  A plentitude of applications in scientific computing requires the
approximation of mappings between Banach spaces. Recently introduced Fourier
Neural Operator (FNO) and Deep Operator Network (DeepONet) can provide this
functionality. For both of these neural operators, the input function is
sampled on a given grid (uniform for FNO), and the output function is
parametrized by a neural network. We argue that this parametrization leads to
1) opaque output that is hard to analyze and 2) systematic bias caused by
aliasing errors in the case of FNO. The alternative, advocated in this article,
is to use Chebyshev and Fourier series for both domain and codomain. The
resulting Spectral Neural Operator (SNO) has transparent output, never suffers
from aliasing, and may include many exact (lossless) operations on functions.
The functionality is based on well-developed fast, and stable algorithms from
spectral methods. The implementation requires only standard numerical linear
algebra. Our benchmarks show that for many operators, SNO is superior to FNO
and DeepONet.
