Robust Inverse Material Design With Physical Guarantees Using the Voigt-Reuss Net

Abstract

We apply the Voigt-Reuss net, a spectrally normalized neural surrogate introduced in [38], for forward and inverse mechanical homogenization with a key guarantee that all predicted effective stiffness tensors satisfy Voigt-Reuss bounds in the Löwner sense during training, inference, and gradient-driven optimization. The approach operates in a bounded spectral space by reparametrizing each effective tensor relative to its Voigt and Reuss bounds, ensuring that all outputs reside within a unit-cube domain and are mapped back via a deterministic inverse transform to physically admissible tensors. For 3D elasticity, a fully connected Voigt-Reuss net is trained on $\sim 1.18$ million high-fidelity homogenization labels of stochastic biphasic microstructures. The model ingests 236 image-derived morphological descriptors and phase parameters that encode bulk and shear moduli, enabling a single surrogate to represent material combinations spanning 4 orders of magnitude. Due to the rotational invariance of the descriptor set, the surrogate recovers the isotropic projection of the effective stiffness ($R^2\ge 0.998$ for isotropy-related components). However, anisotropy-revealing entries remain unlearnable from the available features. At the tensor level, the relative Frobenius error exhibits a median of approximately 1.8% (mean approximately 3.6%) and approaches the irreducible isotropic-projection floor, far outperforming all alternative surrogates considered. In 2D plane-strain elasticity, spectral normalization is integrated with a differentiable microstructure renderer and a convolutional regressor, yielding a surrogate that maps generator parameters to effective stiffness tensors for highly anisotropic and high-contrast composites. Voigt-Reuss net is compared against vanilla and Cholesky regressors trained with identical architectures, data, and training procedures. The unconstrained surrogates frequently violate Voigt/Reuss bounds and even positive definiteness, whereas Voigt-Reuss net produces no violations by design and also enables robust inverse design. Utilizing the surrogate with batched first-order optimization, the approach can match prescribed target tensors and optimize tensor functionals, recovering classical optima while avoiding nonphysical and spurious designs observed with unconstrained surrogates.