  Spectral methods are an important part of scientific computing's arsenal for
solving partial differential equations (PDEs). However, their applicability and
effectiveness depend crucially on the choice of basis functions used to expand
the solution of a PDE. The last decade has seen the emergence of deep learning
as a strong contender in providing efficient representations of complex
functions. In the current work, we present an approach for combining deep
neural networks with spectral methods to solve PDEs. In particular, we use a
deep learning technique known as the Deep Operator Network (DeepONet), to
identify candidate functions on which to expand the solution of PDEs. We have
devised an approach which uses the candidate functions provided by the DeepONet
as a starting point to construct a set of functions which have the following
properties: i) they constitute a basis, 2) they are orthonormal, and 3) they
are hierarchical i.e., akin to Fourier series or orthogonal polynomials. We
have exploited the favorable properties of our custom-made basis functions to
both study their approximation capability and use them to expand the solution
of linear and nonlinear time-dependent PDEs.
