  We propose Physics-Informed Fourier Networks for Electrical Properties (EP)
Tomography (PIFON-EPT), a novel deep learning-based method for EP
reconstruction using noisy and/or incomplete magnetic resonance (MR)
measurements. Our approach leverages the Helmholtz equation to constrain two
networks, responsible for the denoising and completion of the transmit fields,
and the estimation of the object's EP, respectively. We embed a random Fourier
features mapping into our networks to enable efficient learning of
high-frequency details encoded in the transmit fields. We demonstrated the
efficacy of PIFON-EPT through several simulated experiments at 3 and 7 tesla
(T) MR imaging, and showed that our method can reconstruct physically
consistent EP and transmit fields. Specifically, when only $20\%$ of the noisy
measured fields were used as inputs, PIFON-EPT reconstructed the EP of a
phantom with $\leq 5\%$ error, and denoised and completed the measurements with
$\leq 1\%$ error. Additionally, we adapted PIFON-EPT to solve the generalized
Helmholtz equation that accounts for gradients of EP between inhomogeneities.
This yielded improved results at interfaces between different materials without
explicit knowledge of boundary conditions. PIFON-EPT is the first method that
can simultaneously reconstruct EP and transmit fields from incomplete noisy MR
measurements, providing new opportunities for EPT research.
