# PIFON-EPT: MR-Based Electrical Property Tomography Using Physics-Informed Fourier Networks

arXiv: https://arxiv.org/abs/2302.11883 · HTML: https://mlsanity.com/paper/2302.11883/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 75 references that are in the map.
- First posted on arXiv 23 Feb 2023.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [Towards Understanding the Spectral Bias of Deep Learning](https://mlsanity.com/paper/1912.01198/) (2019)
- [The Convergence Rate of Neural Networks for Learned Functions of Different Frequencies](https://mlsanity.com/paper/1906.00425/) (2019)
- [Towards Robust Neural Networks via Close-loop Control](https://mlsanity.com/paper/2102.01862/) (2021)
- [TT-PINN: A Tensor-Compressed Neural PDE Solver for Edge Computing](https://mlsanity.com/paper/2207.01751/) (2022)
- [Contrastive Learning with Consistent Representations](https://mlsanity.com/paper/2302.01541/) (2023)
- [MR-Based Electrical Property Reconstruction Using Physics-Informed Neural Networks](https://mlsanity.com/paper/2210.12584/) (2022)
- [Memory footprint reduction for the FFT-based volume integral equation method via tensor decompositions](https://mlsanity.com/paper/1811.00484/) (2018)
- [A Fast Volume Integral Equation Solver with Linear Basis Functions for the Accurate Computation of Electromagnetic Fields in MRI](https://mlsanity.com/paper/1902.02196/) (2019)
- [Combining Deep Learning and 3D Contrast Source Inversion in MR-based Electrical Properties Tomography](https://mlsanity.com/paper/1908.04542/) (2019)
- [Magnetic-resonance-based electrical property mapping using Global Maxwell Tomography with an 8-channel head coil at 7 Tesla: a simulation study](https://mlsanity.com/paper/2003.09285/) (2020)
- [Physics-Coupled Neural Network Magnetic Resonance Electrical Property Tomography (MREPT) for Conductivity Reconstruction](https://mlsanity.com/paper/2109.12873/) (2021)
- [A hybrid volume-surface integral equation method for rapid electromagnetic simulations in MRI](https://mlsanity.com/paper/2206.12409/) (2022)

## Built on by (most cited first, 4 of 4)

- [Tensor-Compressed Back-Propagation-Free Training for (Physics-Informed) Neural Networks](https://mlsanity.com/paper/2308.09858/) (2023)
- [Real-Time FJ/MAC PDE Solvers via Tensorized, Back-Propagation-Free Optical PINN Training](https://mlsanity.com/paper/2401.00413/) (2023)
- [Scalable Back-Propagation-Free Training of Optical Physics-Informed Neural Networks](https://mlsanity.com/paper/2502.12384/) (2025)
- [Electromagnetic Inverse Scattering from a Single Transmitter](https://mlsanity.com/paper/2506.21349/) (2025)

## Abstract (arXiv)

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 ≤ 5% error, and denoised and completed the measurements with ≤ 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.