\textit{Objective:} In this paper, we introduce Physics-Informed Fourier Networks for Electrical Properties Tomography (PIFON-EPT), a novel deep learning-based method that solves an inverse scattering problem based on noisy and/or incomplete magnetic resonance (MR) measurements. \textit{Methods:} We used two separate fully-connected neural networks, namely $B_1^{+}$ Net and EP Net, to solve the Helmholtz equation in order to learn a de-noised version of the input $B_1^{+}$ maps and estimate the object's EP. A random Fourier features mapping was embedded into $B_1^{+}$ Net, to learn the high-frequency details of $B_1^{+}$ more efficiently. The two neural networks were trained jointly by minimizing the combination of a physics-informed loss and a data mismatch loss via gradient descent. \textit{Results:} We performed several numerical experiments, showing that PIFON-EPT could provide physically consistent reconstructions of the EP and transmit field. Even when only $50\%$ of the noisy MR measurements were used as inputs, our method could still reconstruct the EP and transmit field with average error $2.49\%$, $4.09\%$ and $0.32\%$ for the relative permittivity, conductivity and $B_{1}^{+}$, respectively, over the entire volume of the phantom. The generalized version of PIFON-EPT that accounts for gradients of EP yielded accurate results at 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 EP estimation. \textit{Significance:} PIFON-EPT can efficiently de-noise $B_1^{+}$ maps, which has the potential to improve other MR-based EPT techniques. Furthermore, PIFON-EPT is the first technique that can reconstruct EP and $B_{1}^{+}$ simultaneously from incomplete noisy MR measurements.

翻译：\textit{目的：}本文提出基于物理信息傅里叶网络的电阻抗断层成像方法（PIFON-EPT），这是一种基于含噪和/或不完全磁共振测量数据求解逆散射问题的新型深度学习方法。\textit{方法：}我们采用两个独立的全连接神经网络——$B_1^+$网络和EP网络——求解亥姆霍兹方程，以学习输入$B_1^+$映射的去噪版本并估计目标的电阻抗特性。通过将随机傅里叶特征嵌入$B_1^+$网络，可更高效地学习$B_1^+$的高频细节。两个神经网络通过梯度下降联合最小化物理信息损失与数据失配损失的组合进行训练。\textit{结果：}多项数值实验表明，PIFON-EPT能够提供物理一致的电阻抗特性和发射场重建结果。即使在仅使用$50\%$含噪磁共振测量数据作为输入的情况下，该方法仍能以平均误差$2.49\%$（相对介电常数）、$4.09\%$（电导率）和$0.32\%$（$B_1^+$）重建整个体模体积的电阻抗特性和发射场。考虑电阻抗特性梯度的PIFON-EPT扩展版本在无需任何边界条件的情况下，也能在不同电阻抗特性值区域的界面上获得精确结果。\textit{结论：}本研究验证了PIFON-EPT的可行性，表明其可成为电阻抗特性估计的精确有效方法。\textit{意义：}PIFON-EPT能有效去除$B_1^+$图的噪声，有望改进其他基于磁共振的电阻抗断层成像技术。此外，PIFON-EPT是首个可从不完全含噪磁共振测量数据中同时重建电阻抗特性和$B_1^+$的技术。