\textit{Objective:} In this paper, we introduce Physics-Informed Fourier Networks (PIFONs) for Electrical Properties (EP) Tomography (EPT). Our novel deep learning-based method is capable of learning EPs globally by solving an inverse scattering problem based on noisy and/or incomplete magnetic resonance (MR) measurements. \textit{Methods:} We use two separate fully-connected neural networks, namely $B_1^{+}$ Net and EP Net, to learn the $B_1^{+}$ field and EPs at any location. A random Fourier features mapping is embedded into $B_1^{+}$ Net, which allows it to learn the $B_1^{+}$ field more efficiently. These two neural networks are trained jointly by minimizing the combination of a physics-informed loss and a data mismatch loss via gradient descent. \textit{Results:} We showed that PIFON-EPT could provide physically consistent reconstructions of EPs and transmit field in the whole domain of interest even when half of the noisy MR measurements of the entire volume was missing. The average error was $2.49\%$, $4.09\%$ and $0.32\%$ for the relative permittivity, conductivity and $B_{1}^{+}$, respectively, over the entire volume of the phantom. In experiments that admitted a zero assumption of $B_z$, PIFON-EPT could yield accurate EP predictions near the interface between regions of different EP values without requiring any boundary conditions. \textit{Conclusion:} This work demonstrated the feasibility of PIFON-EPT, suggesting it could be an accurate and effective method for electrical properties estimation. \textit{Significance:} PIFON-EPT can efficiently de-noise MR measurements, which shows the potential to improve other MR-based EPT techniques. Furthermore, it is the first time that MR-based EPT methods can reconstruct the EPs and $B_{1}^{+}$ field simultaneously from incomplete simulated noisy MR measurements.

翻译：\textit{目的：}本文提出物理信息傅里叶网络（PIFONs）用于电阻抗断层成像（EPT）。我们的新型深度学习方法能够通过求解基于含噪和/或不完整磁共振（MR）测量的逆散射问题，实现全局电阻抗参数（EPs）的学习。\textit{方法：}采用两个独立的全连接神经网络——$B_1^{+}$ 网络和EP网络，分别学习任意位置的$B_1^{+}$场和EPs。$B_1^{+}$网络嵌入随机傅里叶特征映射，使其能更高效地学习$B_1^{+}$场。这两个网络通过梯度下降法联合训练，最小化物理信息损失和数据失配损失的组合。\textit{结果：}我们证明，即使整个体积的含噪MR测量数据缺失一半，PIFON-EPT仍能在全部感兴趣区域内提供物理一致的EPs和发射场重建。在体模整个体积上，相对介电常数、电导率和$B_{1}^{+}$的平均误差分别为$2.49\%$、$4.09\%$和$0.32\%$。在允许$B_z$为零假设的实验中，PIFON-EPT无需任何边界条件即可在EP值不同的区域界面附近产生精确的EP预测。\textit{结论：}本工作验证了PIFON-EPT的可行性，表明该方法可成为电阻抗参数估计的准确而有效的手段。\textit{意义：}PIFON-EPT能有效去噪MR测量数据，这显示出改进其他基于MR的EPT技术的潜力。此外，这是首次基于MR的EPT方法能从含噪且不完整的模拟MR测量中同时重建EPs和$B_{1}^{+}$场。