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.

翻译：深度学习方法在数值求解各类偏微分方程方面引起了广泛关注。其中物理信息神经网络（PINN）尤为突出，它将物理原理融入神经网络，将偏微分方程求解过程转化为神经网络的优化问题。针对一系列困难条件下的对流扩散方程（ADE）求解，本文提出了一种新型网络结构。该架构将求解器（PINN方法中使用的多尺度深度神经网络MscaleDNN）与一种称为HCPINN的硬约束技术相结合。该方法通过引入一个包含控制方程残差并惩罚边界与初始条件偏差的损失函数，构造了ADE待求解的修正解形式。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。为缓解神经网络中的"频谱偏差"现象，我们在HCPINN中集成了MscaleDNN的子网络结构与傅里叶诱导激活函数，从而形成了一种混合方法SFHCPINN。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在精度和效率上均优于标准PINN及其嵌入傅里叶特征的子网络版本，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。