CoNO: Complex Neural Operator for Continuous Dynamical Systems

TL;DR

CoNO introduces a complex neural operator that leverages complex fractional Fourier transforms and algebraic properties to improve modeling of continuous dynamical systems, demonstrating superior performance and robustness across various tasks.

Contribution

The paper proposes a novel complex neural operator using fractional Fourier domain parameterization and complex-valued neural networks, enhancing representation and robustness in modeling PDEs.

Findings

Effective capture of PDEs with a single complex Fourier transform

Outperforms state-of-the-art models in robustness and data efficiency

Demonstrates superior performance in zero-shot super-resolution and out-of-distribution tasks

Abstract

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.