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.

翻译：求解奇异摄动微分方程（SPDEs）面临计算挑战，这源于其解在薄层区域内存在急剧变化。深度学习在求解微分方程方面的有效性促使我们采用此类方法来解决SPDEs。本文提出了一种创新的算子学习方法——分量傅里叶神经算子（ComFNO），该方法以傅里叶神经算子（FNO）为基础，同时融入了从渐近分析中获得的宝贵先验知识。我们的方法不仅限于FNO，还可应用于其他神经网络框架，如深度算子网络（DeepONet），从而有望构建出类似的SPDE求解器。在多种类型SPDEs上的实验结果表明，与原始FNO相比，ComFNO显著提高了求解精度。此外，ComFNO对不同数据分布表现出天然的适应性，并在少样本场景下表现良好，这展示了其在实际应用中的优异泛化能力。