Partial-differential-algebraic equations of nonlinear dynamics by physics-informed neural-network: (I) Operator splitting and framework assessment

Several forms for constructing novel physics-informed neural-networks (PINNs) for the solution of partial-differential-algebraic equations (PDAEs) based on derivative operator splitting are proposed, using the nonlinear Kirchhoff rod as a prototype for demonstration. The present work is a natural extension of our review paper (Vu-Quoc and Humer, CMES-Comput Modeling Eng Sci, 137(2):1069–1343, 2023) aiming at both experts and first-time learners of both deep learning and PINN frameworks, among which the open-source DeepXDE (DDE; SIAM Rev, 63(1):208–228, 2021) is likely the most well documented framework with many examples. Yet, we encountered some pathological problems (time shift, amplification, static solutions) 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 (e.g., displacements, slope, finite extension) to higher-level form with more dependent variables (e.g., forces, moments, momenta), 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 also developed a script based on JAX, the High Performance Array Computing library. For the axial motion of elastic bar, while our JAX script did not show the pathological problems of DDE-T (DDE with TensorFlow backend), it is slower than DDE-T. Moreover, 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 (network training) through a normalization/standardization of the network-training process so readers can reproduce our results.