Two axes classify every learned-geometry method: what the network models — functions or operators — and who holds final authority — the learner or the compiler. Three quadrants are crowded; the fourth is nearly empty. This program lives there: the learner is demoted to proposer, and the compiler holds the pen.
The crowded default. The network models a function — coordinates in, values out — and whatever it emits is the artifact. Radiance fields, implicit surfaces, diffusion generators: the representation is the learned object, authority lives entirely in the weights, and invariants, where present at all, are loss terms.
Operator learning. The learned object is the function-to-function map itself — FNO, DeepONet, GINO — the right vocabulary for geometry, with resolution invariance built in. But syntax, support, and evaluation stay fused in one dense tensor: the trained network is the operator, and off-distribution it degrades with nothing to catch it.
Gated approximation. A symbolic or deterministic system holds final say while the learned object remains a function. Physics-penalty hybrids lean toward it without arriving — a violated equation is discouraged, not refused — and neurosymbolic provers reach it only on the narrow domains where a formal verifier already exists.
Operator recovery, not operator learning. The learner recovers the algebraic identity of an operation and parameterizes it; the typed compiler performs it; invariants hold by construction, never by regularization. Drop or randomize the learner and the system still executes — legal, stable, deterministic.