Are Neural Operators Really Neural Operators? Frame Theory Meets Operator Learning

Recently, there has been significant interest in operator learning, i.e. learning mappings between infinite-dimensional function spaces. This has been particularly relevant in the context of learning partial differential equations from data. However, it has been observed that proposed models may not behave as operators when implemented on a computer, questioning the very essence of what operator learning should be. We contend that in addition to defining the operator at the continuous level, some form of continuous-discrete equivalence is necessary for an architecture to genuinely learn the underlying operator, rather than just discretizations of it. To this end, we propose to employ frames, a concept in applied harmonic analysis and signal processing that gives rise to exact and stable discrete representations of continuous signals. Extending these concepts to operators, we introduce a unifying mathematical framework of Representation equivalent Neural Operator (ReNO) to ensure operations at the continuous and discrete level are equivalent. Lack of this equivalence is quantified in terms of aliasing errors. We analyze various existing operator learning architectures to determine whether they fall within this framework, and highlight implications when they fail to do so.