Deep neural networks have garnered widespread attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this study, we probe into solving a class of elliptic partial differential equations(PDEs) with multiple scales by utilizing Fourier-based mixed physics informed neural networks(dubbed FMPINN), its solver is configured as a multi-scale deep neural network. In contrast to 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 caused by 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 obtaining a satisfactory solution of PDEs by minimizing the defined loss through some optimization methods. Additionally, a 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, to validate the efficiency and robustness of the proposed FMPINN algorithm, we present several numerical examples of multi-scale problems in various dimensional Euclidean spaces. These examples cover both low-frequency and high-frequency oscillation cases, demonstrating the effectiveness of our approach. All code and data accompanying this manuscript will be made publicly available at \href{https://github.com/Blue-Giant/FMPINN}{https://github.com/Blue-Giant/FMPINN}.

翻译：深度神经网络因其在工程与科学计算领域的简洁性及灵活性而受到广泛关注。本研究探讨了利用基于傅里叶的混合物理信息神经网络（简称FMPINN）求解一类含多尺度的椭圆型偏微分方程，其求解器被配置为多尺度深度神经网络。与经典PINN方法相比，我们引入了关于偏微分方程粗糙系数的对偶（通量）变量，以避免多尺度偏微分方程中震荡系数所导致的神经切向核矩阵病态问题。因此，除了物理守恒定律外，辅助变量与多尺度系数梯度之间的差异也被纳入代价函数中，并通过优化方法最小化所定义的损失函数，从而获得偏微分方程的满意解。此外，FMPINN引入了适合表征复杂目标函数导数的三角函数激活函数。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，为验证所提FMPINN算法的效率与鲁棒性，我们展示了多个不同维度欧氏空间中多尺度问题的数值算例。这些算例涵盖低频与高频振荡情形，验证了本方法的有效性。本文所有代码和数据集将在\href{https://github.com/Blue-Giant/FMPINN}{https://github.com/Blue-Giant/FMPINN}公开发布。