Spectral Neural Operators

In recent works, the authors introduced a neural operator: a special type of neural networks 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 transforms, a systematic bias is inevitable, owning to aliasing errors. To avoid this bias, we introduce spectral neural operators based on explicit discretization of the domain and the codomain. Although discretization deteriorates the approximation properties, 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.