CoNO: Complex Neural Operator for Continous Dynamical Physical Systems
This addresses a bottleneck in scientific machine learning for modeling complex physical systems, though it appears incremental as an enhancement to existing neural operator frameworks.
The paper tackles the limitation of neural operators in handling non-stationary signals in continuous dynamical systems by introducing the Complex Neural Operator (CoNO), which uses Fractional Fourier Transform and achieves state-of-the-art performance with an average relative gain of 10.9% across seven PDEs.
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.