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.

翻译：本文提出了一种用于各向同性超弹性的新型神经网络框架，该框架在满足通用逼近定理的同时，强制执行必要的物理和数学约束。其两个关键组成部分是输入凸网络架构以及基于变形梯度带符号奇异值基本多项式的表述形式。与先前发表的网络一致，该框架能够严格满足客观性（标架无差异性）和多凸性要求，同时满足角动量平衡和增长条件等进一步约束。然而，与先前网络不同的是，本文证明了所提方法具有通用逼近定理。更明确地说，所提出的网络能够逼近任何满足客观性、各向同性的多凸能量函数（前提是网络规模足够大）。这一特性是通过采用充分必要的客观性、各向同性多凸函数判定准则实现的。与现有方法的对比研究揭示了所提方法的优势，特别是在逼近非多凸能量及计算多凸包络方面。