  Numerical simulation is dominant in solving partial difference 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 remains 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.
