Ye Li

Ye Li, Ting Du, Yiwen Pang et al.

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.

Yuyan Zhou, Ye Li, Lei Feng et al.

Recent studies showed that the generalization of neural networks is correlated with the sharpness of the loss landscape, and flat minima suggests a better generalization ability than sharp minima. In this paper, we propose a novel method called \emph{optimum shifting}, which changes the parameters of a neural network from a sharp minimum to a flatter one while maintaining the same training loss value. Our method is based on the observation that when the input and output of a neural network are fixed, the matrix multiplications within the network can be treated as systems of under-determined linear equations, enabling adjustment of parameters in the solution space, which can be simply accomplished by solving a constrained optimization problem. Furthermore, we introduce a practical stochastic optimum shifting technique utilizing the Neural Collapse theory to reduce computational costs and provide more degrees of freedom for optimum shifting. Extensive experiments (including classification and detection) with various deep neural network architectures on benchmark datasets demonstrate the effectiveness of our method.