Amortized Bayesian Inference of GISAXS Data with Normalizing Flows Maksim Zhdanov_2

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Amortized Bayesian Inference of GISAXS Data with

Normalizing Flows

Maksim Zhdanov

Helmholtz-Zentrum Dresden-Rossendorf

maxxxzdn@gmail.com

Lisa Randolph

European XFEL

Thomas Kluge

Helmholtz-Zentrum Dresden-Rossendorf

Motoaki Nakatsutsumi

European XFEL

Christian Gutt

University of Siegen

Marina Ganeva

Forschungszentrum Jülich

Nico Hoffmann

Helmholtz-Zentrum Dresden-Rossendorf

Abstract

Grazing-Incidence Small-Angle X-ray Scattering (GISAXS) is a modern imaging

technique used in material research to study nanoscale materials. Reconstruction of

the parameters of an imaged object imposes an ill-posed inverse problem that is fur-

ther complicated when only an in-plane GISAXS signal is available. Traditionally

used inference algorithms such as Approximate Bayesian Computation (ABC) rely

on computationally expensive scattering simulation software, rendering analysis

highly time-consuming. We propose a simulation-based framework that combines

variational auto-encoders and normalizing ﬂows to estimate the posterior distribu-

tion of object parameters given its GISAXS data. We apply the inference pipeline

to experimental data and demonstrate that our method reduces the inference cost

by orders of magnitude while producing consistent results with ABC.

1 Introduction

X-ray scattering

Structural and morphological properties of surfaces and multi-layer thin ﬁlms

can be investigated by grazing-incidence small angle X-ray scattering (GISAXS). An incident X-ray

beam hits the sample under grazing incidence, after which its scattering pattern is recorded by a 2D

area detector. The analysis of GISAXS patterns allows one to observe in-situ transient surface and

subsurface density proﬁles with nanometer depth resolution. It is important for studying the growth

of nanomaterials, as demonstrated by Metwalli et al. [1], and for investigating the (sub-)picosecond

surface dynamics of laser-irradiated solids, as shown by Randolph et al. [

]. The reconstruction of

the structural properties is complicated by multiple scattering contributions and the missing phase

information. As the number of layers grows, the scattering patterns get highly complex, rendering the

inference computationally infeasible without prior knowledge about imaged objects.

Related work

Traditionally, a simulated GISAXS pattern of a sample model is generated and

compared with the experimentally measured pattern. Iterative ﬁtting processes adjust the model

parameters by minimizing

χ2

until they match the experimental GISAXS pattern [

]. The need

to execute a forward model iteratively makes a comprehensive analysis of GISAXS images highly

time-consuming, which in turn limits the throughput of large-scale neutron and X-ray facilities. The

need for accelerated inference motivated the spread of machine learning-based algorithms. Cherukara

et al. [

] demonstrated the real-time inversion of Wide-Angle X-ray scattering using generative neural

networks yielding 500 times faster inference compared to standard iterative algorithms. Ikemoto

Preprint. Under review.

arXiv:2210.01543v1 [cs.LG] 4 Oct 2022

Figure 1: A schematic overview of the proposed inference pipeline for GISAXS data (A). In the ﬁrst

step, conditional probability

p(X, z|ζ)

over GISAXS data

and latent variables

given the in-plane

GISAXS signal

is approximated by conditional VAEs (B). The probabilistic model allows us to

compute robust representation

that can be obtained even when only

is given as input. Afterwards,

we approximate posterior distribution

p(y|c)

over object parameters with normalizing ﬂows (D),

yielding fast inference that allows accelerating feedback during experiments (C).

et al. [

] and Liu et al. [

] use convolutional neural networks for the one-step classiﬁcation of

experimental images. At the same time, Van Herck et al. [

] infer the rotation distribution of

nanoparticle arrangements. Similar to our use case, Mironov et al. [

] use convolutional neural

networks with uncertainty quantiﬁcation to estimate ﬁlm parameters from neutron reﬂectivity curves.

Our contribution

In this paper, we develop an inference framework allowing for fast and robust

reconstruction of GISAXS data to accelerate GISAXS data analysis. As the signal-to-noise ratio

(SNR) of experimental images might suffer from distortions caused by grazing-incidence geometry

[

], we mainly focus on using an in-plane scattering signal

as input. Despite the recent success of

discriminative neural networks, such models do not account for the inherent ambiguity of reconstruc-

tion and, as a rule, do not provide a researcher with uncertainty quantiﬁcation. Instead of learning

a function from images to parameters, we use the Bayesian approach and estimate the posterior

distribution of object parameters given the GISAXS data. Our framework has a two-fold structure.

In the ﬁrst step, we learn a robust probabilistic representation of the GISAXS generative process

with variational auto-encoders [

]. Second, we model the posterior distribution via likelihood-free

inference [11] with normalizing ﬂows [12].

2 Methods

Conditional variational auto-encoders

Variational autoencoders (VAE) [

] is a framework to

model the data generation process via approximation of joint probability

p(x, z)

over observed

variables

and latent variables

. It combines a generative model

pθ(x|z)

, an inference model

qφ(z|x)

and a prior

p(z)

, allowing unconditional data generation from a learned distribution model.

Conditional VAEs (CVAEs) [

] is an extension of the framework that models a conditional distri-

bution

p(x, z|y)

. To learn the model parameters

, one maximizes the conditional log-likelihood

log pθ(x|y)

via maximizing the evidence lower bound [

]. Subsequently, one can sample from

conditional distribution

pθ(x|y, z)

where random noise

attributes for the variance in reconstruction

of xfrom y. We model each distribution as Normal distribution with learnable mean and variance.

We deﬁne the in-plane scattering signal (proﬁle) as the average of a central region (see Fig. 1A) of an image

over the lateral dimension of a detector with subtracted parasitic scattering signal from the beamstop.

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AmortizedBayesianInferenceofGISAXSDatawithNormalizingFlowsMaksimZhdanovHelmholtz-ZentrumDresden-Rossendorfmaxxxzdn@gmail.comLisaRandolphEuropeanXFELThomasKlugeHelmholtz-ZentrumDresden-RossendorfMotoakiNakatsutsumiEuropeanXFELChristianGuttUniversityofSiegenMarinaGanevaForschungszentrumJülichNicoHoffm...

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Amortized Bayesian Inference of GISAXS Data with Normalizing Flows Maksim Zhdanov_2

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