Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning
Francesca Bartolucci, Emmanuel de Bézenac, Bogdan Raonić, Roberto Molinaro, Siddhartha Mishra, Rima Alaifari
TL;DR
The paper tackles aliasing as a fundamental obstruction in learning operators between function spaces. It introduces a Representation Equivalent Neural Operators (ReNO) framework that formalizes aliasing via an error $ε$ and recasts neural operators as structure-preserving mappings between Hilbert spaces with a strict continuous/discrete correspondence. The authors provide definitions (1–6), a key result (Proposition 7), and a training-set guideline that together establish a principled path to aliasing-free operator learning, with concrete instantiations in Spectral Neural Operators and Convolutional Neural Operators. This framework highlights a rigorous route to discretization-invariant operator learning and lays groundwork for designing new neural operators that maintain continuous structures during learning and inference, with implications for learning PDE mappings from data.
Abstract
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.