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

2025/02/12 by Gian-Luca Geuken, Patrick Kurzeja, Geuken, Gian-Luca +5 · 6 citations
Engineering · Physics and Astronomy · #68T07 #74B20 #Aeroelasticity and Vibration Control #Artificial Intelligence (cs.AI) #Computational Engineering #Elasticity and Material Modeling #FOS: Computer and information sciences #Finance #I.2.1 #J.2 #Model Reduction and Neural Networks #and Science (cs.CE)

paper · doi:10.48550/arxiv.2502.08534

openalex publication_date 2025/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• A physics-augmented neural network framework for finite strain incompressible viscoelasticity
• A Complement to Neural Networks for Anisotropic Inelasticity at Finite Strains
• COMMET: orders-of-magnitude speed-up in finite element method via batch-vectorized neural constitutive updates
• Unsupervised Material Fingerprinting: Ultra-fast hyperelastic model discovery from full-field experimental measurements
• Hetero-EUCLID: Interpretable model discovery for heterogeneous hyperelastic materials using stress-unsupervised learning
• In search of constitutive conditions in isotropic hyperelasticity: polyconvexity versus true-stress-true-strain monotonicity

Related

• Parametrised polyconvex hyperelasticity with physics-augmented neural networks
• Neural Network Enhanced Polyconvexification of Isotropic Energy Densities in Computational Mechanics
• Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)
• Input Specific Neural Networks
• Polyconvex neural network models of thermoelasticity