Spectral neural operators

In recent works, authors introduced neural operator – a particular type of neural network that can approximate maps between infinite-dimensional spaces. Using numerical and analytical techniques, we will highlight the peculiarities of the training and evaluation of these operators. In particular, we will show that for a broad class of neural operators based on integral transformations, a systematic bias is inevitable, owning to aliasing errors. To avoid this bias, we introduce spectral neural operators based on explicit discretization of domain and codomain. Discretization should decrease the approximation properties, but numerical experiments show that the accuracy of spectral neural operators is often superior to the one of neural operators defined on infinite-dimensional Banach spaces.