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 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 ( 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.