Neural operators extend data-driven models to map between infinite-dimensional functional spaces. These models have successfully solved continuous dynamical systems represented by differential equations, viz weather forecasting, fluid flow, or solid mechanics. However, the existing operators still rely on real space, thereby losing rich representations potentially captured in the complex space by functional transforms. In this paper, we introduce a Complex Neural Operator (CoNO), that parameterizes the integral kernel in the complex fractional Fourier domain. Additionally, the model employing a complex-valued neural network along with aliasing-free activation functions preserves the complex values and complex algebraic properties, thereby enabling improved representation, robustness to noise, and generalization. We show that the model effectively captures the underlying partial differential equation with a single complex fractional Fourier transform. We perform an extensive empirical evaluation of CoNO on several datasets and additional tasks such as zero-shot super-resolution, evaluation of out-of-distribution data, data efficiency, and robustness to noise. CoNO exhibits comparable or superior performance to all the state-of-the-art models in these tasks. Altogether, CoNO presents a robust and superior model for modeling continuous dynamical systems, providing a fillip to scientific machine learning.

翻译：神经算子扩展了数据驱动模型，使其能够映射无限维函数空间之间的对应关系。这类模型已成功求解由微分方程描述的连续动力系统，例如天气预报、流体流动或固体力学领域。然而，现有算子仍依赖于实数空间，从而丢失了函数变换在复数空间中可能捕获的丰富表示。本文提出一种复数神经算子（CoNO），该算子在复数分数阶傅里叶域中对积分核进行参数化。此外，模型采用复值神经网络与无混叠激活函数，能够保持复数值及其代数性质，从而提升表示能力、噪声鲁棒性与泛化性能。我们证明，该模型可通过单一复数分数阶傅里叶变换有效捕捉底层偏微分方程。我们在多个数据集及附加任务（如零样本超分辨率、分布外数据评估、数据效率与噪声鲁棒性）上对CoNO进行了广泛实证评估。CoNO在这些任务中展现出与所有最先进模型相当或更优的性能。总体而言，CoNO为连续动力系统建模提供了一种鲁棒且优越的模型，为科学机器学习注入了新的动力。