  Partial differential equations (PDEs) play a foundational role in modeling
physical phenomena. This study addresses the challenging task of determining
variable coefficients within PDEs from measurement data. We introduce a novel
neural network, "pseudo-differential IAEnet" (pd-IAEnet), which draws
inspiration from pseudo-differential operators. pd-IAEnet achieves
significantly enhanced computational speed and accuracy with fewer parameters
compared to conventional models. Extensive benchmark evaluations are conducted
across a range of inverse problems, including Electrical Impedance Tomography
(EIT), optical tomography, and seismic imaging, consistently demonstrating
pd-IAEnet's superior accuracy. Notably, pd-IAEnet exhibits robustness in the
presence of measurement noise, a critical characteristic for real-world
applications. An exceptional feature is its discretization invariance, enabling
effective training on data from diverse discretization schemes while
maintaining accuracy on different meshes. In summary, pd-IAEnet offers a potent
and efficient solution for addressing inverse PDE problems, contributing to
improved computational efficiency, robustness, and adaptability to a wide array
of data sources.
