  Operator learning trains a neural network to map functions to functions. An
ideal operator learning framework should be mesh-free in the sense that the
training does not require a particular choice of discretization for the input
functions, allows for the input and output functions to be on different
domains, and is able to have different grids between samples. We propose a
mesh-free neural operator for solving parametric partial differential
equations. The basis enhanced learning network (BelNet) projects the input
function into a latent space and reconstructs the output functions. In
particular, we construct part of the network to learn the ``basis'' functions
in the training process. This generalized the networks proposed in Chen and
Chen's universal approximation theory for the nonlinear operators to account
for differences in input and output meshes. Through several challenging
high-contrast and multiscale problems, we show that our approach outperforms
other operator learning methods for these tasks and allows for more freedom in
the sampling and/or discretization process.
