Physics-Informed Neural Networks;Digital Volume Correlation;Navier–Cauchy equation;Displacement measurement;Physics-based regularisation

Digital Volume Correlation (DVC) tracks the interior displacement field of a specimen from paired micro-CT scans, but fails wherever the image lacks texture, leaving gaps in the measurement. This paper reconstructs the full field using a Physics-Informed Neural Network that fits the sparse DVC data while enforcing mechanical equilibrium (the Navier-Cauchy equations, with stress obtained via automatic differentiation) as a regularizer over the unobserved regions — a mesh-free, continuous counterpart to classical equilibrium-gap DVC. A controlled architecture study, evaluated on gap-region error, shows that with gradient-norm loss balancing the equilibrium constraint reduces error below a data-only baseline at every tested sparsity level, with the largest gains where data are sparsest. Applied end-to-end to a real 3D micro-CT rock-compression dataset, both trained models outperform deterministic interpolants on held-out nodes on average, though the improvement the physics term adds over the data-only network is small and within run-to-run variability. On a smooth, noise-free synthetic benchmark, by contrast, a thin-plate-spline interpolant remains the strongest method overall. The paper is explicit about the conditions under which the physics-based regularization pays off and about the limits of the current evidence.

Digital Volume Correlation (DVC) tracks the interior displacement field of a specimen from paired micro-CT scans, but fails wherever the image lacks texture, leaving gaps in the measurement. This paper reconstructs the full field using a Physics-Informed Neural Network that fits the sparse DVC data while enforcing mechanical equilibrium (the Navier-Cauchy equations, with stress obtained via automatic differentiation) as a regularizer over the unobserved regions — a mesh-free, continuous counterpart to classical equilibrium-gap DVC. A controlled architecture study, evaluated on gap-region error, shows that with gradient-norm loss balancing the equilibrium constraint reduces error below a data-only baseline at every tested sparsity level, with the largest gains where data are sparsest. Applied end-to-end to a real 3D micro-CT rock-compression dataset, both trained models outperform deterministic interpolants on held-out nodes on average, though the improvement the physics term adds over the data-only network is small and within run-to-run variability. On a smooth, noise-free synthetic benchmark, by contrast, a thin-plate-spline interpolant remains the strongest method overall. The paper is explicit about the conditions under which the physics-based regularization pays off and about the limits of the current evidence.