Neural operators: Difference between revisions

Revision as of 07:05, 20 November 2023

Revision as of 02:39, 22 November 2023

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== Definition and formulation ==

Architecturally, neural operators are similar to feed-forward neural networks in the sense that they are comprisedcomposed of alternating [[Linear map|linear maps]] and non-linearities. Since neural operators act on and output functions, neural operators have been instead formulated as a sequence of alternating linear [[integral operators]] on function spaces and point-wise non-linearities.<ref name="NO journal" /> Using an analogous architecture to finite-dimensional neural networks, similar [[Universal approximation theorem|universal approximation theorems]] have been proven for neural operators. In particular, it has been shown that neural operators can approximate any continuous operator on a [[Compact space|compact]] set.<ref name="NO journal"/>

Neural operators seek to approximate some operator <math>\mathcal{G} : \mathcal{A} \to \mathcal{U}</math> between function spaces <math>\mathcal{A}</math> and <math>\mathcal{U}</math> by building a parametric map <math>\mathcal{G}_\phi : \mathcal{A} \to \mathcal{U}</math>. Such parametric maps <math>\mathcal{G}_\phi</math> can generally be defined in the form