Physics-informed learning for the inverse problem in resonant ultrasound spectroscopy
Machine Learning
Summary
The authors address the challenge of figuring out elastic constants of materials from limited resonant ultrasound data, which is a complex inverse problem. They reformulate the problem using physics-based variables that capture material properties and geometry, simplifying the task. Their approach incorporates a machine learning model focused on these reduced features, combined with analytical steps to recover final elastic constants. They test their method on cubic crystals and show reasonably accurate reconstructions compared to known values. Overall, they turn a difficult nonlinear problem into a more manageable regression problem respecting physical constraints.
Authors
Alejandro Cubillos Muñoz, Manuela Rivas, Julian Rincon
Abstract
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.