Deep neural networks have received significant attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this work, we probe into solving a class of elliptic PDEs with multiple scales by means of Fourier-based mixed physics-informed neural networks (called FMPINN), and its solver is configured as a multi-scale DNN model. Unlike the classical PINN method, a dual (flux) variable about the rough coefficient of PDEs is introduced to avoid the ill-condition of neural tangent kernel matrix that resulted from the oscillating coefficient of multi-scale PDEs. Therefore, apart from the physical conservation laws, the discrepancy between the auxiliary variables and the gradients of multi-scale coefficients is incorporated into the cost function, then leveraging the optimization method to yield the satisfactory solution of PDEs by minimizing the defined loss. Additionally, a novel trigonometric activation function is introduced for FMPINN, which is suited for representing the derivatives of complex target functions. Handling the input data by Fourier feature mapping will effectively improve the capacity of deep neural networks to solve high-frequency problems. Finally, by introducing several numerical examples of multi-scale problems in various dimensional Euclidean spaces, we validate the efficiency and robustness of the proposed FMPINN algorithm in both low-frequency and high-frequency oscillation cases.

翻译：深度神经网络因其在工程与科学计算领域的简洁性和灵活性而受到广泛关注。本文探索利用基于傅里叶的混合物理信息神经网络（简称FMPINN）求解一类多尺度椭圆型偏微分方程，其求解器被构造成多尺度深度神经网络模型。不同于经典PINN方法，我们引入关于粗糙系数的对偶（通量）变量，以规避多尺度偏微分方程振荡系数导致的神经正切核矩阵病态问题。因此，除物理守恒定律外，辅助变量与多尺度系数梯度之间的差异被纳入损失函数，进而通过优化方法最小化所定义损失以得到偏微分方程的满意解。此外，本文为FMPINN引入一种新颖的三角激活函数，该函数适用于表示复杂目标函数的导数。通过傅里叶特征映射处理输入数据，将有效提升深度神经网络求解高频问题的能力。最后，通过引入多个不同维度欧氏空间中多尺度问题的数值算例，我们验证了所提出的FMPINN算法在低频和高频振荡情形下的有效性与鲁棒性。