
domains of unseen shapes and boundary conditions that are, respectively, $1200$
and $12$ times larger than the domains where training is performed. Since our
framework eliminates the need to re-train, it demonstrates up to 3 orders of
magnitude speedups compared to the state-of-the-art.
