Do attention maps in neural operators learn the Green's function?

Abstract

Attention-based neural operators are often read as if the attention weights were a learned Green's function — a kernel mapping source to response. This work tests that reading. Raw attention maps are compared against the effective operator the trained network actually realizes, across several one-dimensional PDEs. The two do not have to agree. The effective operator can recover inverse-operator structure in cases where the attention map on its own shows nothing of the kind, so the attention weights are not a reliable window onto what the surrogate has learned. Conditioning and resonance limit how far the comparison can be pushed.