  In this paper, we propose a refinement strategy to the well-known
Physics-Informed Neural Networks (PINNs) for solving partial differential
equations (PDEs) based on the concept of Optimal Transport (OT).
  Conventional black-box PINNs solvers have been found to suffer from a host of
issues: spectral bias in fully-connected architectures, unstable gradient
pathologies, as well as difficulties with convergence and accuracy.
  Current network training strategies are agnostic to dimension sizes and rely
on the availability of powerful computing resources to optimize through a large
number of collocation points.
  This is particularly challenging when studying stochastic dynamical systems
with the Fokker-Planck-Kolmogorov Equation (FPKE), a second-order PDE which is
typically solved in high-dimensional state space.
  While we focus exclusively on the stationary form of the FPKE, positivity and
normalization constraints on its solution make it all the more unfavorable to
solve directly using standard PINNs approaches.
  To mitigate the above challenges, we present a novel training strategy for
solving the FPKE using OT-based sampling to supplement the existing PINNs
framework.
  It is an iterative approach that induces a network trained on a small dataset
to add samples to its training dataset from regions where it nominally makes
the most error.
  The new samples are found by solving a linear programming problem at every
iteration.
  The paper is complemented by an experimental evaluation of the proposed
method showing its applicability on a variety of stochastic systems with
nonlinear dynamics.
