# A Machine Learning Inversion Scheme for Determining Interaction from
Scattering

Ming-Ching Chang1 Chi-Huan Tung2,3 Shou-Yi Chang2 Jan-Michael Carrillo4 Bobby
G. Sumpter4 Changwoo Do3 Wei-Ren Chen3 1Department of Computer Science,
University at Albany - State University of New York, Albany, New York 12222,
United States  2Department of Materials Science and Engineering, National
Tsing Hua University, Hsinchu 300044, Taiwan  3Neutron Scattering Division,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States
4Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak
Ridge, Tennessee 37831, United States

###### Abstract

We outline a machine learning strategy for determining the effective
interaction in the condensed phases of matter using scattering. Via a case
study of colloidal suspensions, we showed that the effective potential can be
probabilistically inferred from the scattering spectra without any restriction
imposed by model assumptions. Comparisons to existing parametric approaches
demonstrate the superior performance of this method in accuracy, efficiency,
and applicability. This method can effectively enable quantification of
interaction in highly correlated systems using scattering and diffraction
experiments.

Colloidal interactions, hard-sphere Yukawa potential, Ornstein–Zernike (OZ)
equation, closures, singular value decomposition, kernel ridge regression,
Gaussian process, simulation.

Measuring interaction between particles in condensed matter has been of
paramount interest since it provides a starting point for describing the
statistical properties of the system under consideration. Elastic scattering
and diffraction techniques have played an important role in this continued
endeavor: From the measured spectra, extensive effort has been devoted to
inferring the nature of forces that govern the properties of a variety of
highly correlated disordered systems including dense atomic liquids Schommers
(1983); Aers and Dharma-wardana (1984); Levesque et al. (1985); Lyubartsev and
Laaksonen (1995), ionic liquids March and Tosi (1984) such as electrolytes
Lyubartsev and Laaksonen (1999) and molten salts Babu and Ichiye (1994);
Caccamo (1996), molecular fluids Soper (1996); Yamaguchi et al. (1999),
suspensions of colloids, micelles and emulsions Schmitz (1992a); Nägele
(1996); Likos (2001), solutions of polymers Yethiraj and Schweizer (1992);
Schweizer and Curro (1994), and polyelectrolytes Yethiraj and Shew (1996); Qiu
et al. (2006). Nonetheless, potential inversion based on this experimental
protocol is often hampered by the difficulty of precisely modeling the two-
point static correlation functions, the quantity of interest in radiation
scattering experiments, in terms of the relevant parameter.

In this report we present a non-parametric strategy to circumvent the
intrinsic limitation of existing approaches and demonstrate its feasibility by
a case study of charged colloidal suspensions, a representative soft matter
system. When the long-range electrostatic repulsion dominates over all
distances, the effective interaction between charged colloids can be described
by a hard sphere with a Yukawa tail of screened Coulomb repulsion Pusey
(1991):

$\beta\;V_{HSY}(r)=\begin{cases}\infty,&\text{if }r<D\\\
A\;\frac{\exp\left[-\kappa(r-D)\right]}{r},&\text{otherwise}\end{cases}$ (1)

where $\beta\equiv\frac{1}{k_{B}T}$ is the Boltzmann factor, $A$ the coupling
parameter defined as $\frac{Z^{2}e^{2}}{\epsilon(1+\frac{\kappa D}{2})^{2}}$,
$D$ the colloidal diameter, $Z$ the charge number, $e$ the electric charge,
$\epsilon$ the dielectric constant of solvent, and $\kappa$ the Debye
screening constant. Eq. (1) has been extensively used to model the
electrostatic interaction in a wide variety of charged colloidal systems
including ionized nanoparticles Schmitz (1992a), self-assemblies Zemb and
Charpin (1985), and biological systems Tardieu et al. (1999). The relevant
correlation function is the inter-particle structure factor $S(Q)$ in
reciprocal $Q$ space. One well-adopted approach to determine $V_{HSY}(r)$ from
the measured $S(Q)$ is through the Ornstein-Zernike (OZ) integral equation
Hansen and McDonald (1986); Schmitz (1992b); Nägele (2004):

$h(r_{12})=c(r_{12})+n_{p}\int c(r_{13})h(r_{23})d^{3}r_{3},$ (2)

where $h(r_{12})\equiv g(r_{12})-1$ and $g(r)$ is the pair distribution
function, $c(r_{12})$ is the direct correlation function, and $n_{p}$ is the
particle number density. $S(Q)$ is the Fourier transform of $g(r)$. Since both
$h(r)$ and $c(r)$ are unknown, a second closure equation is required to solve
Eq. (2). Several closures have been developed as approximations Schmitz
(1992b); Klein and D’Aguanno (1996); Nägele (1996). Despite the popularity of
the integral equation approach, existing studies have indicated its
limitations: Because the accuracy of a given closure is not known $a$
$priori$, its validity in any specific phase region needs to be justified
computationally Belloni (1983); Beresford-Smith et al. (1985); Belloni (1985,
1991); Fritz et al. (2000); Banchio and Nägele (2008); Heinen et al. (2011a,
b). Moreover, the convergence behavior of each closure is found to depend on
the complexity of adopted numerical procedures Huš et al. (2013). For highly
charged systems, the extraction of $V_{HSY}(r)$ from the corresponding $S(Q)$
is also known to be compounded by the strong electrostatic interactions Anta
and Lago (2002). As alluded to above, no ideal closure is currently available.

The position we take here is that developing another closure does not
necessarily provide the most effective solution. From a Bayesian perspective
Murphy (2012) we instead sought to solve this inversion problem via a machine
learning (ML) approach based on Gaussian process Rasmussen and Williams
(2006), which defines a distribution over functions as a conceptual extension
of the familiar Gaussian distribution. Using Eq. (1) we computationally
generated an extensive library of $S(Q)$ from the equilibrium fluid phase SI ,
which is defined as the training set $\\{S_{train}(Q)\\}$ in this study. In
the vector space of $S(Q)$, we treated the $A$, $\kappa$ and $n_{p}$ of
$\\{S_{train}(Q)\\}$ as a collection of normally distributed random variables
and accordingly determined the statistical relationship of each variable in
the ML process SI . Using the optimized covariance matrix designed to quantify
data correlation, we were able to probabilistically infer the values of $A$,
$\kappa$ and $n_{p}$ from a given $S(Q)$ without having to rely on a
prescribed parametric equation, such as OZ equation and a complementary
closure, to specify this mathematical relationship deterministically. The
uncertainties associated with the related parameters naturally emerge during
the inference process embedded in the spectral analysis procedure.

Figure 1: (a) The eigenvalues of principal components. As demonstrated in
panel (b), the results of maximum profile likelihood analysis Zhu and Ghodsi
(2006) show that the linear subspace spanned by the first three singular
vectors given in Panel (c) is sufficiently expressive to illustrate the
features of the correlated data points.

Before implementing the ML process, it is critical to first examine the
feasibility of our proposed approach. To uniquely determine the potential
parameters from a given $S(Q)$, a necessary condition to meet is the
separability of dataset in the vector space of $S(Q)$ of dimension 80, the
sampled $Q$ points in our computational trajectory analysis. Here we used a
principal component analysis (PCA) by the singular value decomposition (SVD)
Strang (2016) to extract relevant information of the data distribution in this
high dimensional vector space SI . From the results of singular value analysis
Zhu and Ghodsi (2006) presented in Figs. 1(a) and 1(b), it is confirmed that
the variance of original data is mostly retained by the first three
statistically significant singular value ranks. The vector space
$\mathbb{R}^{3}$ spanned by these three singular vectors, denoted as SVD0,
SVD1 and SVD2 in Fig. 1(c), is therefore sufficiently expressive in capturing
essential features of the correlated data.

Figure 2:  (a) The distributions of training data in the $\mathbb{R}^{3}$
vector space spanned by SVD0, SVD1 and SVD2 given in Fig. 1(c). The magenta
symbols represent the fluid phase regions where the effective interaction can
be precisely determined by solving the OZ equation with MPB-RMSA closure
Heinen et al. (2011a, b). The blacks symbols represent the three highly
correlated charged colloidal suspensions given in Fig. 4. Panels (b)-(d) give
the distributions of $\phi$, $\frac{1}{\kappa D}$ and $\beta A$. The detailed
distributions for $\phi=0.3$ are given as an example in (e).

In this $\mathbb{R}^{3}$ vector space, each $S(Q)$ in $\\{S_{train}(Q)\\}$ is
represented by a point and the distribution of a gridded dataset is given in
Fig. 2. As demonstrated by Fig. 2(a), it is seen to be narrowly distributed
along one-dimension of a twisted three-dimensional manifold which resembles
the shape of a half fern leaf: The elongated central axis represents the data
of of hard-sphere fluids with different volume fraction $\phi$ defined as
$\frac{\pi}{6}n_{p}D^{3}$. Each pinna bursting forth from the central stalk
consists of data points with same $\phi$ but different $\kappa$ and $A$. We
thoroughly examined the data distribution presented in Fig. 2(a) and no
inseparable overlapping was found. This observation, which reflects the one-
to-one mapping between $S(Q)$ and a set of $\phi$, $\frac{1}{\kappa D}$, and
$\beta A$ in colloidal fluid phase, provides initial support for a viable
framework capable of inversely extracting potential parameters from measured
scattering functions. The data points collected from the fluid phase
characterized by $A$ $<$ 3 $k_{B}T$ are further marked by magenta color in
Fig. 2(a). In this phase region the effective interaction of charged colloidal
suspensions can be precisely determined by OZ equation complemented by the
MPB-RMSA closure proposed by Heinen and coworkers Heinen et al. (2011a, b), a
sophisticated approach among the continued efforts Hansen and Hayter (1982);
Snook and Hayter (1992) aiming at improving the MSA closure Hayter and Penfold
(1981). Within probed phase regions, its quantitative accuracy is found to be
equivalent to that of Rogers-Young closure Rogers and Young (1984) but the
computational efficient is significantly improved. One can further label the
data points presented in Fig. 2(a) with the numerical values of $\phi$,
$\frac{1}{\kappa D}$, and $\beta A$ to examine the characteristics of their
distributions. Judging from the results given in Figs. 2(b)-2(d), the
distributions of $\phi$, $\frac{1}{\kappa D}$ and $\beta A$ are all seen to
vary smoothly. This observation suggests that the data points of these three
parameters not only are self-avoiding but also highly correlated with a
certain length scale. One can therefore uniquely extract the potential
parameters from a given $S(Q)$ in its vector space.

Having verified the feasibility of our proposed approach for spectral
inversion, we can now readily demonstrate its numerical reliability. For this
purpose it is instructive to further investigate the difference in
heterogeneity of distribution for $\phi$, $\frac{1}{\kappa D}$, and $\beta A$.
We found that the distribution of $\phi$ is a more slowly varying function in
comparison to that of $\frac{1}{\kappa D}$ and $\beta A$. As exemplified by
Fig. 2(e), for the pinna of $\phi=0.3$, both $\frac{1}{\kappa D}$ and $\beta
A$ are seen to vary characteristically within $|x-x^{\prime}|\sim{1}$ and
$|x-x^{\prime}|\sim{2}$ respectively, where $x$ and $x^{\prime}$ are the
coordinates in the vector space of $S(Q)$. In addition, the corresponding
gradient vectors of $\frac{1}{\kappa D}$ and $\beta A$ are seen to point
perpendicularly across and axially along the blade respectively. Clearly the
numerical accuracy of potential extraction depends on how well this observed
distributional heterogeneity is addressed. On the basis of Gaussian process we
developed a covariance matrix, as the beating heart of our non-parametric
inversion method, to quantitatively describe the statistical relationships of
$\phi$, $\frac{1}{\kappa D}$ and $\beta A$ in the vector space of $S(Q)$ SI .
From $\\{S_{train}(Q)\\}$, the ML process was carried out to determine the
optimal correlation lengths characterizing the covariance matrix to describe
the local topological features of each parameter in the manifold of
$\mathbb{R}^{80}$. For an input $S_{i}(Q)$, its statistical relevant neighbors
can be identified from $\\{S_{train}(Q)\\}$ by comparing the Euclidean
distance to $S_{i}(Q)$ and the optimized correlation length of each parameter.
Therefore the $\phi$, $\frac{1}{\kappa D}$ and $\beta A$ of $S_{i}(Q)$, which
must follow the correlation built-in in the covariance matrix if the system is
in the equilibrium fluid state, can be forecasted probabilistically from those
of its specified neighbors.

Figure 3:  The comparison of the extracted (a) $\phi$, (b) $\frac{1}{\kappa
D}$, and (c) $\ln(\beta A)$ and their corresponding computational references.
The numerical accuracy of reconstruction is quantified by the relative error
$E$ given in each panel.

Another extensive set of $S(Q)$ termed as $\\{S_{test}(Q)\\}$ was simulated
separately from equilibrium fluid phase to gauge the numerical accuracy of our
proposed method. Fig. 3 presents the comparison of input parameters in
simulations and those inverted from $\\{S_{test}(Q)\\}$. All three extracted
parameters are in remarkable accord with their computational inputs, but
$\phi$ is in closer quantitative agreement as indicated by the relative error
$E$. The origin of this varying degree of uncertainty is worth exploiting: As
illustrated in Fig. 2(e), both $\frac{1}{\kappa D}$ and $\beta A$ exhibit
extreme changes within a relatively short Euclidean distance in comparison to
the large-scale variation of $\phi$. This observation suggests that, the
susceptibility of $S(Q)$ towards the variation of potential variables is
inherently determined by their distributions in the vector space of $S(Q)$.
The difference in $E$ can therefore be viewed a reflection of landscape
heterogeneity for different parameters.

Figure 4:  Panels (a)-(c) present examples of $V_{HSY}(r)$ calculated based on
$\phi$, $\kappa$ and $A$ given Fig. 3. The references used in computer
simulations and those determined by ML inversion are presented in dashed and
solid curves respectively. The magnitude of uncertainties calculated based on
the errors in Fig. 3 are represented by the shaded regions. The color scheme
is same as that of Fig. 3. Panels (d)-(f) give the reconstructed $S(Q)$
(insets) and $I(Q)$ calculated from three highly correlated systems (yellow
curves in panels (a)-(c) and upper right most points in Fig. 3). Both
computationally generated reference $S(Q)$ and $I(Q)$ and their ML
reconstructions are indistinguishable within the scales used in these plots.

In Figs. 4(a)-4(c) we give the comparison of $V_{HSY}(r)$ calculated from the
ML-extracted parameters specified by the solid lines in Fig. 3 and their
corresponding references used in simulation. To demonstrate the numerical
accuracy of our approach, the $S(Q)$ of three highly correlated systems, as
indicated by the black symbols in Fig. 2(a), were calculated and compared with
their references in $\\{S_{train}(Q)\\}$. Accordingly the corresponding
coherent intensities of small angle neutron scattering (SANS) $I(Q)$ were also
calculated assuming the system is consisted of monodisperse densely-packed
spherical particles SI . From the results presented in Figs. 3(d)-3(f), it is
clearly seen that the differences between the reference $S(Q)$ (black dashed
curves) and the reconstructed $S(Q)$ (red solid curves) are indeed
indistinguishable on the scale of our plots. This observation demonstrates
that the structural variation of suspending charged particles caused by the
change of $V_{HSY}(r)$ with the magnitude of $E$ given in Figs. 3(a)-3(c)
cannot be detected by elastic scattering techniques. The insets in Figs.
4(d)-4(f) also show that the difference in $I(Q)$ are within the range of
statistical variation of resolution of general SANS instruments. For the three
strongly correlated systems presented in Figs. 4(d)-4(f), integral equation
theories are often unable to determine $\phi$ , $\kappa$ and $A$ from $S(Q)$
in a numerically precise manner Anta and Lago (2002). The validity of our
proposed ML approach for quantitatively extracting $V_{HSY}(r)$ from the
scattering spectra of charged colloidal suspensions over a wide range of
$\phi$ in the equilibrium fluid phase is therefore verified.

In conclusion, we have developed a new ML inversion method, based on the
framework of Gaussian process, to inversely determine effective interaction of
colloidal suspensions from their scattering spectra. By treating the
probability distributions of the relevant potential parameters in the vector
space of scattering function, our non-parametric approach circumvents the
mathematical constraints inherent to deterministic models for spectral
analysis. We demonstrated that our approach offers several advantages over the
existing parametric approaches from the standpoint of numerical accuracy,
computational efficiency SI  and general applicability. Complemented by
computer simulations, our method can be systematically extended for solving
the inversion scattering problems of various colloidal systems characterized
by different effective interactions Asakura and F. Oosawa (1954); Witten and
Pincus (1986); Liu et al. (2005); Bianchi et al. (2011). Moreover, one
recognized challenge in the analysis of scattering data of interacting systems
is to take polydispersity into account adequately Salgi and Rajagopalan
(1993). In this pursuit, our ML strategy promises a new paradigm toward
quantitative characterization of highly interacting systems characterized by
significant variations of interaction potential and particle size which cannot
be addressed precisely by existing decoupling approximations Kotlarchyk and
Chen (1983); Hayter and Penfold (1983); Pusey (1991).

In principle, the applicability of this method is not restricted to structural
studies of colloidal systems. The essential idea underlying this method of
spectral analysis, which does not suffer from the drawbacks of explicit
modeling, allows quantitative extraction of relevant parameters based on which
the targeted systems are computationally constructed. We are optimistic that
our approach will provide a useful toolbox to facilitate the progress in many
important inversion problems of radiation scattering, diffraction, and imaging
Carbajal-Tinoco et al. (1996); Rajagopalan and Rao (1997); Behrens and Grier
(2001); Quesada-Pérez et al. (2001) experiments from strongly correlated
systems which traditionally have been difficult and time consuming to solve.

We thank Y. Shinohara, Y. Wang, P. Falus, L. Porcar, Y. Liu, and S.-H. Chen
for helpful discussions. This research was performed at he Spallation Neutron
Source and the Center for Nanophase Materials Sciences, which are DOE Office
of Science User Facilities operated by Oak Ridge National Laboratory. MD
simulations used resources of the Oak Ridge Leadership Computing Facility,
which is supported by DOE Office of Science under Contract DE-AC05-00OR22725.
C.-H. T. thanks the financial support from the Shull Wollan Center during his
stay at ORNL. BGS acknowledges support by the U.S. Department of Energy,
Office of Science, Office of Basic Energy Sciences Data, Artificial
Intelligence and Machine Learning at DOE Scientific User Facilities Program
under Award Number 34532. M.-C. C. thanks the support provided by the
University at Albany - SUNY.

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