Multiscale LegONet: Learning Coefficient-Dependent Local Spaces for Composable Multiscale Finite Elements with Localized Error Bounds

We introduce multiscale LegONet (Ms-LegONet), a plug-and-play learned block for heterogeneous elliptic problems. One network per spatial dimension is trained once on local coefficient patches and reused, without retraining, across coefficient layouts, sources, boundary data and domain sizes within the trained coefficient class. On a fixed primitive patch the network maps the local coefficient to a zero-boundary space, the only learned object. The multiscale basis functions are the harmonic extensions of a fixed set of boundary traces, each a Galerkin solve in that space. Shared interface traces enforce continuity, larger blocks are assembled by recursive static condensation without any parent-scale network, and a localized Céa bound attributes the error of the learned spaces primitive by primitive. The training labels are harmonic responses of single patches, so no global solve enters the training. Against MsFEM, GMsFEM, the neural-operator element method and an end-to-end Fourier neural operator, Ms-LegONet is the most accurate at equal unknowns: its $L^2$ error is $4$ to $150$ times below the baselines on two-dimensional mosaics of $256$ high-contrast primitives, and its median error stays within $0.1$ percentage points of that of exact local spaces in three-dimensional layered media. For each new coefficient it is $1.6$ to $2.3$ times faster than a fine-grid direct solve in two dimensions. In three dimensions it computes the many-query port response of a chain $30$ to $900$ times faster than finite element solves on the same or a finer mesh, at a relative error of $10^{-4}$ against their $6\%$ and $1.6\%$.