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

Sanath Keshav, Felix Fritzen

TL;DR

The paper introduces Voigt–Reuss net, a spectrally normalized surrogate for forward and inverse elastic homogenization that guarantees physical admissibility by construction. By reparameterizing the Voigt–Reuss gap into a bounded SPD form and predicting eigenvalues in $[0,1]$ plus an orthogonal factor, the approach enforces symmetric positive definiteness and Löwner bounds across training and optimization. It demonstrates two regimes: a 3D descriptor-based pipeline for stochastic biphasic microstructures achieving near-perfect isotropic accuracy (with $R^2\ge 0.998$ for isotropic components and median tensor errors around $1.7\%$) and a 2D end-to-end differentiable pipeline with a differentiable renderer that yields $R^2>0.99$ and subpercent losses while tracking percolation-induced eigenvalue jumps. The framework supports efficient, large-batch inverse design with diverse near-optimal microstructures, all within physically admissible bounds, and is extensible to other elliptic problems and multi-physics settings with rigorous bounds.

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 Löwner 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.

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

TL;DR

The paper introduces Voigt–Reuss net, a spectrally normalized surrogate for forward and inverse elastic homogenization that guarantees physical admissibility by construction. By reparameterizing the Voigt–Reuss gap into a bounded SPD form and predicting eigenvalues in plus an orthogonal factor, the approach enforces symmetric positive definiteness and Löwner bounds across training and optimization. It demonstrates two regimes: a 3D descriptor-based pipeline for stochastic biphasic microstructures achieving near-perfect isotropic accuracy (with for isotropic components and median tensor errors around ) and a 2D end-to-end differentiable pipeline with a differentiable renderer that yields and subpercent losses while tracking percolation-induced eigenvalue jumps. The framework supports efficient, large-batch inverse design with diverse near-optimal microstructures, all within physically admissible bounds, and is extensible to other elliptic problems and multi-physics settings with rigorous bounds.

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 ; the inverse map returns symmetric positive-definite predictions that lie between the bounds in the Löwner sense. In 3D linear elasticity on an open dataset of stochastic biphasic microstructures, a fully connected Voigt-Reuss net trained on FFT-based labels with 236 isotropy-invariant descriptors and three contrast parameters recovers the isotropic projection with near-perfect fidelity (isotropy-related entries: ), while anisotropy-revealing couplings are unidentifiable from -invariant inputs. Tensor-level relative Frobenius errors have median and mean across splits. For 2D plane strain on thresholded trigonometric microstructures, coupling spectral normalization with a differentiable renderer and a CNN yields 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.

Paper Structure

This paper contains 18 sections, 38 equations, 16 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Machine learned effective property forecasting
  4. Objectives and scope.
  5. Notation
  6. Foundations
  7. Löwner bounds for elliptic problems
  8. Spectral normalization
  9. Implementation of the spectral normalization
  10. Application to 3D linear mechanical homogenization
  11. Application to highly anisotropic and high contrast 2D composites
  12. Neural surrogate architecture
  13. Numerical experiments and results
  14. Microstructure to property forecasting
  15. Inverse materials design
  16. ...and 3 more sections

Figures (16)

  • Figure 1: Two approaches for constructing the effective elasticity tensor $\overline{\ul{\ul{C}}}$ from microstructure: (a) direct numerical simulation (DNS) workflow, and (b) a machine-learning surrogate model employing spectral normalization to enforce two-sided Löwner bounds.
  • Figure 2: Examples of 3D microstructures taken from ulm2020_dataset for eight of the nine classes (spatial stochastic graphs omitted here). The inclusion phase is shown in red, and the matrix is shown as a void for visibility. Each class contributes $10{,}000$ samples.
  • Figure 3: Phase contrasts of the bulk and shear moduli for the material pairs from \ref{['tab:binary_material_pairs']}. Each material pair is represented twice, considering the possibility of the two materials being in either phase 1 or phase 2.
  • Figure 4: Lower triangle: Ground truth vs. prediction for all the stiffness components of $\overline{\ul{\ul{C}}}$ (hexbin density). Each tile lists per-component $R^2$. All entries corresponding to isotropy achieve $R^2\ge 0.998$: the axial terms $\{\overline{C}_{11},\overline{C}_{22},\overline{C}_{33}\}$, shear terms $\{\overline{C}_{44},\overline{C}_{55},\overline{C}_{66}\}$, and the normal couplings $\{\overline{C}_{21},\overline{C}_{31},\overline{C}_{32}\}$. The remaining (anisotropy-revealing) components have $R^2\approx 0$.
  • Figure 5: Left: Relative Frobenius error histograms ${\cal E}_\mathsf{F}$ for train/val/test splits (nearly coincident; mean $\approx 3.4\%$, median $\approx 1.7\%$). Right: ECDFs of ${\cal E}_\mathsf{F}$ and two references: isotropic projection, Voigt-Reuss net and the Hill average. Voigt-Reuss net surrogate using isotropy-only descriptors operates very close to the lower (isotropy) limit.
  • ...and 11 more figures

Theorems & Definitions (2)

  • remark 1
  • remark 2