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Separable Physics-Informed Neural Networks for the solution of elasticity problems

Vasiliy A. Es'kin, Danil V. Davydov, Julia V. Gur'eva, Alexey O. Malkhanov, Mikhail E. Smorkalov

TL;DR

This work introduces separable physics-informed neural networks (SPINN) integrated with the deep energy method (DEM) to solve linear elasticity problems. By contrasting PDE-based PINN, SPINN-PDE, and SPINN-DEM across beam and thin-walled geometry tests, the authors show that SPINN-DEM delivers superior convergence speed and accuracy, enables solution on complex geometries, and approaches the efficiency of finite element methods while using far less GPU memory than vanilla PINN. The approach leverages a per-axis separable basis in SPINN and an energy-based loss in DEM to enforce constitutive relations and energy minimization, yielding smoother, physically consistent solutions. The results imply broad applicability to industrial-scale elasticity problems and potential extensions to other continua problems beyond the tested configurations.

Abstract

A method for solving elasticity problems based on separable physics-informed neural networks (SPINN) in conjunction with the deep energy method (DEM) is presented. Numerical experiments have been carried out for a number of problems showing that this method has a significantly higher convergence rate and accuracy than the vanilla physics-informed neural networks (PINN) and even SPINN based on a system of partial differential equations (PDEs). In addition, using the SPINN in the framework of DEM approach it is possible to solve problems of the linear theory of elasticity on complex geometries, which is unachievable with the help of PINNs in frames of partial differential equations. Considered problems are very close to the industrial problems in terms of geometry, loading, and material parameters.

Separable Physics-Informed Neural Networks for the solution of elasticity problems

TL;DR

This work introduces separable physics-informed neural networks (SPINN) integrated with the deep energy method (DEM) to solve linear elasticity problems. By contrasting PDE-based PINN, SPINN-PDE, and SPINN-DEM across beam and thin-walled geometry tests, the authors show that SPINN-DEM delivers superior convergence speed and accuracy, enables solution on complex geometries, and approaches the efficiency of finite element methods while using far less GPU memory than vanilla PINN. The approach leverages a per-axis separable basis in SPINN and an energy-based loss in DEM to enforce constitutive relations and energy minimization, yielding smoother, physically consistent solutions. The results imply broad applicability to industrial-scale elasticity problems and potential extensions to other continua problems beyond the tested configurations.

Abstract

A method for solving elasticity problems based on separable physics-informed neural networks (SPINN) in conjunction with the deep energy method (DEM) is presented. Numerical experiments have been carried out for a number of problems showing that this method has a significantly higher convergence rate and accuracy than the vanilla physics-informed neural networks (PINN) and even SPINN based on a system of partial differential equations (PDEs). In addition, using the SPINN in the framework of DEM approach it is possible to solve problems of the linear theory of elasticity on complex geometries, which is unachievable with the help of PINNs in frames of partial differential equations. Considered problems are very close to the industrial problems in terms of geometry, loading, and material parameters.
Paper Structure (20 sections, 28 equations, 6 figures, 8 tables)

This paper contains 20 sections, 28 equations, 6 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Statement of the Problem
  3. Formulation of the problem on the basis of partial differential equations
  4. Formulation of the problem on the basis of energy minimization
  5. Methods
  6. Physical-Informed approach
  7. Vanilla PINN
  8. Energy PINN
  9. Artificial Neural Network Architectures
  10. Multi-layer perceptron
  11. Separable PINN
  12. Numerical experiments
  13. Beam
  14. PINN
  15. SPINN in the case of the loss based on PDE
  16. ...and 5 more sections

Figures (6)

  • Figure 1: Schematic diagram of the general PINN method.
  • Figure 2: Schematic diagram of the deep energy method for the SPINN approach.
  • Figure 3: Beam with uniform load $\bf T$ in the gravity force $\bf g$
  • Figure 4: Beam problem: (a), (d), (g) and (j) are the predictions of a trained physics-informed neural network; (b), (e), (h) and (k) are reference solutions; (c), (f), (i) and (l) are absolute values of differences between reference and predicted solutions, for the value $u_x$, $u_y$, $u_z$ and $\sigma_{\rm vM}$, respectively.
  • Figure 5: Thin-walled angle with uniform load $\bf T$ in the gravity force $\bf g$
  • ...and 1 more figures