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Quantum-Inspired Solver for Simulating Material Deformations

Mazen Ali, Aser Cortines, Siddhartha Morales, Samuel Mugel, Mireia Olave, Roman Orus, Samuel Palmer, Hodei Usabiaga

TL;DR

This work introduces a quantum-inspired tensor-network (TN) solver for linear elasticity within the finite element method (FEM) framework, achieving exponential reductions in memory and computation by representing displacement, force, and system operators as tensor-train (TT) objects. The authors detail TT-based assembly for the Jacobian, stiffness operator, and force vector in 2D, and solve the resulting system with the AMEn TT solver, comparing results to classical FEM (FEniCS) on a cantilever beam to demonstrate accuracy and significant scaling advantages. Key contributions include a TT representation of the Jacobians and mass/rigidity operators, a TT-assembly workflow, and empirical evidence of logarithmic scaling with respect to degrees of freedom. The work lays a foundation for extending TN-FEM elasticity to 3D, preconditioning, and non-homogeneous or non-linear materials, with potential impact on large-scale material deformation simulations.

Abstract

This paper explores the application of tensor networks (TNs) to the simulation of material deformations within the framework of linear elasticity. Material simulations are essential computational tools extensively used in both academic research and industrial applications. TNs, originally developed in quantum mechanics, have recently shown promise in solving partial differential equations (PDEs) due to their potential for exponential speedups over classical algorithms. Our study successfully employs TNs to solve linear elasticity equations with billions of degrees of freedom, achieving exponential reductions in both memory usage and computational time. These results demonstrate the practical viability of TNs as a powerful classical backend for executing quantum-inspired algorithms with significant efficiency gains. This work is based on our research conducted with IKERLAN.

Quantum-Inspired Solver for Simulating Material Deformations

TL;DR

This work introduces a quantum-inspired tensor-network (TN) solver for linear elasticity within the finite element method (FEM) framework, achieving exponential reductions in memory and computation by representing displacement, force, and system operators as tensor-train (TT) objects. The authors detail TT-based assembly for the Jacobian, stiffness operator, and force vector in 2D, and solve the resulting system with the AMEn TT solver, comparing results to classical FEM (FEniCS) on a cantilever beam to demonstrate accuracy and significant scaling advantages. Key contributions include a TT representation of the Jacobians and mass/rigidity operators, a TT-assembly workflow, and empirical evidence of logarithmic scaling with respect to degrees of freedom. The work lays a foundation for extending TN-FEM elasticity to 3D, preconditioning, and non-homogeneous or non-linear materials, with potential impact on large-scale material deformation simulations.

Abstract

This paper explores the application of tensor networks (TNs) to the simulation of material deformations within the framework of linear elasticity. Material simulations are essential computational tools extensively used in both academic research and industrial applications. TNs, originally developed in quantum mechanics, have recently shown promise in solving partial differential equations (PDEs) due to their potential for exponential speedups over classical algorithms. Our study successfully employs TNs to solve linear elasticity equations with billions of degrees of freedom, achieving exponential reductions in both memory usage and computational time. These results demonstrate the practical viability of TNs as a powerful classical backend for executing quantum-inspired algorithms with significant efficiency gains. This work is based on our research conducted with IKERLAN.
Paper Structure (13 sections, 23 equations, 8 figures, 2 algorithms)

This paper contains 13 sections, 23 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Previous Work
  3. This Work
  4. Linear Elasticity
  5. Finite Element Methods
  6. Tensor Networks
  7. Methods: TN-FEM Elasticity Solver
  8. Jacobians
  9. Stiffness Operator
  10. Force Vector
  11. Algorithm
  12. Results
  13. Conclusion

Figures (8)

  • Figure 1: QR-decomposition of matrix $A$.
  • Figure 2: Tensor network diagrams.
  • Figure 3: Quadrilateral discretization in powers of 4. As an example we show a $4^2$ discretization, i.e., we have 16 grid points, including boundary points, 32 d.o.f. (16 for each vector field component) and 9 elements.
  • Figure 4: Deformation of the bar. For this plot, we used $4^8$ grid points.
  • Figure 5: The FEniCS and the TN-FEM solver relative error for the maximum displacement.
  • ...and 3 more figures