Deep learning methods have gained considerable interest in the numerical solution of various partial differential equations (PDEs). One particular focus is on physics-informed neural networks (PINNs), which integrate physical principles into neural networks. This transforms the process of solving PDEs into optimization problems for neural networks. In order 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, which is a multi-scale deep neural network (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 advection-diffusion equations (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 advection-diffusion equations (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.

翻译：深度学习方法在各类偏微分方程的数值求解中引起了广泛关注，其中物理信息神经网络（PINNs）将物理原理融入神经网络，将偏微分方程的求解转化为神经网络的优化问题。针对一系列具有挑战性的对流扩散方程，本文提出了一种新型网络结构。该结构将基于PINN方法的多尺度深度神经网络（MscaleDNN）求解器与一种称为HCPINN的硬约束技术相结合。该方法通过利用包含控制方程残差及边界初始条件偏差惩罚的损失函数，改进了对流扩散方程解的表达形式。通过自动满足边界约束，该方法提升了PINN技术的精度与效率。为应对神经网络中的"频谱偏置"现象，HCPINN中引入了MscaleDNN的子网络结构与傅里叶诱导激活函数，形成了一种名为SFHCPINN的混合方法。通过涉及不同维度对流扩散方程的多项数值实验，验证了SFHCPINN的有效性。数值结果表明，SFHCPINN在精度与效率上均优于标准PINN及其含傅里叶特征嵌入的子网络版本，能够有效处理对流扩散方程中的复杂边界条件与高频场景。