新神经网络可精准模拟弹性材料能量，且能逼近任意物理合理函数。
Input convex neural networks: universal approximation theorem and implementation for isotropic polyconvex hyperelastic energies
- 基于输入凸网络与变形梯度奇异值多项式构建
- 证明了其可逼近任意各向同性多凸能量函数
- 适合需严格满足物理约束的材料建模研究
本文提出一种新型神经网络框架，用于各向同性超弹性材料建模，可在满足必要物理与数学约束的同时实现通用逼近。核心是输入凸网络结构与变形梯度奇异值符号多项式的表达方式。该方法严格满足坐标系无关性、多凸性，以及角动量守恒、生长条件等额外约束。与已有工作不同，本文首次证明了该方法的通用逼近定理：只要网络足够大，即可逼近任意坐标系无关、各向同性、多凸的能量函数。这一能力源于对这类函数充要条件的精确刻画。对比实验表明，该方法在逼近非多凸能量及计算多凸包方面具有显著优势。
原文摘要 · Abstract (English)
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.
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