Radial Basis Operator Networks

Abstract

Operator networks are designed to approximate nonlinear operators, which provide mappings between infinite-dimensional spaces such as function spaces. These networks are playing an increasingly important role in machine learning, with their most notable contributions in the field of scientific computing. Their significance stems from their ability to handle the type of data often encountered in scientific applications. For instance, in climate modeling or fluid dynamics, input data typically consists of discretized continuous fields (like temperature distributions or velocity fields). We introduce the radial basis operator network (RBON), which represents a significant advancement as the first operator network capable of learning an operator in both the time domain and frequency domain when adjusted to accept complex-valued inputs. Despite the small, single hidden-layer structure, the RBON boasts small $L^2$ relative test error for both in- and out-of-distribution data (OOD) of less than $1\times 10^{-7}$ in some benchmark cases. Moreover, the RBON maintains small error on OOD data from entirely different function classes from the training data.