The Nonlocal Neural Operator: Universal Approximation

Neural operator architectures approximate operators between infinite-dimensional Banach spaces of functions. They are gaining increased attention in computational science and engineering, due to their potential both to accelerate traditional numerical methods and to enable data-driven discovery. A popular variant of neural operators is the Fourier neural operator (FNO). Previous analysis proving universal operator approximation theorems for FNOs resorts to use of an unbounded number of Fourier modes and limits the basic form of the method to problems with periodic geometry. Prior work relies on intuition from traditional numerical methods, and interprets the FNO as a nonstandard and highly nonlinear spectral method. The present work challenges this point of view in two ways: (i) the work introduces a new broad class of operator approximators, termed nonlocal neural operators (NNOs), which allow for operator approximation between functions defined on arbitrary geometries, and includes the FNO as a special case; and (ii) analysis of the NNOs shows that, provided this architecture includes computation of a spatial average (corresponding to retaining only a single Fourier mode in the special case of the FNO) it benefits from universal approximation. It is demonstrated that this theoretical result unifies the analysis of a wide range of neural operator architectures. Furthermore, it sheds new light on the role of nonlocality, and its interaction with nonlinearity, thereby paving the way for a more systematic exploration of nonlocality, both through the development of new operator learning architectures and the analysis of existing and new architectures.