CoNO: Complex Neural Operator for Continous Dynamical Physical Systems

TL;DR

CoNO introduces a complex-valued neural operator using Fractional Fourier Transform to effectively model non-stationary signals in continuous dynamical systems, achieving state-of-the-art results across various PDE tasks.

Contribution

The paper proposes CoNO, a novel neural operator with a fractional Fourier transform-based kernel, demonstrating universal approximation and superior empirical performance on challenging PDEs.

Findings

Achieves an average relative gain of 10.9% over existing models.

Outperforms competitors in zero-shot super-resolution and noise robustness.

Learns effectively from only 60% of the training data.

Abstract

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.