Tensor-Train Methods for 3D Linear Elasticity: Block and Global Operator Representations with Solver Performance Analysis

What the math gives to ML

The paper's transferable asset is not elasticity itself, but its exploitation of tensor-product structure through tensor-train (TT) representations of coupled operators, vectors, and solutions. The same construction can replace dense 3D-grid neural operators, coordinate-grid predictors, or large tensorized linear layers with sequences of small contractions whose cost scales with TT rank rather than the full Cartesian grid size. The block-versus-monolithic operator distinction also suggests preserving physically or semantically meaningful channel blocks instead of forcing one global low-rank approximation. The strongest first experiment is a TT neural operator or coordinate-grid projection with controlled TT-rank truncation, measuring memory, throughput, and accuracy at equal parameter budgets.