A novel paradigm for solving PDEs: multi-scale neural computing

Numerical simulation is dominant in solving partial differential equations (PDEs), but balancing fine-grained grids with low computational costs is challenging. Recently, solving PDEs with neural networks (NNs) has gained interest, yet cost-effectiveness and high accuracy remain a challenge. This work introduces a novel paradigm for solving PDEs, called multi-scale neural computing (MSNC), considering spectral bias of NNs and local approximation properties in the finite difference method (FDM). The MSNC decomposes the solution with a NN for efficient capture of global scale and the FDM for detailed description of local scale, aiming to balance costs and accuracy. Demonstrated advantages include higher accuracy (10 times for 1D PDEs, 20 times for 2D PDEs) and lower costs (4 times for 1D PDEs, 16 times for 2D PDEs) than the standard FDM. The MSNC also exhibits stable convergence and rigorous boundary condition satisfaction, showcasing the potential for hybrid of NN and numerical method.