\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：} 本文提出基于物理信息傅里叶网络（Physics-Informed Fourier Networks, PIFONs）的电阻抗成像（Electrical Properties Tomography, EPT）方法。该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，全局学习电阻抗（Electrical Properties, EPs）。\textit{方法：} 我们采用两个独立的全连接神经网络（即$B_1^{+}$网络和EP网络）分别学习任意位置的$B_1^{+}$场和电阻抗。通过将随机傅里叶特征映射嵌入$B_1^{+}$网络，网络能够更高效地学习$B_1^{+}$场。两个网络通过最小化物理信息损失与数据失配损失的组合（基于梯度下降法）进行联合训练。\textit{结果：} 实验表明，即使缺失全部体积中一半的含噪MR测量数据，PIFON-EPT仍能在整个感兴趣区域提供物理一致的电阻抗和发射场重建。在仿体全域内，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电阻抗值不同区域的界面附近实现精确的电阻抗预测。\textit{结论：} 本研究验证了PIFON-EPT的可行性，表明其可作为电阻抗估计的精确且有效方法。\textit{意义：} PIFON-EPT能够高效去噪MR测量，具有改进其他基于MR的EPT技术的潜力。此外，这是基于MR的EPT方法首次能够从不完整的模拟含噪MR测量中同时重建电阻抗和$B_{1}^{+}$场。