Component Fourier Neural Operator for Singularly Perturbed Differential Equations

Solving Singularly Perturbed Differential Equations (SPDEs) poses computational challenges arising from the rapid transitions in their solutions within thin regions. The effectiveness of deep learning in addressing differential equations motivates us to employ these methods for solving SPDEs. In this manuscript, we introduce Component Fourier Neural Operator (ComFNO), an innovative operator learning method that builds upon Fourier Neural Operator (FNO), while simultaneously incorporating valuable prior knowledge obtained from asymptotic analysis. Our approach is not limited to FNO and can be applied to other neural network frameworks, such as Deep Operator Network (DeepONet), leading to potential similar SPDEs solvers. Experimental results across diverse classes of SPDEs demonstrate that ComFNO significantly improves accuracy compared to vanilla FNO. Furthermore, ComFNO exhibits natural adaptability to diverse data distributions and performs well in few-shot scenarios, showcasing its excellent generalization ability in practical situations. With the surge of deep learning, efforts have been directed toward employing artificial neural networks for solving partial differential equations (PDEs) (Roos, Stynes, and Tobiska 2008), particularly in the field of physics-informed machine learning (Bar-Sinai et al. 2019; Greenfeld et al. 2019; Karniadakis et al. 2021) . Notably, operator learning techniques like FNO (Li et al. 2020) and DeepONet (Lu et al. 2021) have gained attention for their ability to learn operators between infinite-dimensional functional spaces. However, when addressing SPDEs, standard methods like