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.

翻译：我们提出了一种具有严格物理保证的谱归一化代理模型，用于正向和逆向力学均匀化。利用Voigt-Reuss界限，我们通过类Cholesky算子分解其差值，并学习一个特征值位于$[0,1]$区间内的无量纲、对称半正定表示；逆向映射返回的对称正定预测在Löwner意义下位于界限之间。在一个随机双相微观结构开放数据集上进行的三维线弹性分析中，一个全连接的Voigt-Reuss网络基于$>\!7.5\times 10^{5}$个FFT计算标签进行训练，使用236个各向同性不变描述符和三个对比参数，以近乎完美的保真度恢复了各向同性投影（各向同性相关项：$R^2 \ge 0.998$），而揭示各向异性的耦合项无法从$SO(3)$不变输入中识别。张量级相对Frobenius误差在不同数据划分上的中位数约为$1.7\%$，平均值约为$3.4\%$。对于基于阈值化三角函数的微观结构进行的二维平面应变分析，将谱归一化与可微分渲染器及CNN结合，在所有分量上实现了$R^2>0.99$，归一化损失低于百分之一，准确追踪了渗流引起的特征值跃变，并对分布外图像表现出鲁棒的泛化能力。将参数化微观结构作为设计变量，使用单一代理模型进行批量一阶优化，可在百分之几的误差内匹配目标张量，并返回多样化的近最优设计。总体而言，Voigt-Reuss网络统一了精确、物理可接受的正向预测与大规模、约束一致的逆向设计，并适用于椭圆算子和多物理场耦合场景。