Component Fourier Neural Operator for Singularly Perturbed Differential Equations
This addresses computational challenges in solving SPDEs for applications in physics and engineering, representing an incremental improvement over existing operator learning methods.
The paper tackled solving singularly perturbed differential equations (SPDEs) by introducing Component Fourier Neural Operator (ComFNO), which improved accuracy compared to vanilla FNO, as demonstrated in experiments across diverse SPDE classes.
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.