Physics-informed learning for the inverse problem in resonant ultrasound spectroscopy

Alejandro Cubillos Muñoz; Manuela Rivas; Julian Rincon

cond-mat.mtrl-sciAug 27, 2026

Physics-informed learning for the inverse problem in resonant ultrasound spectroscopy

Authors: Alejandro Cubillos Muñoz, Manuela Rivas, Julian Rincon

Organizations: Department of Physics, Universidad de los Andes, Bogotá, D.C. 111711, Colombia

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.

Figures & tables

Panel of Figure 1 — Figure 1: Correlation structure before and after the physics-informed reduction. The left panel shows the raw isotropic variables, including sample dimensions, density, and squared resonance-frequency features; the strong correlations among the frequency features reveal substantial redundancy in the raw spectral representation. The right panel shows the reduced isotropic features, built from the aspect-ratio variables $\eta,\beta$ and the scale-free spectral ratios $\xi_{j}$ . The transformed variables exhibit substantially weaker pairwise correlations, indicating that the dominant raw collinearity has been removed while retaining spectral-shape information.

Figure 2: Reduced isotropic spectral ratios as functions of

G/K

, expressed through the angular coordinate ( 21 ). The ratios

\xi_{0},\ldots,\xi_{4}

are shown for fixed shape coordinates

(\eta,\beta)=(0.31,0.5)\pi

. The ratios occupy different ranges and respond differently as

G/K

changes. This illustrates how the scale-free ratios retain complementary spectral-shape information after the common elastic magnitude has been removed.

Raw quantity	Reduced quantity	Physical role
$L_{x},L_{y},L_{z}$	$R,\eta,\beta$	Size/shape split
$\omega_{j}$	$\lambda_{j}=m\omega_{j}^{2}/R$	Size scaling
$\lambda_{j}$ (isotropic)	$\xi_{j}=\lambda_{j}/\lambda_{j+1}$	Elastic scaling
$\lambda_{j}$ (cubic)	$\chi_{j}=(\lambda_{j}-\lambda_{j-1})/\lambda_{n_{\omega}-1}$	Spectral gaps
$K,G$	$(K,G)\to(M,\phi_{K})$	Stability coords
$C_{11},C_{12},C_{44}$	$(\mathcal{K},a,\mu)\to(M,\phi_{\mathcal{K}},\phi_{a})$	Stability coords

Table 1: Physics-informed transformations used to reduce the inverse problem before the regression model is introduced.

Figure 3: Physics-informed inverse-problem pipeline for cubic elastic-constant reconstruction. Raw resonance, mass, and geometry data are transformed into size-normalized spectral variables, scale-free spectral features, and shape coordinates. The feed-forward neural network predicts only the bounded angular variables

(\phi_{\mathcal{K}},\phi_{a})

. A reference forward calculation fixes the elastic scale, after which the final constants

C_{11}

C_{12}

, and

C_{44}

are reconstructed analytically.

Figure 4: Parity plots for the full cubic benchmark in the inverse pipeline. Each panel compares generated values on the horizontal axis with reconstructed values on the vertical axis for one independent cubic elastic constant; the diagonal line represents perfect agreement. In this benchmark, the sample dimensions vary within the domain used by the physics-informed parameterization, while the elastic constants span the thermodynamically stable cubic domain. The agreement is strongest for

C_{44}

and broadest for

C_{12}

. The broader spread of the

C_{12}

distribution reflects the sensitivity of

C_{12}=\mathcal{K}-a

to propagated errors in the reconstructed variables

\mathcal{K}

and

a

Figure 5: Absolute percentage-error distributions for the fixed-geometry benchmarks. The left panel shows the cubic reconstruction errors for

C_{11}

C_{12}

, and

C_{44}

; the right panel shows the isotropic reconstruction errors for

K

and

G

. Both panels use the

3{:}4{:}5

aspect ratio as a controlled comparison, so that the aspect-ratio dependence present in the full benchmark is removed. The corresponding MAPE values are

4.14(3.87)\%

8.31(8.50)\%

, and

2.44(2.86)\%

for

C_{11}

C_{12}

, and

C_{44}

, respectively, and

4.0(3.6)\%

and

0.4(0.3)\%

for

K

and

G

, respectively.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Quantity	Symbol	Range or value	Comment
Sample lengths	$L_{x},L_{y},L_{z}$	$0.1$ – $1~\mathrm{cm}$	Variables before $R,\eta,\beta$ conversion
Density	$\rho$	$0.2$ – $10~\mathrm{g}/\mathrm{cm}^{3}$	Sample mass computation
Isotropic bulk, shear moduli	$K,G$	$0.3$ – $5.6~\mathrm{Tdyn}/\mathrm{cm}^{2}$	Raw data; sampled with $K,G>0$
Cubic elastic constants	$C_{11},C_{12},C_{44}$	not sampled directly	Born-stable via $\mathcal{K},a,\mu>0$
Angular targets	$\phi_{K},\phi_{\mathcal{K}},\phi_{a}$	$0$ – $\pi/2$	Stability-respecting ratio coordinates
Isotropic spectral features	$\xi_{0},\ldots,\xi_{4}$	$5$ ratios	Consecutive normalized-eigenvalue ratios

Appendix

Table 2: Sampling domain used to generate the synthetic RUS datasets. The table summarizes the dimensional and material ranges, angular targets, and retained reduced spectral features used in the isotropic and cubic benchmarks.

Figure 6: Probability distribution of the generated resonance frequencies on a logarithmic scale.

Figure 7: Correlation matrix of the reduced cubic features. The variables include the normalized spectral spacings $\{\chi_{j}\}$ and the aspect-ratio coordinates $(\eta,\beta)$ used as inputs for the cubic angular model. The reduced features show substantially weaker pairwise correlations than the raw spectral representation, indicating that the physics-informed transformation also removes the dominant collinearity in the cubic case.

Angular model	Split	MAE	RMSE	$R^{2}$
	Train	$0.024$	$0.038$	$0.982$
Isotropic	Validation	$0.024$	$0.039$	$0.981$
	Test	$0.024$	$0.040$	$0.980$
	Train	$0.018$	$0.036$	$0.973$
Cubic	Validation	$0.028$	$0.061$	$0.924$
	Test	$0.028$	$0.062$	$0.922$

Appendix

Table 3: Train, validation, and test metrics for the angular regression models. The isotropic model predicts the scaled angular variable

\phi_{K}

, and the cubic model predicts

(\phi_{\mathcal{K}},\phi_{a})

Figure 8: Absolute percentage-error distributions for the full cubic and isotropic benchmarks. The left panel shows the cubic reconstruction errors for

C_{11}

C_{12}

, and

C_{44}

; the right panel shows the isotropic reconstruction errors for

K

and

G

. Unlike the fixed-geometry benchmark in Fig. 5 , these distributions retain the aspect-ratio dependence present in the full benchmark.

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Estimating spatially heterogeneous elastic properties from low-resolution displacement measurements is a severely ill-posed inverse elasticity problem because low resolution obscures spatial details needed to distinguish heterogeneous property variations, and small measurement perturbations or fitting errors are amplified through inverse estimation. Existing inverse methods often rely on high-fidelity observations and manually prespecified loss weights, limiting their adaptability and making them sensitive to noise and resolution degradation. We propose a Probabilistic Inverse Elasticity Physics-Informed Neural Network (PIE-PINN) framework for robust estimation of Young's modulus and Poisson's ratio from noisy, low-resolution displacement data. PIE-PINN models displacement observation, strain-discrepancy, and equilibrium residuals using Laplace distributions within a unified probabilistic model. To improve robustness, the framework combines a B-spline-guided displacement network with a hierarchical half-Cauchy model for displacement residual scales. The B-spline provides a smooth global representation of the displacement field, while the neural network correction captures local variations. The hierarchical scale model adaptively downweights severe displacement fitting errors, enabling more robust recovery of the latent mean displacement field. An alternating maximum-likelihood training strategy updates the mean through weighted residual minimization and updates the scales to adjust the loss weights. Systematic case studies across varying noise levels and observation resolutions demonstrate the robustness of PIE-PINN.

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