Recently, operator learning, or learning mappings between
infinite-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 necessitate
discretization in the transition to computer implementations. This step can
compromise their integrity, often causing them to deviate from the underlying
operators. This research offers a fresh take on neural operators with a
framework Representation equivalent Neural Operators (ReNO) designed to address
these issues. At its core is the concept of operator aliasing, which measures
inconsistency between neural operators and their discrete representations. We
explore this for widely-used operator learning techniques. Our findings detail
how aliasing introduces errors when handling different discretizations and
grids and loss of crucial continuous structures. More generally, this framework
not only sheds light on existing challenges but, given its constructive and
broad nature, also potentially offers tools for developing new neural
operators.Comment: 28 page