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Input convex neural networks: universal approximation theorem and implementation for isotropic polyconvex hyperelastic energies

Gian-Luca Geuken, Patrick Kurzeja, David Wiedemann, Jörn Mosler

TL;DR

The paper tackles learning isotropic hyperelastic energies under stringent physical and mathematical constraints, notably frame-indifference and polyconvexity, while retaining universal approximation capabilities. It introduces the Convex Signed Singular Value Neural Network (CSSV-NN), which operates on signed singular values and enforces Pi_3-invariance by averaging over permutations, with an ICNN-based convex architecture ensuring polyconvexity. A universal approximation theorem is proved for frame-indifferent, isotropic polyconvex energies, built on convex-ICNN results extended to singular-value representations. Numerical results demonstrate CSSV-NN’s superior accuracy for both polyconvex and non-polyconvex energies and its ability to approximate polyconvex hulls, including regimes with linear energy growth, underscoring its potential as a physics-informed surrogate for data-driven constitutive modeling.

Abstract

This paper presents a novel framework of neural networks for isotropic hyperelasticity that enforces necessary physical and mathematical constraints while simultaneously satisfying the universal approximation theorem. The two key ingredients are an input convex network architecture and a formulation in the elementary polynomials of the signed singular values of the deformation gradient. In line with previously published networks, it can rigorously capture frame-indifference and polyconvexity - as well as further constraints like balance of angular momentum and growth conditions. However and in contrast to previous networks, a universal approximation theorem for the proposed approach is proven. To be more explicit, the proposed network can approximate any frame-indifferent, isotropic polyconvex energy (provided the network is large enough). This is possible by working with a sufficient and necessary criterion for frame-indifferent, isotropic polyconvex functions. Comparative studies with existing approaches identify the advantages of the proposed method, particularly in approximating non-polyconvex energies as well as computing polyconvex hulls.

Input convex neural networks: universal approximation theorem and implementation for isotropic polyconvex hyperelastic energies

TL;DR

The paper tackles learning isotropic hyperelastic energies under stringent physical and mathematical constraints, notably frame-indifference and polyconvexity, while retaining universal approximation capabilities. It introduces the Convex Signed Singular Value Neural Network (CSSV-NN), which operates on signed singular values and enforces Pi_3-invariance by averaging over permutations, with an ICNN-based convex architecture ensuring polyconvexity. A universal approximation theorem is proved for frame-indifferent, isotropic polyconvex energies, built on convex-ICNN results extended to singular-value representations. Numerical results demonstrate CSSV-NN’s superior accuracy for both polyconvex and non-polyconvex energies and its ability to approximate polyconvex hulls, including regimes with linear energy growth, underscoring its potential as a physics-informed surrogate for data-driven constitutive modeling.

Abstract

This paper presents a novel framework of neural networks for isotropic hyperelasticity that enforces necessary physical and mathematical constraints while simultaneously satisfying the universal approximation theorem. The two key ingredients are an input convex network architecture and a formulation in the elementary polynomials of the signed singular values of the deformation gradient. In line with previously published networks, it can rigorously capture frame-indifference and polyconvexity - as well as further constraints like balance of angular momentum and growth conditions. However and in contrast to previous networks, a universal approximation theorem for the proposed approach is proven. To be more explicit, the proposed network can approximate any frame-indifferent, isotropic polyconvex energy (provided the network is large enough). This is possible by working with a sufficient and necessary criterion for frame-indifferent, isotropic polyconvex functions. Comparative studies with existing approaches identify the advantages of the proposed method, particularly in approximating non-polyconvex energies as well as computing polyconvex hulls.

Paper Structure

This paper contains 19 sections, 3 theorems, 28 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation for physics-based neural networks in constitutive modeling
  3. Challenges of physical constraints for hyperelastic modeling
  4. Open challenge of a universal, frame-indifferent, isotropic polyconvex ANN approximation for hyperelasticity
  5. Designing frame-indifferent and polyconvex functions
  6. Approach 1: Anisotropic polyconvex hyperelasticity
  7. Approach 2: Isotropic polyconvex hyperelasticity based on invariants
  8. Approach 3: Isotropic polyconvex hyperelasticity based on signed singular values
  9. Physics-based convex signed singular value neural network for isotropic hyperelasticity
  10. Universal approximation theorem for frame-indifferent, isotropic polyconvex functions
  11. Algorithmic implementation
  12. Network architecture
  13. Sobolev training
  14. Approximation of polyconvex hulls
  15. Numerical results
  16. ...and 4 more sections

Key Result

Theorem 1

A $\Pi_3$-invariant function $\widetilde{\Psi}\colon \, \mathbb{R}^3 \rightarrow \mathbb{R}_\infty$ is singular value polyconvex (and thus $\Psi$ is polyconvex) if and only if there exists a lower semicontinuous and convex function $\overline{\Psi}\colon \mathbb{R}^7 \rightarrow \mathbb{R}_\infty$ f

Figures (5)

  • Figure 1: Illustration of the convex signed singular value neural network (CSSV-NN) framework.
  • Figure 2: Mean squared error of the training data set from each architecture and random initializations for the polyconvex signed singular value energy (left, Eq. \ref{['eq:ssvenergy']}) and the non-polyconvex Hencky energy (right, Eq. \ref{['eq:henckyenergy']}) after training.
  • Figure : Polyconvex signed singular value energy (Eq. \ref{['eq:ssvenergy']})
  • Figure : Non-polyconvex Hencky energy (Eq. \ref{['eq:henckyenergy']})
  • Figure : Approximation of a polyconvex hull (Eq. \ref{['eq:nematic energy']})

Theorems & Definitions (7)

  • Theorem 1: Singular value polyconvexity, cf. wiedemann2023
  • Remark 1
  • Lemma 2: Universal approximation theorem for convex functions
  • proof
  • Theorem 3: Universal approximation theorem for singular value polyconvex functions
  • proof
  • Remark 2