  The solution of a partial differential equation can be obtained by computing
the inverse operator map between the input and the solution space. Towards this
end, we introduce a \textit{multiwavelet-based neural operator learning scheme}
that compresses the associated operator's kernel using fine-grained wavelets.
By explicitly embedding the inverse multiwavelet filters, we learn the
projection of the kernel onto fixed multiwavelet polynomial bases. The
projected kernel is trained at multiple scales derived from using repeated
computation of multiwavelet transform. This allows learning the complex
dependencies at various scales and results in a resolution-independent scheme.
Compare to the prior works, we exploit the fundamental properties of the
operator's kernel which enable numerically efficient representation. We perform
experiments on the Korteweg-de Vries (KdV) equation, Burgers' equation, Darcy
Flow, and Navier-Stokes equation. Compared with the existing neural operator
approaches, our model shows significantly higher accuracy and achieves
state-of-the-art in a range of datasets. For the time-varying equations, the
proposed method exhibits a ($2X-10X$) improvement ($0.0018$ ($0.0033$) relative
$L2$ error for Burgers' (KdV) equation). By learning the mappings between
function spaces, the proposed method has the ability to find the solution of a
high-resolution input after learning from lower-resolution data.
