Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning
This work addresses a fundamental issue in operator learning for researchers and practitioners, offering a constructive framework to improve accuracy and consistency in applications like partial differential equations, though it is incremental in building on existing neural operator techniques.
The paper tackles the problem of discretization errors in neural operators, which cause deviations from the underlying continuous operators, by introducing the Representation Equivalent Neural Operators (ReNO) framework and the concept of operator aliasing. The result is a framework that identifies aliasing-induced errors in existing methods and provides tools for developing new, alias-free neural operators.
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.