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

arXiv: https://arxiv.org/abs/2511.11388 · HTML: https://mlsanity.com/paper/2511.11388/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 46 references that are in the map.
- First posted on arXiv 14 Nov 2025.

## Builds on (most cited first, 13 of 13)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Inverse-design of nonlinear mechanical metamaterials via video denoising diffusion models](https://mlsanity.com/paper/2305.19836/) (2023)
- [Neural networks meet hyperelasticity: A guide to enforcing physics](https://mlsanity.com/paper/2302.02403/) (2023)
- [Unifying the design space and optimizing linear and nonlinear truss metamaterials by generative modeling](https://mlsanity.com/paper/2306.14773/) (2023)
- [A numerical method for computing the overall response of nonlinear composites with complex microstructure](https://mlsanity.com/paper/2012.08962/) (2020)
- [Neural networks meet hyperelasticity: A monotonic approach](https://mlsanity.com/paper/2501.02670/) (2025)
- [Learning Homogenization for Elliptic Operators](https://mlsanity.com/paper/2306.12006/) (2023)
- [Data-Driven Microstructure Property Relations](https://mlsanity.com/paper/1903.10841/) (2019)
- [Deep learning-aided inverse design of porous metamaterials](https://mlsanity.com/paper/2507.17907/) (2025)
- [Universal Fourier Neural Operators for periodic homogenization problems in linear elasticity](https://mlsanity.com/paper/2507.12233/) (2025)
- [Spectral Normalization and Voigt-Reuss net: A universal approach to microstructure-property forecasting with physical guarantees](https://mlsanity.com/paper/2504.00712/) (2025)
- [Data-efficient inverse design of spinodoid metamaterials](https://mlsanity.com/paper/2505.03415/) (2025)

## Abstract (arXiv)

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.