面向共振超声谱逆问题的物理信息学习方法
Physics-informed learning for the inverse problem in resonant ultrasound spectroscopy
浏览论文内容
中文总结 AI 辅助
该研究针对共振超声谱的弹性常数逆问题，提出物理信息学习方法，通过构建约束逆等谱问题引入低维变量，结合回归模型与解析重构，在立方及各向同性基准上实现了高精度弹性常数推断。
中文摘要 AI 辅助
从共振超声谱推断弹性常数是基于有限谱数据的非线性且通常超定的逆问题。我们将瑞利-里兹逆问题构建为物理可接受弹性张量集合上的约束逆等谱问题，这为可接受弹性流形上的逆映射引入了有效低维变量：长度与弹性尺度、长宽比坐标、无尺度谱特征以及符合稳定性要求的弹性比率。我们利用这些变量构建了物理信息学习流程，其中回归模型仅作用于降维后的谱与几何特征，而尺度恢复与最终弹性常数重构通过解析方式实现。对于完整立方基准，C₁₁、C₁₂、C₄₄的重构常数平均绝对误差（MAE）分别为20.37（35.15）GPa、24.30（41.33）GPa、2.13（3.66）GPa；在固定几何基准中，对应立方平均绝对百分比误差（MAPE）分别为4.14（3.87）%、8.31（8.50）%、2.44（2.86）%，而各向同性基准的体积模量和剪切模量的MAPE分别为4.0（3.6）%和0.4（0.3）%。该逆问题随后转变为适配胡克弹性的几何、尺度、晶体对称性及热力学稳定性的变量约束回归问题。
英文摘要
Inferring elastic constants from resonant ultrasound spectra is a nonlinear and typically overdetermined inverse problem based on finite spectral data. We formulate the Rayleigh-Ritz inverse problem as a constrained inverse-isospectral problem on the set of physically admissible elasticity tensors. This induces effective low-dimensional variables for the inverse map on the admissible elasticity manifold: length and elastic scales, aspect-ratio coordinates, scale-free spectral features, and stability-respecting elastic ratios. We use these variables to construct a physics-informed learning pipeline in which a regression model acts only on reduced spectral and geometric features, while scale recovery and final elastic-constant reconstruction are imposed analytically. For the full cubic benchmark, the reconstructed constants have MAE values of $20.37(35.15)$, $24.30(41.33)$, and $2.13(3.66)~\mathrm{GPa}$ for $C_{11}$, $C_{12}$, and $C_{44}$. In the fixed-geometry benchmark, the corresponding cubic MAPE values are $4.14(3.87)\%$, $8.31(8.50)\%$, and $2.44(2.86)\%$, while the isotropic values are $4.0(3.6)\%$ and $0.4(0.3)\%$ for the bulk and shear moduli. The inverse problem then becomes a constrained regression problem in variables adapted to the geometry, scaling, crystal symmetry, and thermodynamic stability of Hookean elasticity.
