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Recently, operator learning, or learning mappings betweeninfinite-dimensional function spaces, has garnered significant attention,notably in relation to learning partial differential equations from data.Conceptually clear when outlined on paper, neural operators necessitatediscretization in the transition to computer implementations. This step cancompromise their integrity, often causing them to deviate from the underlyingoperators. This research offers a fresh take on neural operators with aframework Representation equivalent Neural Operators (ReNO) designed to addressthese issues. At its core is the concept of operator aliasing, which measuresinconsistency between neural operators and their discrete representations. Weexplore this for widely-used operator learning techniques. Our findings detailhow aliasing introduces errors when handling different discretizations andgrids and loss of crucial continuous structures. More generally, this frameworknot only sheds light on existing challenges but, given its constructive andbroad nature, also potentially offers tools for developing new neuraloperators.

Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning