  Many representative graph neural networks, e.g., GPR-GNN and ChebNet,
approximate graph convolutions with graph spectral filters. However, existing
work either applies predefined filter weights or learns them without necessary
constraints, which may lead to oversimplified or ill-posed filters. To overcome
these issues, we propose BernNet, a novel graph neural network with theoretical
support that provides a simple but effective scheme for designing and learning
arbitrary graph spectral filters. In particular, for any filter over the
normalized Laplacian spectrum of a graph, our BernNet estimates it by an
order-$K$ Bernstein polynomial approximation and designs its spectral property
by setting the coefficients of the Bernstein basis. Moreover, we can learn the
coefficients (and the corresponding filter weights) based on observed graphs
and their associated signals and thus achieve the BernNet specialized for the
data. Our experiments demonstrate that BernNet can learn arbitrary spectral
filters, including complicated band-rejection and comb filters, and it achieves
superior performance in real-world graph modeling tasks. Code is available at
https://github.com/ivam-he/BernNet.
