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Partial-differential-algebraic equations of nonlinear dynamics by Physics-Informed Neural-Network: (I) Operator splitting and framework assessment

Loc Vu-Quoc, Alexander Humer

Abstract

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.

Partial-differential-algebraic equations of nonlinear dynamics by Physics-Informed Neural-Network: (I) Operator splitting and framework assessment

Abstract

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.
Paper Structure (30 sections, 111 equations, 52 figures, 2 tables)

This paper contains 30 sections, 111 equations, 52 figures, 2 tables.

Table of Contents

  1. Introduction
  2. PDAEs: Beam formulations with no shear
  3. Geometrically-exact beam with no shear, Kirchhoff rod
  4. Kinematics, finite extension, constitutive relations
  5. Balance of momenta, non-dimensional form
  6. Constitutive relations, non-dimensional form
  7. Equations of motion, non-dimensional form
  8. Input parameters: Boundary and initial conditions, distributed load
  9. Inconsistency, reduction from geometrically-exact beam with shear
  10. Small deformation, Euler-Bernoulli beam
  11. PINN formulations
  12. Inputs
  13. Form 1: No operator splitting
  14. Form 2a: Split time derivatives
  15. Form 2b: Split space derivatives
  16. ...and 15 more sections

Figures (52)

  • Figure 1: Geometrically-exact beam without shear deformation. Section \ref{['sc:Kirchhoff-rod-derivation']}. Shear forces introduced for equilibrium. Section \ref{['sc:inconsistency-Kirchhoff-rod-Euler-Bernoulli-beam']}: Inconsistency in Kirchhoff rod (large deformation) and Euler-Bernoulli (small deformation) beam theories. Figure \ref{['fig:geom-exact-with-shear']}, geometrically-exact beam with shear deformation.
  • Figure 2: Geometrically-exact beam with shear deformation. The deformed configuration (solid line) is superposed on the initial configuration (dotted line) of unit length (which could be multiplied by $dX$). Shear deformation is the difference between the angle $\alpha$ of the deformed centroidal line and the rotation $\theta$ of the cross section. Spatial strains $\{ {\gamma} _{1}{} , {\gamma} _{2}{}\}$ and material strains $\{ {\Gamma} _{1}{} , {\Gamma} _{2}{}\}$, such that $\boldsymbol{\gamma} = {\gamma} _{1}{} {{\hbox{\boldmath $e$}}} _{1}{} + {\gamma} _{2}{} {{\hbox{\boldmath $e$}}} _{2}{} = {\Gamma} _{1}{} {{\hbox{\boldmath $t$}}} _{1}{} + {\Gamma} _{2}{} {{\hbox{\boldmath $t$}}} _{2}{}$. See simo1986dynamicsI and Figure \ref{['fig:geom-exact-no-shear']} for geometrically-exact beam with no shear.
  • Figure 3: https://deepxde.readthedocs.io/en/latest/-https://deepxde.readthedocs.io/en/latest/user/installation.html?highlight=backend. Pinned-pinned elastic bar, axial motion. Model summary. Feedforward neural network (fnn). Remark \ref{['rm:network-params']}. Section \ref{['sc:axial-Form-1']}, Form 1. Dense-connection layers, each between two consecutive layers of neurons. Network: Remark \ref{['rm:parameter-names']}, n_inp=2, W=64, H=4, n_out=1, 12737 parameters. Six neuron layers (1 input, 4 hidden, 1 output), five connection layers (pairs of consecutive neuron layers). (23721R1-1)
  • Figure 4: Optimization learning-rate scheduling. Section \ref{['sc:optimization-learning-rate-scheduling']}. Cycles and periods.
  • Figure 5: Axial motion of elastic bar. Section \ref{['sc:axial-motion']}. Mathematica solutions. Eq. \ref{['eq:eom-euler-bernoulli-axial']} with slenderness $\mathcal{s} = 1$ and distributed load ${ \overline{f} {}} _{X}{} = 1/2$. Two vibration periods for each set of boundary conditions. Left: Pinned-pinned bar, with $\overline{X} {} \in \left[0, 1\right]$ and $\overline{t} {} \in \left[0, 4\right]$. Right: Pinned-free bar, with $\overline{X} {} \in \left[0, 1\right]$ and $\overline{t} {} \in \left[0, 8\right]$.
  • ...and 47 more figures

Theorems & Definitions (32)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 5.1
  • Remark 5.2
  • Remark 5.3
  • Remark 5.4
  • Remark 5.5
  • ...and 22 more