  Deep learning methods have gained considerable interest in the numerical
solution of various partial differential equations (PDEs). One particular focus
is physics-informed neural networks (PINN), which integrate physical principles
into neural networks. This transforms the process of solving PDEs into
optimization problems for neural networks. To address a collection of
advection-diffusion equations (ADE) in a range of difficult circumstances, this
paper proposes a novel network structure. This architecture integrates the
solver, a multi-scale deep neural networks (MscaleDNN) utilized in the PINN
method, with a hard constraint technique known as HCPINN. This method
introduces a revised formulation of the desired solution for ADE by utilizing a
loss function that incorporates the residuals of the governing equation and
penalizes any deviations from the specified boundary and initial constraints.
By surpassing the boundary constraints automatically, this method improves the
accuracy and efficiency of the PINN technique. To address the ``spectral bias''
phenomenon in neural networks, a subnetwork structure of MscaleDNN and a
Fourier-induced activation function are incorporated into the HCPINN, resulting
in a hybrid approach called SFHCPINN. The effectiveness of SFHCPINN is
demonstrated through various numerical experiments involving ADE in different
dimensions. The numerical results indicate that SFHCPINN outperforms both
standard PINN and its subnetwork version with Fourier feature embedding. It
achieves remarkable accuracy and efficiency while effectively handling complex
boundary conditions and high-frequency scenarios in ADE.
