  Several forms for constructing novel physics-informed neural-networks (PINN)
for the solution of partial-differential-algebraic equations based on
derivative operator splitting are proposed, using the nonlinear Kirchhoff rod
as a prototype for demonstration. The open-source DeepXDE is likely the most
well documented framework with many examples. Yet, we encountered some
pathological problems and proposed novel methods to resolve them. Among these
novel methods are the PDE forms, which evolve from the lower-level form with
fewer unknown dependent variables to higher-level form with more dependent
variables, in addition to those from lower-level forms. Traditionally, the
highest-level form, the balance-of-momenta form, is the starting point for
(hand) deriving the lowest-level form through a tedious (and error prone)
process of successive substitutions. The next step in a finite element method
is to discretize the lowest-level form upon forming a weak form and
linearization with appropriate interpolation functions, followed by their
implementation in a code and testing. The time-consuming tedium in all of these
steps could be bypassed by applying the proposed novel PINN directly to the
highest-level form. We developed a script based on JAX. While our JAX script
did not show the pathological problems of DDE-T (DDE with TensorFlow backend),
it is slower than DDE-T. That DDE-T itself being more efficient in higher-level
form than in lower-level form makes working directly with higher-level form
even more attractive in addition to the advantages mentioned further above.
Since coming up with an appropriate learning-rate schedule for a good solution
is more art than science, we systematically codified in detail our experience
running optimization through a normalization/standardization of the
network-training process so readers can reproduce our results.
