  Neural operators extend data-driven models to map between
infinite-dimensional functional spaces. While these operators perform
effectively in either the time or frequency domain, their performance may be
limited when applied to non-stationary spatial or temporal signals whose
frequency characteristics change with time. Here, we introduce Complex Neural
Operator (CoNO) that parameterizes the integral kernel using Fractional Fourier
Transform (FrFT), better representing non-stationary signals in a
complex-valued domain. Theoretically, we prove the universal approximation
capability of CoNO. We perform an extensive empirical evaluation of CoNO on
seven challenging partial differential equations (PDEs), including regular
grids, structured meshes, and point clouds. Empirically, CoNO consistently
attains state-of-the-art performance, showcasing an average relative gain of
10.9%. Further, CoNO exhibits superior performance, outperforming all other
models in additional tasks such as zero-shot super-resolution and robustness to
noise. CoNO also exhibits the ability to learn from small amounts of data --
giving the same performance as the next best model with just 60% of the
training data. Altogether, CoNO presents a robust and superior model for
modeling continuous dynamical systems, providing a fillip to scientific machine
learning.
