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Deep neural networks have garnered widespread attention due to theirsimplicity and flexibility in the fields of engineering and scientificcalculation. In this study, we probe into solving a class of elliptic partialdifferential equations(PDEs) with multiple scales by utilizing Fourier-basedmixed physics informed neural networks(dubbed FMPINN), its solver is configuredas 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 toavoid the ill-condition of neural tangent kernel matrix caused by theoscillating coefficient of multi-scale PDEs. Therefore, apart from the physicalconservation laws, the discrepancy between the auxiliary variables and thegradients of multi-scale coefficients is incorporated into the cost function,then obtaining a satisfactory solution of PDEs by minimizing the defined lossthrough some optimization methods. Additionally, a trigonometric activationfunction is introduced for FMPINN, which is suited for representing thederivatives of complex target functions. Handling the input data by Fourierfeature mapping will effectively improve the capacity of deep neural networksto solve high-frequency problems. Finally, to validate the efficiency androbustness of the proposed FMPINN algorithm, we present several numericalexamples of multi-scale problems in various dimensional Euclidean spaces. Theseexamples cover both low-frequency and high-frequency oscillation cases,demonstrating the effectiveness of our approach. All code and data accompanyingthis manuscript will be made publicly available at\href{https://github.com/Blue-Giant/FMPINN}{https://github.com/Blue-Giant/FMPINN}.

Solving a class of multi-scale elliptic PDEs by means of Fourier-based mixed physics informed neural networks