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Neural operators extend data-driven models to map betweeninfinite-dimensional functional spaces. These models have successfully solvedcontinuous dynamical systems represented by differential equations, viz weatherforecasting, fluid flow, or solid mechanics. However, the existing operatorsstill rely on real space, thereby losing rich representations potentiallycaptured in the complex space by functional transforms. In this paper, weintroduce a Complex Neural Operator (CoNO), that parameterizes the integralkernel in the complex fractional Fourier domain. Additionally, the modelemploying a complex-valued neural network along with aliasing-free activationfunctions preserves the complex values and complex algebraic properties,thereby enabling improved representation, robustness to noise, andgeneralization. We show that the model effectively captures the underlyingpartial differential equation with a single complex fractional Fouriertransform. We perform an extensive empirical evaluation of CoNO on severaldatasets and additional tasks such as zero-shot super-resolution, evaluation ofout-of-distribution data, data efficiency, and robustness to noise. CoNOexhibits comparable or superior performance to all the state-of-the-art modelsin these tasks. Altogether, CoNO presents a robust and superior model formodeling continuous dynamical systems, providing a fillip to scientific machinelearning.

CoNO: Complex Neural Operator for Continuous Dynamical Systems