††thanks: These authors contributed equally to this work.††thanks: These
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# Computational Reverse Engineering Analysis of Scattering Experiments Method
for Interpretation of 2D Small-Angle Scattering Profiles (CREASE-2D)

Sri Vishnuvardhan Reddy Akepati  Data Science Program, University of Delaware,
Newark, DE 19716 USA.  Nitant Gupta  Department of Chemical and Biomolecular
Engineering, University of Delaware, Newark, DE 19716 USA.  Arthi Jayaraman
Department of Chemical and Biomolecular Engineering, University of Delaware,
Newark, DE 19716 USA.  Department of Materials Science and Engineering,
University of Delaware, Newark, DE 19716 USA.

###### Abstract

Characterization of structural diversity within soft materials is key for
engineering new materials for various applications. Small-angle scattering
(SAS) is a widely used characterization technique that provides structural
information in soft materials at varying length scales (nm to microns). The
typical output of a SAS measurement is the scattered intensity $I(\mathbf{q})$
as a function of $\mathbf{q}$, the scattered wavevector with respect to the
incident wave. The scattered wavevector $\mathbf{q}$ is represented by its
magnitude $|\mathbf{q}|\equiv q$ (in inverse distance units) and azimuthal
angle $\theta$. While structures with isotropic spatial arrangement can be
interpreted by analysis of azimuthally averaged one-dimensional (1D)
scattering profile, to understand anisotropic spatial arrangements, one has to
interpret the two-dimensional (2D) scattering profile, $I(q,\theta)$. Manual
interpretation of such 2D profiles is complicated, and usually involves
fitting of approximate analytical models to azimuthally averaged sections of
the 2D profile. In this paper, we present a new method called CREASE-2D that
interprets, without any azimuthal averaging, the entire 2D scattering profile,
$I(q,\theta)$, and outputs the relevant structural features. CREASE-2D is an
extension of the ‘computational reverse engineering analysis for scatting
experiments’ (CREASE) method that has been used successfully to analyze 1D SAS
profiles for a variety of soft materials. CREASE uses a genetic algorithm for
optimization and an artificial neural network (ANN) as the surrogate machine
learning (ML) model for fast calculation of 1D ‘computed’ scattering profiles
that are then compared to the experimental 1D scattering profiles in the
optimization. CREASE-2D goes beyond CREASE by enabling analysis of 2D
scattering profiles, which is far more challenging to interpret than the
azimuthally averaged 1D profiles. Further, we use XGBoost as the surrogate ML
model in CREASE-2D, in place of ANNs, to relate structural features to the
$I(q,\theta)$ profile. The CREASE-2D workflow identifies the structural
features whose computed $I(q,\theta)$ profiles, calculated using the surrogate
ML model, match the input experimental $I(q,\theta)$. We test the performance
of CREASE-2D by using as input a variety of in silico 2D scattering profiles
whose structural features are known to us. We demonstrate that CREASE-2D works
well by showing that for every one of these input in silico 2D scattering
profiles, CREASE-2D converges towards the correct structural features. We
expect this CREASE-2D method will be a valuable tool for materials’
researchers who need direct interpretation of the 2D scattering profiles in
contrast to analyzing azimuthally averaged 1D $I(q)$ vs. $q$ profiles that can
lose important information related to structural anisotropy.

Researchers studying soft materials, namely polymers, colloids, liquid
crystals, gels and chemical formulations, aim to establish molecular design-
structure-property relationships to engineer new materials with improved
physical properties. Towards this goal, microscopy and scattering are two
prominent characterization techniques for understanding the structure formed
within such soft materials. Microscopy techniques that are commonly used for
soft materials include optical microscopy, to probe structures with length
scales above 10 microns, and scanning-electron/transmission-electron/atomic-
force microscopy (SEM/TEM/AFM) to probe structures with features below 10
microns. Such microscopy methods can reveal the pertinent structural features
in the area of the material that is imaged, albeit only over a narrow range of
length scales. Furthermore, microscopy only outputs a two-dimensional (2D)
projection of the structure and the depth information can be non-trivial to
interpret. In contrast, bulk structural characterization techniques that rely
on scattering of light (X-rays/visible/infrared) or neutrons are able to
reveal three dimensional (3D) structural information across multiple length
scales. In particular, for soft materials, small angle X-ray scattering (SAXS)
and small angle neutron scattering (SANS) techniques 39, 18, 26, 28, 35, 4,
10, 32 are used widely to elucidate spatial distributions of amorphous (i.e.,
not crystalline) ordered or disordered structures at various length scales.

A typical SAXS or SANS measurement captures the scattered intensity
$I(\mathbf{q})$ as a function of the scattered wavevector $\mathbf{q}$ with
respect to the incident wave, expressed by its magnitude $|\mathbf{q}|\equiv
q$ (in inverse distance units) and the azimuthal angle $\theta$. For materials
that have isotropic structural arrangements, the patterns found in the 2D
SAXS/SANS profiles, $I(q,\theta)$, are expected to exhibit spherical or
cylindrical symmetry. Analysis of such symmetric scattering profiles involves
integrating over all azimuthal angles and fitting analytical models to the
one-dimensional (1D) form of the scattering profile - $I(q)$ vs. $q$. Presence
of a peak in these 1D scattering profiles at a certain $q$ value indicates
presence of structural correlations at length scales around $2\pi/q$, either
due to the dimensions of the constituent particles (i.e., form factor, $P(q)$)
or the arrangement of particles that influences their inter-particle spacing
(i.e., structure factor, $S(q)$). Therefore, even when a structure consists of
anisotropic particles that are devoid of any inter-particle orientational
order, the 1D scattering profile can reveal most of the relevant structural
details about the material. Further, in case of dilute solutions (e.g.,
amphiphilic polymer solutions at low polymer concentrations), the form factor
of the primary particle (e.g., assembled micelles) can be analyzed using
shape-dependent or shape-independent models to obtain the dimensions of the
primary particles.33 However, in cases where there is significant dispersity
in dimensions of the primary particles, such models can be poor approximations
and the resulting analysis can be flawed. In case of concentrated solutions
(e.g., amphiphilic polymer solutions at high concentrations), in addition to
the form factor of the primary particle, one has to analyze the structure
factor which holds the information about spatial arrangement of the primary
particles (e.g., inter-particle or inter-micelle arrangement). In such cases,
if one assumes that the form factor remains constant with changing
concentration, then one can use the analyzed form factor obtained at dilute
concentration to interpret the structure factor. Analytical structure factor
models like the ‘sticky hard sphere’ or ‘Percus Yevick’ 21 can be used to
interpret isotropic structures with low dispersity. However, in the case of
systems where the values of primary particle’s shape and size or entire
distributions of shape and size of primary particles change with concentration
or systems in which the structure develops anisotropy during processing or
rheological measurements11, 17, 34, the interpretation of these scattering
profiles can be challenging.

To circumvent these challenges with traditional approaches involving manual
fitting with shape-dependent or independent models that can be approximate or
incorrect in some cases, there is a need for other analysis approaches.
Additionally, the surge in high-throughput measurements and the quest for
artificial intelligence (AI) driven manufacturing demand analyses methods that
can be fast and automated in interpreting scattering profiles, and
complementary characterization results, as and when the measurement is done.
We direct readers to a recent perspective by Anker et al. that covers many
ongoing developments and studies within this topic of fast computational
analysis of scattering and spectroscopic measurements in materials sciences.
2. The challenges for computational methods being developed for fast and/or
automated scattering analyses in the area of synthetic soft materials are
different from inorganic hard materials or biological molecules. This is
because (non-biological) soft materials structures tend to be mostly
amorphous, often exhibiting significant dispersity in structural dimensions,
unlike the precise crystalline order seen in inorganic materials or secondary
and tertiary structures of proteins,14, 9, 13 . To address this specific need
in the area of soft materials with amorphous structures, Jayaraman and
coworkers recently developed the ‘Computational Reverse Engineering Analysis
of Scattering Experiments’ (CREASE) method.3, 24, 25, 40, 41, 43, 44, 23, 42

The CREASE method outputs the features or descriptors of the three-dimensional
(3D) structures that produce ‘computed’ scattering profiles
$I_{\text{comp}}(q)$ which closely resemble the scattering profile obtained in
experiments $I_{\text{exp}}(q)$. Rather than iterating exhaustively over 3D
structures themselves, a computationally intensive and slow process, in CREASE
the optimization cycle iterates over a lower dimensional representation of the
3D structures. We call these lower-dimensional descriptors of structure as
_structural features_ ; as we use genetic algorithm for optimization, in the
jargon of evolutionary algorithms, we refer to these structural features
as“genes”.

In a typical GA optimization loop, an initial population of “individuals” is
generated, where each individual has a unique set of structural features or
“genes”. The structural features can have single values of structural
parameters or encode parameters representing distributions of structural
parameters. For each individual, these structural features are converted to a
computed scattering profile using a surrogate machine learning (ML) model. The
computed scattering profile $I_{\text{comp}}(q)$ of each individual is then
compared to the input scattering profile $I_{\text{exp}}(q)$. The extent of
match between the $I_{\text{comp}}(q)$ and $I_{\text{exp}}(q)$ is calculated
as a fitness value for that individual. After the fitness value has been
calculated for all individuals in a generation, then a new “generation” of
individuals is created based on the current generation’s fitness ranking and
genetic operations like“pairing” and “mutations”6. As the optimization
proceeds, with each new“generation”, the individuals progressively exhibit
better fitness values, i.e., improvement in the match between their
$I_{\text{comp}}(q)$ and the input $I_{\text{exp}}(q)$. At the end of the GA
cycle, upon convergence in fitness values, CREASE outputs multiple individuals
(i.e., sets of structural features) that all have the mutually similar
scattering profiles that also match the input scattering profile. If the GA
results consist of multiple distinct sets of structural features, then one
would either use their domain knowledge or guidance for imaging techniques
and/or molecular modeling and simulations to remove the “individual(s)” that
are deemed unphysical and keep only those “individual(s)” that are physically
possible.

Within the optimization loop, the use of surrogate ML models for calculation
of $I_{\text{comp}}(q)$ for each individual has significantly accelerated the
computational speed of CREASE. Traditionally, for 3D structures with known
positional coordinates of each particle or constituents of each particle, one
would use the computationally intensive Debye scattering equation to calculate
scattering profiles. To accelerate this step of calculating
$I_{\text{comp}}(q)$, in recent CREASE studies, Jayaraman and coworkers
introduced the idea of using a surrogate ML model (e.g., artificial neural
networks or ANN) that connects the structural features (i.e., lower
dimensional representation of the 3D structure) to its
$I_{\text{comp}}(q)$.41, 43, 25 Using this machine learning enhanced CREASE
(ML-CREASE) one can interpret input scattering profiles fast and on modest
computational resources as described in Refs. 24, 25, 43

Fig. 1: CREASE-2D method workflow used in this paper. For proving the
CREASE-2D works correctly, we used as input an in-silico 2D scattering profile
generated from a 3D structure with predefined set of structural features. Only
the 2D scattering profile, which we call “experimental” scattering profile or
$I_{\text{exp}}(q,\theta)$ is used as the input to the CREASE-2D method. The
genetic algorithm (GA) optimizes towards structural features whose
$I_{\text{comp}}(q,\theta)$ closely resembles $I_{\text{exp}}(q,\theta)$. By
comparing the optimized structural features to the ones used to create the
$I_{\text{exp}}(q,\theta)$ we can show that CREASE-2D works well.

There have been multiple soft materials systems where CREASE has been used
successfully to analyze the 1D scattering profiles, and in many cases, CREASE
has performed better than existing analytical models. For example, CREASE has
been used to analyze the form factor of assembled structures in dilute
solutions; for example, spherical micelles3, cylindrical micelles40, 41 and
vesicles44 formed by novel polymers or macromolecules in solution. In these
cases, the existing analytical models were either too approximate3 or could
not handle dispersity in structural dimensions well44. In some cases, CREASE
was used to test hypotheses of potential assembled structures in the solution
which could not have been done with analytical models alone. 27 CREASE has
also been used to interpret the scattering profiles of concentrated particle
systems where the form of the particle was known a priori (e.g., simple
spherical particles); in this case, CREASE was used to understand the extent
of mixing within binary nanoparticle mixtures 23, 25. CREASE has also been
extended to the ‘P(q) and S(q) CREASE’ version that can analyze both the form
and the structure factors of the primary particles simultaneously24. This
‘P(q) and S(q) CREASE’ method was used to understand a system of silica
particles coated with surfactant (core-shell particles) at varying temperature
and salt concentrations. Both temperature and salts affect the cationic
surfactants in the shell around the particles and as a result, the form of the
surfactant-coated particles and inter-particle structure. 29

While all of the above applications of CREASE involved isotropic structures
and the 1D scattering profile from experiments as input, in this paper we have
extended the CREASE method to CREASE-2D that can analyze 2D scattering
profiles directly and in turn, enable interpretation of structures that may
have anisotropy.

The input to CREASE-2D is a 2D scattering profile coming from structures that
have some orientational order within the material’s structure either produced
by processing conditions (e.g., shear or field-induced alignment of domains)
and/or because of the form of primary particle (e.g., ellipsoidal domain). In
such cases, characterization of structural anisotropy requires the use of the
2D SAS profiles $I_{\text{exp}}(q,\theta)$ which hold information of the
length scales of arrangements that can vary along various azimuthal angles,
$\theta$.

In this paper, we present all the relevant details of CREASE-2D method
development and demonstrate its successful application by correctly outputting
the structural features of the the 3D structures that gave rise to the input
in-silico ($I_{\text{exp}}(q,\theta)$).

###  CREASE-2D: Overview and Development

Figure 1 provides an overview of the CREASE-2D workflow presented in this
paper. The overall development of CREASE-2D involves four key steps:

  1. 1.

Generating a dataset of 3D structures having an extensive variation of all
important structural features. The structural features that we demonstrate in
this study are distributions of domain sizes, shapes, orientational order, and
volume fraction of domains that produce the scattering;

  2. 2.

Computing the 2D scattering profiles for each of those 3D structures;

  3. 3.

Using the combined dataset of structural features and their computed 2D
scattering to train the surrogate ML model that will output a computed
scattering profile for an input of structural features; and

  4. 4.

Incorporating the trained ML model within the GA optimization loop to fulfill
the CREASE-2D workflow.

While step (4) above enables a smooth and fast execution of the CREASE-2D
method, steps (1) - (3) are necessary for an accurate and reliable
representation of experimentally relevant structural configurations and their
scattering profiles. The amount and quality of data generated in steps (1) and
(2) will also determine the accuracy of the surrogate ML model in step (3),
which in turn dictates the efficacy of the CREASE-2D optimization. Before we
describe each of the above steps in more detail, we note the similarities and
differences between CREASE-2D and prior implementations of CREASE.

Similar to previous implementations, CREASE-2D method also uses a GA to
optimize structural features. The GA loop proceeds in a similar manner as in
previous uses of CREASE and stops when fitness of the individuals converges,
i.e., an individual’s computed 2D scattering profile
$I_{\text{comp}}(q,\theta)$ matches the input profile
$I_{\text{exp}}(q,\theta)$. One difference between the previous CREASE
implementation and CREASE-2D is in the choice of the surrogate ML model to
calculate the $I_{\text{comp}}(q,\theta)$ for each individual. The surrogate
ML model needed in this case not only needs to handle input in the form of a
table having multidimensional variation of its structural features, but also
output a 2D scattering profile rather than a 1D scattering curve of
$I_{\text{comp}}(q)$ vs $q$. More details are provided in the steps (3) and
(4) sub-sections below.

### Step (1): Generating a dataset of 3D structures with varying structural
features

To develop a reliable surrogate ML model for linking structural features to
the 2D scattering profile, we need a training dataset that contains sufficient
samples of 3D structures with all potential variations in structural features
that influence their computed 2D scattering profiles. This sub-section
describes this process of generating such a dataset.

In principle, structural features condense the detailed representation (e.g.,
x, y, and z coordinates of all particles) of the 3D structure to a few
numerical values that pertain to the distributions of parameters describing
the 3D structure. In ML jargon, structural features are similar to the lower
dimensional latent space variables encoding a higher dimensional input
function. Our philosophy is that the structural features should be information
that a soft materials researcher would understand and find relevant. By
relevance we mean that the interested structural features would be ones (e.g.,
shapes, sizes, and spatial arrangement of the domains, extent of
mixing/demixing within domains/between domains, orientational alignment of
domains, grain boundaries, etc.) that will likely control properties/function
of the soft material. Thus, we choose not to have automatically encoded latent
space variables that lack a physical meaning and not easily interpreted by
human, and instead define our own structural features using our soft materials
domain knowledge.

To demonstrate our choices of structural features for a representative example
of soft materials with structural anisotropy, we consider a model system of
spheroidal particles with well-defined distributions of shapes, sizes, and
orientations (shown schematically in Figure 1), along with variations in the
particles’ packing fractions in the material. Generation of such 3D structures
is facilitated by our recently developed (open-source) computational method
called CASGAP (Computational Approach for Structure Generation of Anisotropic
Particles)20. In the original manuscript 20, we demonstrated the versatility
of the CASGAP method to generate 3D structures for user-provided distribution
of particle sizes, shapes, and orientations at or close to the target volume
fraction. Accordingly, the CASGAP method uses parameters $R_{\mu}$,
$R_{\sigma}$, $\gamma_{\mu}$, $\gamma_{\sigma}$, and $\kappa$ to generate the
3D anisotropic structure with a target $\phi_{\text{target}}$. These
structural descriptors serve as the structural features for use in this
development of the CREASE-2D workflow. While the detailed description of these
structural features can be found in the original manuscript 20, we review some
relevant details below:

  1. 1.

The particle sizes and shapes are expressed by the spheroidal volumetric
radius $R=\sqrt[3]{a^{2}c}$ and the spheroidal aspect ratio $\gamma=c/a$,
where $a$ and $c$ are the lengths of the semi-minor and semi-major axes of the
spheroid, respectively. As done in the original manuscript20, the variations
in size and shape are modeled by a log-normal distribution, each with their
means ($R_{\mu}$, $\gamma_{\mu}$) and standard deviations ($R_{\sigma}$,
$\gamma_{\sigma}$). These quantities provide us with the first four structural
features for CREASE-2D.

  2. 2.

The orientations in the structure is quantified by a 3D vector pointing along
the major axis $\mathbf{V}$ of the spheroid. With such description of
orientations, we adopt the 3D von Mises-Fisher (vMF) distribution (see details
in Ref. 20) to model the distribution of orientational order expressed
succinctly by the $\kappa$ parameter. The $\kappa$ parameter is a measure of
the inverse-dispersity in orientation and is defined around a preferred
orientation $\mathbf{\Lambda}$. $\kappa=0$ indicates complete lack of
orientational order (i.e., $\mathbf{V}$ is uniformly distributed on the
surface of a sphere) and $\kappa\rightarrow\infty$ indicates perfect
orientational order (i.e., $\mathbf{V}=\mathbf{\Lambda}$). Relying on the
premise that for an anisotropic structure, the principal axes of anisotropy
can be aligned with the laboratory frame of reference during scattering
measurements such that $\mathbf{\Lambda}=\mathbf{\hat{x}}$, enables us to use
only $\kappa$ as the fifth structural feature.

  3. 3.

Lastly, the concentration of particles is quantified by the volume fraction of
particles, $\phi$. If dense particle configurations is desired, a trade-off is
observed between the computational time for structure generation and the value
of volume fraction achieved in that time. The CASGAP method is designed with
this trade-off in mind and can be terminated at any point of the structure
generation while maintaining a structure that adheres to the desired structure
features’ distribution. However, in such cases of early termination, the
actual volume fraction $\phi$ may not reach the value of
$\phi_{\text{target}}$ leading to $\phi\leq\phi_{\text{target}}$. With such an
expectation, we use the actual $\phi$ evaluated after the structure is
generated as the sixth structural feature since the scattering profile
computed in step (2) (described in the next sub-section) can be significantly
influenced by the actual volume fraction of the particles.

Fig. 2: (a) Histograms and scatter plots describing how the mean and standard
deviations of the volumetric radius $R$ is varied in each of the 3000 samples
in the dataset. (b) Similar to (a) but for the mean and standard deviations of
the aspect ratio $\gamma$. (c) Histograms describing the distribution in the
orientational anisotropy parameter $\kappa$ for the von-Mises Fisher
distribution20. The histogram in the range of $0.1\leq\kappa\leq 10$ is shown
separately (on top) to indicate the distribution of nearly $50\%$ samples
uniformly drawn from this range. (d) Histogram describing the distribution in
the volume fraction $\phi$ of the generated structures. (e-g) Representative
snapshots of 3D structures drawn from the dataset, showing size, shape, and
orientation variations, respectively. Use of different colors facilitate easy
distinction of individual particles visually. The structures in (e) and (f),
correspond to points highlighted in the scatter plots of (a) and (b),
respectively. The detailed information about their structural features are
provided in Table 1. Table  1: Values of Structural Features For Select Few
Samples (in the same order as) shown in Figure 2.

Sample ID | $R_{\mu}$ | $R_{\sigma}$ | $\gamma_{\mu}$ | $\gamma_{\sigma}$ | $\kappa$ | $\phi$
---|---|---|---|---|---|---
31 | 3.07 | 0.00101 | 7.48 | 0.263 | $3.11\times 10^{-1}$ | 0.245
332 | 30.0 | 0.00185 | 0.785 | 0.0352 | $4.41\times 10^{-1}$ | 0.389
2941 | 10.2 | 0.576 | 1.28 | 0.200 | $5.89\times 10^{-10}$ | 0.361
1291 | 10.2 | 3.38 | 0.488 | 0.176 | $6.31\times 10^{0}$ | 0.401
1770 | 17.4 | 1.87 | 0.100 | 0.000 | $2.18\times 10^{-2}$ | 0.237
1172 | 3.57 | 0.201 | 9.96 | 0.00558 | $1.36\times 10^{-1}$ | 0.194
2291 | 19.5 | 2.19 | 1.00 | 0.0358 | $1.35\times 10^{8}$ | 0.422
949 | 12.9 | 0.802 | 1.00 | 0.591 | $3.83\times 10^{4}$ | 0.409
40 | 7.80 | 1.50 | 0.511 | 0.0696 | $\mathbf{1.09\times 10^{-10}}$ | 0.430
585 | 6.80 | 1.63 | 0.881 | 0.0669 | $\mathbf{1.23\times 10^{-1}}$ | 0.461
172 | 14.1 | 3.69 | 0.654 | 0.122 | $\mathbf{9.70\times 10^{0}}$ | 0.459
2495 | 3.64 | 0.0662 | 2.99 | 1.24 | $\mathbf{9.68\times 10^{9}}$ | 0.357

Bolded text is used to highlight the relevant structural features depicted in
Figure 2.

Leveraging the computational efficiency of the CASGAP method, we generate a
dataset of 3000 three-dimensional structures. This dataset has a numerical
index from $1-3000$ used as their Sample ID along with numerical values of all
their structural features. We share some examples from this dataset openly on
Zenodo.1 In Figure 2 we describe how each of the structural features are
varied. Since each structural feature represents a physically relevant
quantity with significant influence over the morphology of the particles,
these could not simply be varied using a uniform distribution over their
respective ranges. As a result, some of these quantities have a normal-like or
a skewed distribution in their chosen ranges as shown in the plots in Figure
2a-d. The numerical details of the random sampling, which is a version of
Monte Carlo sampling, is discussed in detail in the Supporting Information
Section S1. We represent all our structures by a cubic representative volume
of length $L=300$ distance units (in this study 1 distance unit corresponds to
1 Å, but this correspondence can be changed to a different length-scale, as
desired). In Figure 2e-g and the accompanying Table 1 provide some
representative structure snapshots along with their structural features. Some
extreme values of structural features have been indicated in Table 1 with bold
font; we selected the Sample IDs with these extreme values of structural
features to visualize their effects on the overall structure.

The mean volumetric radius $R_{\mu}$ is nearly uniformly sampled over a range
of $3$ Å to $30$ Å, representing a variation of $1\%~{}L$ to $10\%~{}L$ (as
shown in the histogram of Figure 2a). To keep the size variation reasonable
within the prescribed log-normal distributions, the standard deviation of
volumetric radius is controlled by the mean value, such that whenever
$R_{\mu}$ approaches its extreme values, i.e., $3$ Å or $30$ Å,
$R_{\sigma}\rightarrow 0$. This is shown in the scatter plot of Figure 2a,
where an envelop shape over the $R_{\sigma}$ distribution is observed. In
Figure 2e Sample 31 and Sample 332 depict the structure when $R_{\mu}$’s are
$\sim 3$ and $\sim 30$, respectively. While Sample 2941 and Sample 1291 (with
similar $R_{\mu}$) depict the extreme values of $R_{\sigma}$.

Figure 2b shows the variation in aspect ratios in range of $1/10$ to $10$.
Since this is a ratio, the values below 1 (representing oblate spheroids) are
analogous, by a reciprocal relationship, to those above 1 (representing
prolate spheroids). To ensure fair sampling of both these shape types, the
values are nearly uniformly sampled over the logarithmic scale between $1/10$
to $10$ as shown in the histogram of Figure 2b. Here, too, we ensure that
whenever the mean aspect ratio $\gamma_{\mu}$ approaches the extremes,
$\gamma_{\sigma}$ approaches 0 as seen from the scatter plot in Figure 2b. In
Figure 2e Sample 1770 and Sample 1172 depict the structure when
$\gamma_{\mu}$’s are $\sim 0.1$ and $\sim 10$, respectively. While Sample 2291
and Sample 949 (with similar $\gamma_{\mu}$) depict the extreme values of
$\gamma_{\sigma}$.

To vary the degree of orientational order, the $\kappa$-parameter (Figure 2c)
can be varied by sampling equally from 4 intervals defined by the end points:
$10^{-10}$, $0.1$, $1$, $10$ and $10^{10}$. Here values $10^{-10}\approx 0$
and $10^{10}\approx\infty$, are chosen to sample the perfectly isotropic and
anisotropic structures, respectively. Structures from each of these intervals
are in Figure 2e with Sample IDs 40, 585, 172 and 2495, where $\kappa$ values
are nearly $10^{-10}$, $0.1$, $10$ and $10^{10}$, respectively.

In Figure 2d the histogram shows the variation in $\phi$ for the entire
dataset. Unlike all other structural features, the distribution of $\phi$ is
not prescribed but is a result of CASGAP structure generation with
$\phi_{\text{target}}=0.5$ as explained previously. If a stricter control over
$\phi$ is desired, more samples at lower $\phi$ can easily be generated and
added to the dataset to change the shape of the distribution.

Having the dataset of 3D structures, we calculate each of their 2D scattering
profiles in Step (2). The 2D scattering profiles and the structural features
then become the intended “output” and “input” data for training and testing
the surrogate ML model in Step (3).

### Step (2): Calculating 2D Scattering Profile for Each 3D Structure

Fig. 3: (a-f) From left to right, representative snapshots of a structure with
Sample ID denoted on the top, the Cartesian form of their scattering intensity
profile $I_{\text{comp}}(q,\theta)$ with axes $q$ and $\theta$, and the polar
form of their scattering intensity profile $I_{\text{comp}}(q_{x},q_{y})$ with
axes $q_{x}$ and $q_{y}$. Here, $q_{x}$ and $q_{y}$ are components of the
scattering vector $\mathbf{q}$ that are (reciprocally) aligned with the
laboratory frame axes $x$ and $y$, respectively. The polar form of the
scattering intensity profile maintains the logarithmic scaling of the
scattering vector magnitudes, and as a result the center of the profile is not
at $q=0$ but truncated to $q=10^{-2}$.

In all previous implementations of the CREASE method, we used the pairwise
Debye scattering equation to calculate the scattering intensity contribution
of $N$ particles with known form factors $f_{1,2,...,N}(q)$ as follows:

$I_{\text{comp}}(q)=\frac{1}{V}\sum_{n=1}^{N}\sum_{m=1}^{N}f_{n}(q)f_{m}(q)\frac{\sin(qr_{nm})}{qr_{nm}}$
(1)

The above equation is only applicable for isotropic arrangement of particles,
and is obtained by integration over all possible orientations of the
scattering vector $\mathbf{q}$, which is equivalent to azimuthal averaging
performed on the experimental 2D scattering profiles. Notably, this equation
has a double-summation term which necessitates pairwise consideration of
particles and their contributions to the scattering profiles, and has the
effect of making the scattering calculations computationally intensive and
harder to parallelize. Together with the time required to generate structures,
the additional time needed to perform scattering calculations for that
structure, makes them unfit for use directly within the CREASE workflow. This
motivated the need for surrogate ML models that are time-efficient in the
prediction of $I_{\text{comp}}(q)$ (as described in step (3) sub-section).

For CREASE-2D implementation, we compute the 2D scattering intensity
$I_{\text{comp}}(\mathbf{q})$ from the scattering amplitude
$A_{\text{comp}}(\mathbf{q})$ as
$I_{\text{comp}}(\mathbf{q})={1}/{V}\left|A_{\text{comp}}(\mathbf{q})\right|^{2}$.
Here, the $A_{\text{comp}}(\mathbf{q})$ is the complex Fourier transform of
the fluctuation in the scattering length density $\Delta\rho_{n}$15, 19, 5,
and is expressed as:

$A_{\text{comp}}(\mathbf{q})=\sum_{n=1}^{N}\Delta\rho_{n}v_{n}f_{n}(\mathbf{q})\exp\left(-i~{}\mathbf{q}\cdot\mathbf{r}_{n}\right)$
(2)

The above expression only has a single summation term, which significantly
reduces the computational complexity of the scattering calculation from
$O(N^{2})$ to $O(N)$, and has enabled the computation to be parallelized; this
was also noted by Brisard et al. as the ‘simple sums’ computation.5 For our
model system, due to the simplicity of the ellipsoidal shape, computation of
equation 2, is further simplified with the anisotropic form factor
$f_{n}(\mathbf{q})$ of a spheroid, which can also be obtained from Pedersen’s
tabulation of analytical form factors.32 $f_{n}(\mathbf{q})$ is provided as:

$f_{n}(\mathbf{q})\equiv
f_{n}(q,\theta)=\frac{j_{1}(q~{}r_{n}(\theta))}{q~{}r_{n}(\theta)}.$ (3)

In the above equation $j_{1}(\cdot)$ is the first spherical Bessel function,
and $r_{n}(\theta)$ is an effective radius (of particle $n$) that depends on
the direction ($\theta$) of the scattering vector $\mathbf{q}$. A more
detailed expression for the analytical form factor can be found in the
Supporting Information Section S2. We note that for shapes of particles that
are complex, without easily available analytical forms of shape, one can
calculate the entire 2D scattering profile by placing point scatterers in the
box and using sufficient number of point scatterers to resolve the particle
shapes and particle-particle spatial arrangements. We are currently finalizing
a computational efficient, GPU-based code, to calculate 2D scattering profiles
for any shape of the particle using this scatterer approach; we will share
that as open-source code on https://github.com/arthiayaraman-lab .

As the structure is contained in the shape of a cubical box of length $L$, the
scattering calculations can be heavily dominated by the form factor of the
cubical box, referred to as the ‘finite size effects’ in literature5. These
finite size effects greatly obscure the 2D scattering profile of the
structure, and makes it hard to interpret their variation purely due to the
structural features. By accounting for the volume fraction of each particle,
the form factor of the box can be subtracted as a correction to the scattering
profile of the structure. We have adapted the correction scheme described by
Brisard et al.5 to remove these finite size effects as discussed in the
Supporting Information Section S2. Some simplifications like considering the
full shape of the particles at the boundaries can be made and work well as
long as the cubic box size is much larger than the size of the particles,
which in our case is below $10\%L$.

Figure 3a-f provides some representative examples of scattering profile
variations computed using equation 3 after applying the finite size effects
correction. In each panel a structure denoted by their Sample ID is shown
together with two representations of the computed scattering profiles, which
are obtained as the color-coded intensity plots for each $q$ and $\theta$
value. The left scattering plots in Figure 3 are referred to as the Cartesian
scattering intensity plots - $I_{\text{comp}}(q,\theta)$, with axes $q$ and
$\theta$. The right scattering plots show the polar scattering intensity
$I_{\text{comp}}(q_{x},q_{y})$, with axes $q_{x}$ and $q_{y}$, two components
of the scattering vector $\mathbf{q}$. The polar form is easily recognizable
to the soft materials experts, and is the typical representation of 2D
scattering profiles directly produced from SAXS/SANS measurements. Both the
“Cartesian” and “polar” forms of 2D scattering profiles are numerically
equivalent and only differ in their visual representation. We have included
the polar scattering intensity plots as a reference to the reader for easier
comparison to relevant experimental scattering profiles. However, for the
further analysis, the Cartesian representation provides a convenient and
straightforward representation of the numerical data, since only half of the
complete plot $\theta=0^{\circ}$ to $\theta=180^{\circ}$ needs to be
represented due to the inversion symmetry with the other half of the profile,
i.e. $I(\mathbf{q})=I(\mathbf{-q})$. As demonstrated further, the Cartesian
representation can be easily serialized to obtain the complete training and
testing data in a tabular form that is convenient for training the surrogate
ML model as described in the next sub-section.

The structures chosen in Figure 3 are used to demonstrate how structural
variations can influence the scattering profile. For example, Figure 3a and 3b
each demonstrate an isotropic scattering profile, while having different
shapes of the individual particles; more spherical in the former and
disordered (low $\kappa$) prolate-spheroidal in the latter. Figure 3c shows
weakly aligned structure (intermediate $\kappa$), while Figure 3d-f show
highly aligned structures (high $\kappa$). Another distinguishing effect is
the change in the intensity at low $q$ for Figure 3d and Figure 3e; this is
due to the drastic change in average size of the particles.

### Step (3): Training the Machine Learning (ML) Model to Link Structural
Features to Computed Scattering Profile

With a streamlined implementation of steps (1) and (2), the dataset of 3000 3D
structures and their corresponding 2D scattering profiles is ready for the ML
model training and testing (or validation). 80% of the data (2400 structures)
are used for training the ML model and remaining 20% (600 structures) are used
for validation of the ML model’s performance.

Fig. 4: (a) Correlation matrix for all 6 structural features along with $q$,
$\theta$ and $I(\mathbf{q})$ values, that together form the dataset for
training and validation. (b) The importance histogram for each feature
evaluated by the model after training. (c) Learning curve during training of
the surrogate ML model, where R2 error of the training (black) and validation
(green) data entries is plotted against the number of data entries. (d)
Performance of the ML model using the R2 and the Structural Similarity Index
Measure (SSIM) scores for all 3000 samples in the dataset. We note that the
index in the x-axis for this plot runs from 1 to 3000 but it is different from
Sample IDs; the index distinguishes the randomly selected 2400 samples used
for training and 600 samples for the validation. (e-h) Original and predicted
scattering profiles for a selected few samples from the validation dataset,
each marked with their R2 and SSIM scores.

In our efforts to apply an appropriate ML method that predicts the 2D
scattering profile from a given set of structural features, we identify the
need to use a supervised ML approach where continuous-valued quantities can be
predicted from a small set of other continuous parameters. Traditionally, both
deep learning (DL) and ensemble learning approaches have been successfully
applied to achieve these tasks.

With the many DL approaches, one can create a generative model that is
conditionally trained on all the structural features. However, successful
training of generative models requires a lot more data than provided in our
dataset, roughly estimated to be well above 10,000 - 100,000 images for model
training alone.12, 36 On the other hand, ensemble learning methods combine the
prediction of multiple standalone models, to create an overall ‘ensemble’
predictive model that is more accurate than the individual predictions from
the standalone models. Many ensemble learning approaches can be easily
implemented using decision trees which are simpler to work with than neural
networks, and have been shown to perform exceptionally well, outperforming
neural networks31 for tabular data, as is also the case for
$I_{\text{comp}}(q,\theta)$. Motivated by these advantages, we use a decision
tree-based ML model to predict the value of $I_{\text{comp}}(q,\theta)$ for
each of the 6 structural features and the given $q$ and $\theta$ values.

In the realm of decision tree-based ML models, especially when dealing with
tabular data, boosting ML techniques have gained popularity.37, 8, 16, 45 This
is because boosting ML techniques combine groups of weak predictive learners
sequentially and correct previous models’ training loss to form a strong
ensemble model. Here we choose XGBoost algorithm,7 which stands for eXtreme
Gradient Boosting, to be the surrogate ML model in CREASE-2D, due to its
exceptional performance and lower scope of over-fitting.

XGBoost is a generalized algorithm that can be implemented to solve a wide
range of problems. During training, XGBoost assigns weights to all features it
trains on, referred to as feature importance and accordingly adjusts the
construction of decision trees. XGBoost also offers a wide range of hyper-
parameters that can be fine-tuned to a diverse set of training data. In our
work, we utilize these advantages of XGBoost to train the surrogate ML model
that outputs a 2D scattering intensity for the input of structural features
and $(q,\theta)$ values.

To use XGBoost algorithm, the training dataset is reformatted into a table,
where each row contains all the 6 structural features as fields, combined with
a serialized representation of the scattering profiles. Thus, each training
dataset row reads as: $R_{\mu}$, $R_{\sigma}$, $\gamma_{\mu}$,
$\gamma_{\sigma}$, $\kappa$, $\phi$, $q$, $\theta$, and $I(q,\theta)$. During
serialization of the dataset, the resolution of the scattering profile can
have a dominant effect on the efficiency of training. This is because a higher
resolution will result in better quality of the data, but also increases the
computational overhead and memory requirements during training. The 2D
scattering profiles calculated in step (2) are generated over a ($q$,
$\theta$) grid of $501\times 181=90681$ data points, which amounts to over 200
million points for all 2400 samples in the training dataset. In principle one
could use all these points to train the surrogate ML model, if the user has
outstanding computational resources with limitless memory. For users with
modest computational resources (including cost-effective subscriptions to
Google-Colab) sub-sampling of the data is deemed necessary. We therefore
adopted a grid-based sub-sampling approach where we uniformly sample every 4th
$q$ value and every 5th $\theta$ value to obtain a ($q$, $\theta$) grid of
$127\times 37=4662$ data points. This results in around 11 million tabular
entries for the 2400 samples that can be handled reasonably well by the ML
model.

To tune the architecture of the decision trees in the XGBoost model, Bayesian
search optimization38 with cross validation is performed over a large range of
hyper-parameters to identify their best configuration that provides reliable
accuracy in the predicted 2D scattering profiles. More details about
configurations of Bayesian optimization is provided in the Supporting
Information Section S3. As an example, after this optimization, we find that
the predicted intensity values are the most reliable when for each decision
tree and for each node of a decision tree only $90\%$ and $80\%$ of the
structural features are randomly sampled, respectively. Other hyper-parameters
which determine the learning rate, step-size, maximum depth of the decision
tree, etc. are also optimized and described in more detail in the Supporting
Information Section S3 along with their optimum value that are used to train
the ML model. Careful tuning of these hyper-parameters is essential for
achieving optimal model performance and avoiding over-fitting on the given
dataset. Bayesian optimization of the hyper-parameters takes just over an hour
to optimize, when using the V100 GPUs with 51 GB RAM as provided by our Google
Colab Pro subscription. Once the tuned hyper parameters are obtained, the
XGBoost model is trained on CPUs within 10 minutes.

To understand the data, we present the correlation matrix in Figure 4a and to
understand how the ML model interprets the data after training we present the
histogram that measures the feature importance in Figure 4b. The correlation
matrix indicates weak correlations between the means and standard deviations
of $R$ and $\gamma$, possibly due to the way these values are sampled, as
indicated in Step (1). Some correlations are also observed for $\phi$ and all
remaining structural features; as noted above in the CASGAP structure
generation, the volume fraction $\phi$ value is not directly varied during
structure generation and is only evaluated after the structure is generated.
The strongest (inverse) correlation is observed between the scattering
intensity $I(q,\theta)$ and the magnitude of the wavevector $q$; this is
expected as the scattering intensity values display a drastic dependence on
the $q$ values. Consequently, after training, the ML model assigns the highest
importance to $q$, as shown in Figure 4b. Figure 4c shows the learning curve
where the performance is measured using the R2 error, which is a normalized
version of the mean squared error (MSE), and is plotted against the number of
data entries that the model has already used for training. Both the training
and the validation errors are found to converge quickly to beyond 99.5%
indicating that the surrogate ML model does not over fit the training data.

In Figure 4d we evaluate the performance of the surrogate ML model using two
metrics for all 3000 samples, where we have assigned an index (different from
Sample ID) to separate the training samples from the validation (or test)
samples. The first metric is the R2 error evaluated in a similar way as done
during ML model training. These R2 scores provide information about the
prediction accuracy of the ML model at each $q$ and $\theta$ value on an
individual basis (i.e., without necessarily considering the local context). We
find that the R2 values converge to 0.995 and do not differ much for training
vs. validation samples, indicating excellent prediction accuracy of the ML
model. However, to also evaluate the performance of the ML model to output the
entire 2D scattering profile we need another metric that takes into account
the performance of the model at all values in the local vicinity of a $q$ and
$\theta$. For this reason, we choose the structural similarity index (SSIM)
scores which infers the structural differences between the two scattering
profiles, by using image-based characteristics like luminescence, contrast,
and pattern; these quantities are derived from the mean, variance, and
covariance information of the local pixel data. An SSIM score near 1 indicates
a good performance of the ML model in predicting the entire scattering profile
for a given set of structural features. In Figure 4d the SSIM scores converge
to above $\sim 0.8$ indicating a reliable prediction accuracy of the ML model.

A visual comparison between the original and the ML predicted scattering
profiles is also shown in Figure 4(e-h), along with their R2 and SSIM scores.
We note that among all the Sample IDs, Sample 1910 shown in this figure has
the least SSIM score. A more detailed comparison between the original and
predicted profiles is provided in the Supporting Information Section S4, by
overlaying their 1D scattering profiles at a few selected $\theta$ values to
further demonstrate the similarities in the two profiles. These results
demonstrate that the trained surrogate ML model performs reasonably well. It
is important to note that the quality of the surrogate model training will
impact how well CREASE-2D performs. We encourage users of CREASE-2D to invest
the appropriate time for the ML model training, and to ensure that poor
training and testing do not manifest as poor analyses from CREASE-2D.

### Step (4): Optimization within Genetic Algorithm (GA) in CREASE-2D

Fig. 5: Performance of Genetic Algorithm (GA) in CREASE-2D. (a) Structural
Similarity Index or SSIM scores for all 600 samples input to CREASE-2D; SSIM
quantifies the similarity between the GA-optimized or ‘best’
$I_{\text{comp}}(q,\theta)$ at the end of the GA loop with the input
$I_{\text{exp}}(q,\theta)$. (b) The comparison of GA-optimized values of the
normalized “gene” or structural features and the original value of the
structural feature, normalized to represent a target gene value for all 600
samples tested with CREASE-2D. (c-d) Two selected samples - Sample IDs 1910
and 1076- out of the 600 samples tested with CREASE-2D. We show visual
comparison of the input scattering profile and outputs from three independent
GA runs and plot their corresponding evolution of structural feature
predictions during each GA run for Sample IDs 1076 and 1910. The solid colored
curves in the plots in (c) and (d) are the three GA runs and the black dashed
curve is the value of the structural feature corresponding to the original
scattering, with the exact value of that structural feature denoted in text.

The final step in CREASE-2D implementation is to put together the predictive
capacity and the speed of the surrogate ML model within the genetic algorithm
(GA) optimization loop. We refer the reader to previous CREASE publications3,
24, 25, 40, 41, 43, 44, 23, 42 for detailed implementations of the GA
optimization loop in the successful execution of the CREASE (1D) method. In
the current implementation, one major distinction is the use of a continuous
parameter GA in contrast to the binary GA used in the previous work. The
continuous parameter GA is better suited for evolving “genes” that represent
continuous parameters, and has a more straightforward interpretation of the
crossover and mutation operations.22 As noted before, the 6 structural
features ($R_{\mu}$, $R_{\sigma}$, $\gamma_{\mu}$, $\gamma_{\sigma}$,
$\kappa$, $\phi$) are represented as 6 corresponding “genes” and every
“individual” has a unique set of values for these genes in the GA optimization
loop . We first normalize values of the genes, using a scheme similar to the
one used to obtain their randomized distribution; for more detail see
Supporting Information Section S5. The normalization schemes assigns a value
between $0-1$ as the value of the gene and has a monotonic one-to-one
correspondence with the value of the corresponding structural feature.

For every “individual” with a unique set of genes, a scattering profile is
predicted from the surrogate ML model using the individual’s structural
features as the input. All individuals in each generation are then ranked by
their “fitness” value which is quantified by the structural similarity index
(SSIM) of the individual’s computed scattering profile with respect to the
experimental input scattering profile. The objective of the GA optimization
loop is to improve the fitness of an individual; in other words, improve the
SSIM score of its computed scattering profile $I_{\text{comp}}(q,\theta)$ as
compared to $I_{\text{exp}}(q,\theta)$.

The other important considerations in the implementation of the GA
optimization loop is the choice of the number of individuals to sample in each
generation (i.e., the population size) as well as the selection procedures for
determining individuals that move to the next generation. In our
implementation, we use a fixed population of 100 individuals per generation
that are always ranked according to their fitness. In each generation, the top
30 individuals with the highest fitness are selected. These 30 individuals
serve as parents who are randomly paired to form 70 children using a single-
point cross-over method. Subsequently, the 30 parents and 70 children together
form the 100 individuals for the next generation. For these 100 individuals,
the next set of operations are related to mutation. The top two elite
individuals’ gene values are kept unchanged as they progress to the next
generation. The remaining 98 individuals undergo adaptive mutation, where the
mutation probability and step size is varied based on the L2 distance (or the
squared Euclidean distance) of the individual from the mean value of all
individuals. Adaptive mutation is usually recommended to prevent the GA from
converging too quickly to a local minimum, and to have sufficient diversity in
the genes and individuals in the population.30 With this next generation of
100 individuals, the GA optimization loop is then continued. As the number of
generations increases, the fitness of a generation should converge, and upon
convergence the GA loop can be stopped.

### Performance of CREASE-2D

To evaluate the performance of the CREASE-2D method, we use all the 600 test
samples out of the 3000 test samples (the reader will recall that out of the
dataset of 3000 samples, 2400 samples are used to train the surrogate model)
and run GA five separate times for each sample. We run five GA separate runs
for each sample to check how different the output structural features from
CREASE-2D are for each sample’s input scattering profile; this allows us to
understand degeneracy in optimized GA solutions. In Figure 5a the fitness
measured in the form the SSIM scores for all 600 samples tested in CREASE-2D
are all found to in the 0.7 to 0.9 range. We note that this SSIM score range
is similar to the performance of the surrogate ML model indicating a reliable
match between the input and output scattering profiles. For each sample, the
standard deviations in the SSIM scores from its five GA replicates are small
(within 1% of the value shown) and thus, not shown in the plot for clarity. We
note that the CREASE-2D method performs only as well as the surrogate ML model
and it should not be expected to outperform the prediction accuracy of the
surrogate ML model.

In Figures 5b, we compare how well CREASE-2D predicts each structural
parameter value for all 600 test sample whose structural features we (but not
CREASE-2D) know a priori. To make an effective visual comparison, we use the
gene values directly instead of the structural features in these plots. For
some structural features, especially $\phi$, $R_{\mu}$ and $\gamma_{\mu}$
(i.e., volume fraction and means of particle size and shape distributions) the
accuracy of prediction is high, as indicated by the clustering of points close
to red line with unit slope. For $R_{\sigma}$ and $\gamma_{\sigma}$ (which
measure of the dispersity in particle size and shape) the prediction accuracy
is low, despite having a high SSIM score. This indicates that precise values
of extent of dispersity in the particle size and shape have a minimal impact
on the variation of the scattering intensity; this is in line with
observations in experiments that presence of dispersity broadens peaks of
scattering profile but does not alter the shape of the profile with the value
of dispersity. As a result, for $R_{\sigma}$ and $\gamma_{\sigma}$, CREASE-2D
method is dealing with larger degeneracy in solutions. For $\kappa$, that
quantifies orientational order, the accuracy is high only for samples that
have high values of $\kappa$, and the accuracy is low for samples with lower
$\kappa$ values (i.e., low orientational order). As one would expect, at low
values of $\kappa$ which represent isotropic ordering of anisotropic
particles, the precise numerical value of $\kappa$ value has minimal impact on
the scattering intensities.

To further demonstrate the performance of CREASE-2D method for the four
representative samples (same as from Figure 4), in Figure 5c and 5d the
results for Sample IDs 1076 and 1910 are presented and the results for Sample
IDs 1097 and 2176 are provided in Supporting Information Section S5. The
evolution of their fitness values is also presented in Supporting Information
Section S5. In Figure 5 we show the predicted scattering profiles for the best
outputs from 3 out of the 5 GA runs per system along with the evolution of
structural features over 1000 generations from those 3 GA runs. In each run,
the GA loop converges closely to the original value of the structural feature
in the first few generations, as indicated by the convergence of the curves to
the dashed line (the numbers in the plots denote the target structural feature
value of that sample). We note that one GA optimization loop with 1000
generations of 100 individuals uses 30-45 minutes in real-time to complete
when implemented on a single-(CPU) core laptop/computer with modest hardware.

In conclusion, we have developed a new CREASE-2D method that analyzes 2D
scattering profiles as is without any averaging along all or few angles, and
outputs relevant structural features like domain size and shape distribution,
extent of orientational order in the structure, and packing fraction of the
domains in the structure. The development of CREASE-2D relied on the
generation of dataset with 3000 samples each with a desired set of structural
features and the corresponding 3D structures generated using CASGAP 20 and
corresponding computed 2D scattering profile. This dataset enabled training of
a surrogate XGBoost-based model that outputs 2D scattering profile for a given
set of structural features. Using this surrogate ML model within a genetic
algorithm (GA) optimization loop, we are able to identify all the structural
features (and reconstruct 3D real-space configurations, if needed) that
produce a scattering profile that matches the input 2D scattering profile. We
believe soft materials researchers who aim to understand how macroscopic
properties (e.g., rheology, flow) depend on the structural anisotropy and the
hierarchy of structural length scales within the materials will find this
CREASE-2D method useful. CREASE-2D enables users to analyze the output of
scattering experiments holistically without having to use approximate
analytical models to fit to averaged 1D profiles or limiting to analyzing only
averaged angular sections of the 2D profiles.

## Acknowledgements

We are grateful for financial support from National Science Foundation (NSF)
Grant number: 2105744 (for SA and AJ) and MURI AFOSR Grant number MURI-FA
9550-18-1-0142 (for NG and AJ) to complete this method development work. We
also acknowledge the use of DARWIN computing system for some of the computing
in this work: DARWIN – A Resource for Computational and Data-intensive
Research at the University of Delaware and in the Delaware Region, Rudolf
Eigenmann, Benjamin E. Bagozzi, Arthi Jayaraman, William Totten, and Cathy H.
Wu, University of Delaware, 2021.

## Supplementary Information

The Supporting Information is available free of charge at website-to-be-added.
Includes additional views of results presented in the main paper as well as
specific details about some of the steps in the method development.

## Code and Data Availability

Open-source code for CREASE-2D is available on
https://github.com/arthijayaraman-lab/ The dataset discussed in the paper are
available on https://zenodo.org/records/10534943 The open-source code for 3D
structure creation method, CASGAP, is also available on
https://github.com/arthijayaraman-lab/

## Author Contributions

AJ, SA, and NG developed the ideas for this CREASE-2D method. AJ received
funding to support this project. NG and SA generated the dataset of structures
and scattering profiles. SA developed the surrogate XG-Boost ML model to train
on the dataset and tested the model. NG and SA created the CREASE-2D workflow
and implemented the genetic algorithm optimization. AJ, NG, and SA wrote and
edited the manuscript.

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