用物理约束的代理模型实现材料逆向设计，精度高且保证物理解释性。
Robust inverse material design with physical guarantees using the Voigt-Reuss Net
- 基于Voigt-Reuss界构造谱归一化代理模型，保证预测结果在物理边界内。
- 3D各向同性材料恢复精度超0.998，张量误差中位数约1.7%、均值约3.4%。
- 支持批量优化生成多样近优设计方案，适合材料设计与多物理场问题。
我们提出一种谱归一化的代理模型，用于正向与逆向机械均匀化，具有严格的物理保证。通过利用Voigt-Reuss上下界，采用类似Cholesky的算子分解其差异，学习一个无量纲、对称半正定表示，特征值位于[0,1]区间；逆映射返回对称正定预测，满足Löwner序下的边界约束。在包含超过7.5×10⁵个FFT标签的开放3D线弹性随机双相微结构数据集上，使用236个各向同性不变描述符和三个对比参数训练的全连接Voigt-Reuss网络，对各向同性投影的恢复达到近乎完美的保真度（各向同性项的R² ≥ 0.998），而仅依赖SO(3)不变输入时无法识别各向异性耦合。跨划分的张量级相对Frobenius误差中位数约为1.7%，均值约为3.4%。在阈值三角函数微结构的2D平面应变场景中，结合可微渲染器与CNN，R² > 0.99覆盖所有分量，归一化损失低于百分之一，准确追踪渗流诱导的特征值跃迁，并对分布外图像具备强泛化能力。将参数化微结构作为设计变量，使用单个代理模型进行批量一阶优化，可在几百分比内匹配目标张量，并生成多样近优设计方案。总体而言，Voigt-Reuss网络统一了高精度、物理解释性正向预测与大规模、约束一致的逆向设计，适用于椭圆型算子及多物理场场景。
原文摘要 · Abstract (English)
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.
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