Probabilistic Parameter Estimation Using a Gaussian Mixture Density Network: Application to X-ray Reflectivity Data Curve Fitting

TL;DR

This paper introduces a Gaussian mixture density network (MDN) approach for probabilistic parameter estimation in X-ray reflectivity data analysis, significantly reducing analysis time and providing confidence intervals and multiple solutions for complex data.

Contribution

The paper demonstrates the application of MDN for probabilistic prediction in XRR analysis, enabling confidence interval estimation and multimodal solution clustering, which improves efficiency and robustness.

Findings

MDN can estimate best-fit parameters and confidence intervals effectively.

The method provides multiple probable solutions for multimodal distributions.

It reduces analysis time compared to traditional iterative methods.

Abstract

X-ray reflectivity (XRR) is widely used for thin-film structure analysis, and XRR data analysis involves minimizing the difference between an XRR curve calculated from model parameters describing the thin-film structure. This analysis takes a certain amount of time because it involves many unavoidable iterations. However, the recently introduced artificial neural network (ANN) method can dramatically reduce the analysis time in the case of repeated analyses of similar samples. Here, we demonstrate the analysis of XRR data using a mixture density network (MDN), which enables probabilistic prediction while maintaining the advantages of an ANN. First, under the assumption of a unimodal probability distribution of the output parameter, the trained MDN can estimate the best-fit parameter and, at the same time, estimate the confidence interval (CI) corresponding to the error bar of the best-fit parameter. The CI obtained in this manner is similar to that obtained using the Neumann process, a well-known statistical method. Next, the MDN method provides several possible solutions for each parameter in the case of a multimodal distribution of the output parameters. An unsupervised machine learning method is used to cluster possible parameter sets in order of high probability. Determining the true value by examining the candidates of the parameter sets obtained in this manner can help solve the inherent inverse problem associated with scattering data.