Robust inverse material design with physical guarantees using the Voigt-Reuss Net

We propose a spectrally normalized surrogate for forward and inverse mechanical homogenization with hard physical guarantees. Leveraging the Voigt-Reuss bounds, we factor their difference via a Cholesky-like operator and learn a dimensionless, symmetric positive semi-definite representation with eigenvalues in [0,1]; the inverse map returns symmetric positive-definite predictions that lie between the bounds in the L"owner sense. In 3D linear elasticity on an open dataset of stochastic biphasic microstructures, a fully connected Voigt-Reuss net trained on >!7.5× 105 FFT-based labels with 236 isotropy-invariant descriptors and three contrast parameters recovers the isotropic projection with near-perfect fidelity (isotropy-related entries: R2 ≥ 0.998), while anisotropy-revealing couplings are unidentifiable from SO(3)-invariant inputs. Tensor-level relative Frobenius errors have median ≈ 1.7% and mean ≈ 3.4% across splits. For 2D plane strain on thresholded trigonometric microstructures, coupling spectral normalization with a differentiable renderer and a CNN yields R2>0.99 on all components, subpercent normalized losses, accurate tracking of percolation-induced eigenvalue jumps, and robust generalization to out-of-distribution images. Treating the parametric microstructure as design variables, batched first-order optimization with a single surrogate matches target tensors within a few percent and returns diverse near-optimal designs. Overall, the Voigt-Reuss net unifies accurate, physically admissible forward prediction with large-batch, constraint-consistent inverse design, and is generic to elliptic operators and coupled-physics settings.