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Partial differential equations (PDEs) play a foundational role in modelingphysical phenomena. This study addresses the challenging task of determiningvariable coefficients within PDEs from measurement data. We introduce a novelneural network, "pseudo-differential IAEnet" (pd-IAEnet), which drawsinspiration from pseudo-differential operators. pd-IAEnet achievessignificantly enhanced computational speed and accuracy with fewer parameterscompared to conventional models. Extensive benchmark evaluations are conductedacross a range of inverse problems, including Electrical Impedance Tomography(EIT), optical tomography, and seismic imaging, consistently demonstratingpd-IAEnet's superior accuracy. Notably, pd-IAEnet exhibits robustness in thepresence of measurement noise, a critical characteristic for real-worldapplications. An exceptional feature is its discretization invariance, enablingeffective training on data from diverse discretization schemes whilemaintaining accuracy on different meshes. In summary, pd-IAEnet offers a potentand efficient solution for addressing inverse PDE problems, contributing toimproved computational efficiency, robustness, and adaptability to a wide arrayof data sources.

Pseudo-differential integral autoencoder network for inverse PDE operators