Stacked tensorial neural networks for reduced-order modeling of a
  parametric partial differential equation
Caleb G. Wagner
Tensorial neural networks (TNNs) combine the successes of multilinear algebra
with those of deep learning to enable extremely efficient reduced-order models
of high-dimensional problems. Here, I describe a deep neural network
architecture that fuses multiple TNNs into a larger network, intended to solve
a broader class of problems than a single TNN. I evaluate this architecture,
referred to as a "stacked tensorial neural network" (STNN), on a parametric PDE
with three independent variables and three parameters. The three parameters
correspond to one PDE coefficient and two quantities describing the domain
geometry. The STNN provides an accurate reduced-order description of the
solution manifold over a wide range of parameters. There is also evidence of
meaningful generalization to parameter values outside its training data.
Finally, while the STNN architecture is relatively simple and problem agnostic,
it can be regularized to incorporate problem-specific features like symmetries
and physical modeling assumptions.