Deep neural networks have received widespread 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 Partial Differential Equations (PDEs) with multiple scales by means of Fourier-based mixed physics informed neural networks(dubbed FMPINN), the solver of FMPINN is configured as a multi-scale deep neural networks. 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 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, it 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。本文研究了利用基于傅里叶的混合物理信息神经网络（简称FMPINN）求解一类具有多尺度的椭圆型偏微分方程（PDEs），FMPINN求解器被配置为多尺度深度神经网络。与经典PINN方法不同，我们引入了一个关于PDE粗糙系数的对偶（通量）变量，以避免由多尺度PDE振荡系数导致的神经正切核矩阵病态问题。因此，除了物理守恒定律之外，辅助变量与多尺度系数梯度之间的差异被纳入代价函数中，进而通过某些优化方法最小化所定义的损失函数，获得PDE的满意解。此外，FMPINN引入了一种适合表示复杂目标函数导数的三角激活函数。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，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高效性和鲁棒性。