# Neural network analysis of neutron and X-ray reflectivity data: Incorporating prior knowledge for tackling the phase problem

arXiv: https://arxiv.org/abs/2307.05364 · HTML: https://mlsanity.com/paper/2307.05364/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 93 references that are in the map.
- First posted on arXiv 28 Jun 2023.

## Builds on (most cited first, 18 of 18)

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- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [1D Convolutional Neural Networks and Applications: A Survey](https://mlsanity.com/paper/1905.03554/) (2019)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [Learning two-phase microstructure evolution using neural operators and autoencoder architectures](https://mlsanity.com/paper/2204.07230/) (2022)
- [Wav2Letter: an End-to-End ConvNet-based Speech Recognition System](https://mlsanity.com/paper/1609.03193/) (2016)
- [Multi-Rate VAE: Train Once, Get the Full Rate-Distortion Curve](https://mlsanity.com/paper/2212.03905/) (2022)
- [Closing the loop: Autonomous experiments enabled by machine-learning-based online data analysis in synchrotron beamline environments](https://mlsanity.com/paper/2306.11899/) (2023)
- [Fast Fitting of Reflectivity Data of Growing Thin Films Using Neural Networks](https://mlsanity.com/paper/1910.02898/) (2019)

## Built on by (most cited first, 1 of 1)

- [Towards real-time surrogate-free Bayesian inversion for neutron reflectometry](https://mlsanity.com/paper/2509.06924/) (2025)

## Abstract (arXiv)

Due to the lack of phase information, determining the physical parameters of multilayer thin films from measured neutron and X-ray reflectivity curves is, on a fundamental level, an underdetermined inverse problem. This so-called phase problem poses limitations on standard neural networks, constraining the range and number of considered parameters in previous machine learning solutions. To overcome this, we present an approach that utilizes prior knowledge to regularize the training process over larger parameter spaces. We demonstrate the effectiveness of our method in various scenarios, including multilayer structures with box model parameterization and a physics-inspired special parameterization of the scattering length density profile for a multilayer structure. By leveraging the input of prior knowledge, we can improve the training dynamics and address the underdetermined ("ill-posed") nature of the problem. In contrast to previous methods, our approach scales favorably when increasing the complexity of the inverse problem, working properly even for a 5-layer multilayer model and an N-layer periodic multilayer model with up to 17 open parameters.