Abstract: Partial differential equations(PDEs) are crucial mathematical tools for describing real-world systems,and solving them is key for predicting and analyzing system behavior.Analytical solutions for PDEs are often difficult to obtain,and numerical methods are typically used for approximate solutions.However,numerical solutions for parameterized PDEs can be inefficient.In recent years,the use of deep learning for solving PDEs has shown its advantages in addressing these issues,and particularly the Fourier Neural Operator(FNO) has proven effective.However,FNO only captures global information through convolution in the frequency domain and struggles with multi-scale information of PDEs.To address this challenge,a locally-enhanced FNO model is proposed,incorporating a parallel multi-size convolution module in the Fourier layer to enhance the model’s capability to capture local multi-scale information.Behind the linear layer,a multi-branch feature fusion module is introduced,enhancing the model’s ability to integrate multi-channel information by elevating the data across different channels.Experimental results demonstrate that the model reduces errors by 30.9% in solving Burgers’ equation,18.5% in Darcy Flow equations,and 5.5% in Navier-Stokes equations.

Key words: Deep learning, Partial differential equations, Fourier neural operator, Multi-size convolution, Multi-branch feature fusion, Multiscale PDE