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

TL;DR

This paper introduces the Voigt-Reuss Net, a spectral normalization-based surrogate model for robust inverse material design with physical guarantees, achieving high accuracy in 3D and 2D elasticity problems.

Contribution

The paper presents a novel neural network architecture that enforces physical bounds and guarantees in inverse material design, unifying forward prediction and inverse design in a single framework.

Findings

Near-perfect fidelity in 3D isotropic predictions (R^2 ≥ 0.998)

Subpercent normalized errors in tensor predictions (~1.7% median)

High accuracy and robustness in 2D microstructure inverse design (R^2 > 0.99)

Abstract

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 $>\!7.5\times 10^{5}$ FFT-based labels with 236 isotropy-invariant descriptors and three contrast parameters recovers the isotropic projection with near-perfect fidelity (isotropy-related entries: $R^2 \ge 0.998$), while anisotropy-revealing couplings are unidentifiable from $SO(3)$-invariant inputs. Tensor-level relative Frobenius errors have median $\approx 1.7\%$ and mean $\approx 3.4\%$ across splits. For 2D plane strain on thresholded trigonometric microstructures, coupling spectral normalization with a differentiable renderer and a CNN yields $R^2>0.99$ 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.