将渐近分析先验融入FNO，提升奇异摄动方程求解精度
Component Fourier Neural Operator for Singularly Perturbed Differential Equations
- 在FNO基础上引入分量傅里叶模块，融合渐近分析先验知识
- 在多类奇异摄动方程上相比原FNO显著提升精度
- 对不同数据分布和少量样本均表现良好，适合实际应用
求解奇异摄动微分方程（SPDEs）面临计算挑战，因其解在薄区域存在快速变化。深度学习在微分方程求解中的有效性激发了我们采用此类方法求解SPDEs。本文提出组件傅里叶神经算子（ComFNO），一种基于傅里叶神经算子（FNO）的创新算子学习方法，同时融入由渐近分析获得的重要先验知识。该方法不仅适用于FNO，还可推广至DeepONet等其他神经网络框架，有望构建类似SPDEs求解器。在多种类型的SPDEs上进行实验表明，ComFNO相比原FNO显著提升准确性。此外，ComFNO对不同数据分布具有天然适应性，在少样本场景下表现优异，展现出出色的泛化能力。
原文摘要 · Abstract (English)
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.
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