[2]Matthew E. Helgeson [1]Michael D. Graham

1]Department of Chemical and Biological Engineering, University of Wisconsin-
Madison, Madison, Wisconsin, 53706, USA

2]Department of Chemical Engineering, University of California, Santa Barbara,
California, 93106-5080, USA

# Scattering-Informed Microstructure Prediction during Lagrangian Evolution
(SIMPLE) – A data-driven framework for modeling complex fluids in flow

Charles D. Young  Patrick T. Corona  Anukta Datta<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [

###### Abstract

An overarching challenge in rheology is to develop constitutive models for
complex fluids for which we lack accurate first principles theory. A further
challenge is that most experiments probing dynamical structure and rheology do
so only in very simple flow fields that are not characteristic of the complex
deformation histories experienced by material in a processing application.  A
recently-developed experimental methodology holds potential to overcome this
challenge by incorporating a fluidic four-roll mill (FFoRM) into scanning
small-angle x-ray scattering instrumentation (sSAXS) [Corona, P. T. et al.
_Sci. Rep._ 8, 15559 (2018); Corona, P. T. et al. _Phys. Rev. Mater_ 6, 045603
(2022)] to rapidly generate large data sets of scattering intensity for
complex fluids along diverse Lagrangian flow histories. To exploit this
uniquely rich FFoRM-sSAXS data, we propose a machine learning framework,
_Scattering-Informed Microstructure Prediction under Lagrangian Evolution_
(SIMPLE), which uses FFoRM-sSAXS data to learn an evolution equation for the
scattering intensity and an associated tensorial differential constitutive
equation for the stress. The framework incorporates material frame
indifference and invariance to arbitrary rotations by data preprocessing. We
use autoencoders to find an efficient reduced order model for the scattering
intensity and neural network ordinary differential equations to predict the
time evolution of the model coordinates. The framework is validated on a
synthetic FFoRM-sSAXS data set for a dilute rigid rod suspension. The model
accurately predicts microstructural evolution and rheology for flows that
differ significantly from those used in training. SIMPLE is compatible with
but does not require material-specific constraints or assumptions.

###### keywords:

rheo-scattering, constitutive modeling, physically-informed machine learning,
structure-property relationships

##  1 Introduction

There is a critical need to develop rational design approaches for the flow-
processing of microstructured fluids and soft materials. In particular,
researchers have recently recognized the important role that the complexity of
processing flows plays in the development of these materials [20]. Unlike
prototypical flows (simple shearing or elongation) in which the type and rate
of deformation are uniform everywhere in the flow, most industrially-relevant
processing flows involve complex non-uniform flow fields, in which fluid
elements experience significant variations in deformation type and rate as
they move along the process. For example, in film casting processes, the fluid
experiences a transition from low-rate shearing to extension and then to high-
rate shearing as it passes across a coating blade [20]. Such complex flows can
significantly influence the final structure and performance of thin films
obtained from these and other processes.

Although researchers have begun to leverage this understanding to optimize
flow processes toward desired structures, most approaches remain primitive –
involving empirical searching of parameters until optimal structure or
performance is achieved. This is because process-structure relationships are
often developed from ex situ measurements, and correlated to independently
specified process parameters (e.g., blade height, substrate velocity, etc.).
However, this approach is process-specific, and as such is unable to
anticipate new flow processes that could dramatically open the range or
optimization of processed structures available to a material. This has imposed
a significant bottleneck in development of flow-processed materials in
important technologies such as printed electronics [62, 60], photonics [18],
and solar cells [24].

An alternative and much more powerful approach would be founded on detailed
modeling of flow-microstructure interactions to predict what a material of
interest might do in any possible process flow. In such a framework, one would
seek to parameterize not a process, but a flow history of a material element
as it moves through the process. The natural setting for describing such flow
histories is the Lagrangian frame, which maps an arbitrary flow, (i.e., an
Eulerian spatial distribution of velocities $\bm{v}(\bm{x})$) onto a set of
time-dependent trajectories involving variations in the velocity gradient,
$\bm{\nabla}\bm{v}(\mathcal{T})$, where $\mathcal{T}$ is elapsed time along a
streamline. In this frame, both the flow itself as well as the structural
evolution of the material can be uniquely determined through time-variations
in the components of $\bm{\nabla v}$, thereby collapsing all possible
processing flows to a set of Lagrangian excursions in these low-dimensional
variables. To be useful, this approach would require: (1) measurements of
microstructure and velocity gradient along these Lagrangian trajectories, (2)
a modeling framework that could take this information to generate an evolution
equation for the microstructure and ultimately the stress tensor.

Newly developed spatially resolved scattering measurements in complex flows
hold exceptional promise toward the first of these requirements. In
particular, Helgeson and coworkers recently adapted a “fluidic four roll mill”
(FFoRM), a version of Taylor’s original four roll mill [72] due to Mueller and
co-workers [37], for beamline scattering measurements including high-
resolution scanning small angle X-ray scattering (sSAXS) [14, 16]. The
velocity field in these flows can be quantified by particle tracking
velocimetry (PTV). Unlike other fluidic devices for in situ scattering [25,
46, 47, 49, 5, 23, 69, 61, 56, 50], the FFoRM can achieve a range of quasi-2D
stagnation point flows for various rheologically complex fluids, enabling a
wide array of flow histories (and microstructures in them) to be mapped from
the same device (Fig. 1). This FFoRM-sSAXS method is therefore ideal for
generating the large, diverse flow-microstructure data sets required to
achieve our aims. Helgeson and collaborators have already generated large data
sets [16], and have developed state-of-the-art scattering models to draw
physical insights from them [15, 17, 13]. For example, FFoRM-sSAXS on rod-like
nanoparticle suspensions was used to distinguish the influence of
interparticle interactions on flow-induced alignment [16]. Furthermore,
although this has not yet been done, in principle it is possible, with a
variational data assimilation method [1], to use the measured velocity fields
to infer the particle contribution to the stress tensor, as elaborated in Sec.
2.

With regard to the second requirement, one approach would be to start with
classical physics-based approaches for development of constitutive models.
There are some clear successes for this approach, for example in highly
entangled polymers [6, 22, 65] and concentrated hard sphere Brownian particle
suspensions [57, 58, 59]. Despite these prominent successes, first principles
rheological theory has failed to keep up with the ever-expanding set of
materials whose flow processing is crucial for emerging technologies. This is
because the development of microconstitutive models is slow, requiring
iterative testing between theory and experiment to propose and validate
physical theories for a material’s structural dynamics. It is also biasing, in
that rheological experiments are limited to situations in which the fluid is
subjected to a homogeneous deformation type and rate. As a result, while a
model may perform accurately in rheometric flows, it may do quite poorly in
more general flows. A prototypical example is non-dilute suspensions of
elongated (rod-like, ellipsoidal, etc.) Brownian particles. Although an
accurate dilute theory has been known for some time [6], a successful non-
dilute theory is lacking due to the complicated coupling between orientational
dynamics and many-body interactions. Dynamical theories [22, 21, 19] involving
assumptions of mean-field interactions between particles and infinite aspect
ratios to treat the many-body problem fail to quantitatively capture the
features of flow-induced alignment and shear thinning even for steady state
homogeneous shear flows in the semi-dilute regime [36]. Even when one resolves
particle orientations exactly, theories for the rheology of these systems [2,
30] provide accurate predictions only for linear, homogeneous flows, with
extensions to complex flows of the type we wish to address an active area of
research [27].

Given the incredibly rich data available from the FFoRM-sSAXS experiment,
data-driven modeling provides an alternate approach that could dramatically
accelerate predictive design of flow-processed materials. In Newtonian fluid
mechanics, data assimilation and machine learning approaches have had
significant impact [7]. In seminal work, Ling et al. [41] used turbulence
simulations, neural networks (NNs), and model representations constrained by
Galilean invariance to compute expressions for the Reynolds stress tensor
(i.e. a closure model) to facilitate Reynolds-averaged Navier-Stokes
simulations of turbulence. Such advances are enabled by the large data sets
that can be generated either by direct numerical simulation (DNS) or
spatiotemporally resolved velocimetry experiments for Newtonian flows.

In rheology and non-Newtonian fluid mechanics, comparatively few applications
of machine learning have been reported. Some have used rheometric data to
generate parameters for given constitutive relations [52, 53, 55, 64]. Others
have used a generalized upper convected Maxwell model with a nonlinear frame
indifferent term in the form of a neural network, again trained on rheometric
data [39]. Related work involved learning evolution equations for a set of
conformation tensors as conformational descriptors [38]. Thakur et al. [73]
described a physics-informed deep learning approach that uses a given velocity
field to select a best-fit constitutive model and parameters (from a set of
standard models). Despite these advances, the fields of rheology and non-
Newtonian fluid mechanics have yet to embrace, exploit and advance data-driven
methods for modeling and data assimilation.

Several data-driven methods have been developed for efficient, scalable
molecular dynamics simulations with ab initio accuracy from atomistic
coordinates of a DFT simulation [3, 11, 77, 78, 68]. These approaches
essentially generate an accurate and generalizeable empirical force field
which accounts for molecular symmetries, such as translation and rotation. So
far these methods have primarily been applied to small molecules, but in
principle they could be used for macromolecular soft matter systems. However,
they do not perform dimension reduction, and so may not be suitable to
spatially extended domains and long time scales required in soft matter
simulations. Lei et al. [38] proposed a model designed for non-Newtonian
fluids which discovers coarse-grained frame-invariant configuration tensors
based on the Fokker-Planck equation. The model discovers an appropriate
coarse-grained description based on molecular coordinates, such that it can be
extended to non-linear chain architectures [26]. However, the model does not
explicitly account for intra- or inter-molecular hydrodynamic interactions,
other than those arising from the simulation training data. An alternative
coarse-graining approach is the Mori-Zwanzig formalism, in which the challenge
is to learn the memory kernel for a generalized Langevin equation describing a
stochastic observable. Several data-driven models have implemented this
approach for the dynamics of small molecules and dilute polymer solutions [29,
51, 66, 63].

The above approaches rely on molecular simulation data, which is often
computationally prohibitive to collect for novel materials of interest,
particularly when screening a library of materials. While computational
approaches are being developed to accelerate this process [34], it is often
more practical to use experimental data. These data often come in the form of
two-dimensional (2D) images or scattering patterns, which has motivated the
use of convolutional NN (CNN) autoencoders (AEs). These have been use to
featurize microstructural images and classify the resulting latent space, as
shown for synthetic data [48], electronic microscopy of metals [42], and x-ray
photon correlation spectroscopy of colloidal glass relaxation after cessation
of shear flow [31]. Beltran-Villegas et al. [4] applied genetic algorithms and
molecular simulations to recover the real-space structure of self-assembled
amphiphilic polymer solutions from small-angle scattering patterns. Corona et
al. [16] used small-angle scattering to fingerprint the deformation of rod
suspension under mixed shear and rotational flows in a microfluidic four-roll
mill [14]. Chen and coworkers have made substantial progress in analyzing
scattering data for fingerprinting nonequilibrium dynamics [67], inferring the
potential of mean force [9], and inversion to the microscopic distribution
function [32, 33].

Beyond developing physical insight from microstructural measurements, it
remains a significant challenge to determine to bulk mechanical response
without training labels. Corona et al. [15] used Bayesian estimation to infer
the orientation probability distribution function (OPDF) of a semidilute rigid
rod suspension from small-angle scattering measurements. The OPDF was then
used to determine the particle contribution to the stress. However, this
approach is difficult to generalize to deformable materials in which the
underlying microscopic distribution function is not on the surface of a
sphere.

Figure 1: Overview of the FFoRM-sSAXS method (adapted from [16]). (a)
Photograph of the FFoRM geometry and schematic with dimensions illustrating
the scanning sSAXS measurement. (b) Operation of the FFoRMin three distinct
quasi-2D flows, controlled by the direction and magnitude of imposed flow
rates (black arrows), and the resulting multi-modal Eulerian FFoRM-sSAXS
spatial data. (c) Conversion of the Eulerian data along streamlines in (b)
produces corresponding Lagrangian trajectories of flow and structural data.

In the present work, we introduce a framework, which we call “Scattering-
Informed Microstructure Prediction during Lagrangian Evolution (SIMPLE)”, to
leverage the large, high-dimensional data sets produced by FFoRM-sSAXS to
develop a completely new capability — a data-driven workflow to rapidly learn
a general frame-indifferent constitutive model that can then be used to
simulate that fluid in an arbitrary 2D flow. The overall structure of SIMPLE
is illustrated in Figure 2. This paper will focus on development of the
computational aspects of the data-driven constitutive modeling using synthetic
scattering data determined from Brownian dynamics simulations of a dilute
suspension of Brownian rigid rods [28] in a FFoRM flow field generated with
the open source CFD code OpenFOAM. Future work will address the data-
assimilation problem for inferring stress from velocity data and the
development of models from true experimental scattering data for complex fluid
systems.

The remainder of the paper introduces the SIMPLE methodology. In Sec. 2, we
describe the framework and the data set we use to evaluate the quantitative
and qualitative accuracy of the microstructural and rheological predictions.
In Sec. 3.1, we quantify the performance of the model on interpolated flow
protocols, which are not used in training but are within the training data
range. Then in Sec. 3.2, we test the trained model on extrapolation to unseen
flow protocols. Finally, in Sec. 3.3 we demonstrate the ability of NNs to
learn the particle contribution to the stress from the scattering intensity.

Figure 2: SIMPLE constitutive equation learning framework for FFoRM-sSAXS
data. Scattering intensity and velocity measurements are first collected and
transformed to the Lagrangian frame (green box). Large data sets enabled by
FFoRM-sSAXS are used to train data-driven operations (blue boxes) of dimension
reduction and time evolution. Orange boxes are known operations accounting for
frame indifference, phase alignment, and material symmetries.

##  2 Methodology

The aim of the data-driven SIMPLE framework is to use FFoRM-sSAXS data to
develop microconstitutive models for complex fluids. This framework, shown in
Figure 2, will infer stress information from velocity data (we do not address
this issue here), extract the essential microstructural degrees of freedom
from scattering data, and determine evolution equations for the microstructure
and stress using velocity gradient and scattering data along streamlines.

###  2.1 Initial data analysis

Data from the FFoRM-sSAXS experiments will be steady-state two-dimensional
velocity fields $\bm{v}(\bm{x})$ and spatially-resolved scattering patterns
$I(\bm{q},\bm{x})$. From each velocity field, a large number of Lagrangian
particle trajectories will be generated by choosing points $\mathbf{X}_{0}$ in
the domain and integrating
$d\mathbf{X}/d\mathcal{T}=\bm{v}(\mathbf{X}(\mathcal{T}))$ to yield
trajectories $\mathbf{X}(\mathbf{X}_{0},\mathcal{T})$. Along each trajectory
the rate of strain $\mathbf{E}(\mathbf{X}(\mathbf{X}_{0},\mathcal{T}))$ and
vorticity tensor $\bm{\Omega}(\mathbf{X}(\mathbf{X}_{0},\mathcal{T}))$ are
computed. Relatedly, we can determine the scattering pattern along each
trajectory, $I(\bm{q},\mathbf{X}(\mathbf{X}_{0}(\mathcal{T})))$. In general we
will suppress the dependence of Lagrangian quantities on their position
$\mathbf{X}(\mathbf{X}_{0},\mathcal{T})$. Recall that in the present work, we
use synthetic data from Brownian dynamics simulations on trajectories in a
flow field determined by CFD as detailed in Sec. 2.6, rather than true
experimental data.

###  2.2 Accounting for material frame indifference and rotational
equivariance

Even for a purely data-driven framework, key symmetries that data obey can be
built into the reduced representation. For constitutive models, the key
symmetry is material frame indifference, which reflects the physical idea that
rigid rotations should not induce any relative motion within the
microstructure. In the present work we enforce this by working in a reference
frame corotating with the fluid element. The importance of including this
physical constraint cannot be overstated – indeed a widespread current theme
in scientific ML is building symmetries into representations and
architectures; otherwise neural networks require vast amounts of data to learn
them, and even then only do so approximately.

To achieve this for scattering data, we compute the unitary rotation matrix Q
which relates the rotating frame basis vectors $\bm{e}_{i}^{R}$ to the
stationary laboratory frame basis vectors $\bm{e}_{i}$ via
$\bm{e}_{i}=\textbf{Q}\cdot\bm{e}_{i}^{R}$. The rotation of the reference
frame is determined from the vorticity of the flow
$\bm{\Omega}=(\bm{\nabla}\bm{v}-\bm{\nabla}\bm{v}^{T})/2$ as

$\frac{d\textbf{Q}}{d\mathcal{T}}=\bm{\Omega}^{T}\cdot\textbf{Q}$ (1)

From the rotation matrix, we compute a rotation angle
$\theta_{\bm{\Omega}}=\textrm{atan2}(\textrm{Q}_{xy},\textrm{Q}_{xx})$. This
is used to perform a change of basis between the laboratory frame observation
of the scattering intensity $I(\bm{q})$ and the co-rotating frame
$I_{\bm{\Omega}}(\bm{q})=\mathcal{R}(I(\bm{q}),\theta_{\bm{\Omega}})$, where
$\mathcal{R}(\cdot,\theta)$ is the rotation operation. In the current study,
we consider the flow to be 2D in the $xy$ plane and the scattering intensity
to be observed as the 2D projection $I(q_{x},q_{y})$ on a uniform grid. The
range and number of wavevectors in both directions is $q\in[0,0.1]$ Å-1 and
$n_{q}=100$. This is lower resolution than many scattering experiments
(typical resolutions are $n_{q}=128$ for neutron scattering, $n_{q}=1475$ for
the PILATUS 2M detector at the Paul Scherrer Institute used in [16], and
$n_{q}=1024$ for the LiX beamline at NSLS-II (Brookhaven National
Laboratory)). The data are radially truncated to
$r=\sqrt{q_{x}^{2}+q_{y}^{2}}<0.1$ Å-1 so the image can be rotated without
loss of information. An example of this rotation and the rotation matrix in
steady shear flow is shown in Fig. 3.

Figure 3: Pre-processing of startup steady shear flow ($\textrm{Pe}=13.6$)
data before data-driven operations. The lab frame data (a-c) are first
transformed to the co-rotating frame (d-f) using the rotation matrix Q
determined from the vorticity tensor $\bm{\Omega}$. The data are then aligned
by the phase of the first azimuthal Fourier mode of the data. g) Time
evolution of the relevant rotation matrix components and the corotating frame
phase $\alpha$. For comparison, we include the time dependence of the
laboratory frame phase $\alpha_{lab}$. The dashed vertical lines correspond to
the snapshot times.

Material frame indifference is a consequence of physical assumptions about the
evolution of microstructure. Another symmetry that applies more generally is
simple equivariance under rotations. For example, the scattering pattern for a
material under extensional flow along the $x$-direction should be a rotation
of the same pattern for the fluid stretched along the $y$-direction. A brute
force solution to approximate this symmetry is to augment the training data
with rotated representations, but this means dramatically increasing the size
of the data set while still only approximately satisfying the symmetry
relation. While modern neural network architectures are making progress
towards satisfying these conditions by construction [75], it is often simpler
and more effective to apply these symmetry groups to the data before further
processing [40, 43, 76]. Following the latter approaches, we further
preprocess the scattering images by computing a rotational phase angle and
then rotating the images to the the same phase. We compute the radially
averaged azimuthal dependence of the scattering intensity where
$\Theta=\textrm{atan2}(q_{y},q_{x})$ [74, 8].

$I_{\bm{\Omega}}(m_{r},n_{\Theta})=\frac{1}{A_{m_{r},n_{\Theta}}}\sum_{r\in
R(m_{r}),\Theta\in\theta(n_{\Theta})}I_{\bm{\Omega}}(r,\Theta).$ (2)

$A$ is the number of pixels in a segment of radial range $R$ and azimuthal
range $\theta$, and $m_{r}$ and $n_{\Theta}$ are the radial and azimuthal
indices respectively. We use $N_{\Theta}=128$ azimuthal segments and a radial
range $r\in[0.03,0.05]$ Å-1. The phase is computed from the first mode of
discrete Fourier transform of the azimuthal average
$z=\mathcal{F}(I_{\bm{\Omega}}(n_{\Theta}))$ as
$\alpha=\textrm{atan2}(\textrm{Im}[z_{1}],\textrm{Re}[z_{1}])$. The phase
aligned data are generated by the rotation
$I_{\bm{\Omega},\alpha}=\mathcal{R}(I_{\bm{\Omega}},\alpha)$. Thus, the
rotational evolution is captured in the co-rotating frame phase $\alpha$, and
the deformation pattern is captured by the phase-aligned scattering intensity.
The corotating phase under steady shear flow is compared to the laboratory
frame phase in Fig. 3.

###  2.3 Dimension reduction

The machine learning task is now to determine the evolution of the scattering
pattern and phase given an input deformation. However, the data are still
high-dimensional ($I(\bm{q})\in\mathbb{R}^{d},d=n_{q}^{2}=10^{4}$, and up to
$d\approx 10^{6}$ for the experimental data collected on the beamlines listed
above), which would incur significant computational expense in training NN
time evolution models and deploying the model for structure prediction.
Furthermore, we do not expect $n_{q}^{2}$ dimensions to be necessary for
representation of the microstructure of the material [48]. We therefore seek a
reduced order representation of the data to overcome these issues. We follow
recent methods which use undercomplete autoencoders (AEs) to discover
invariant manifolds on which data lie in chaotic and turbulent dynamical
systems [43, 45].

Before passing the data to the autoencoder, we use principal components
analysis (PCA) to perform an initial coarse dimension reduction step. PCA is a
common linear dimension reduction technique by which the data are transformed
to an orthogonal basis defined by the eigenvectors of the data covariance, and
truncated to retain directions in which the variance (“energy”) is large.
There are two notable advantages to performing PCA before the autoencoder.
First, the resulting input to the AE is lower dimensional, reducing the
expense of training the NN. Second, the ordering of PCA modes places lower-
variance fluctuations in the truncated trailing modes, which reduces noise
resulting from experimental uncertainty or sampling limitations. We compute
the PCA basis vectors by singular value decomposition of $M$ snapshots of the
scattering intensity, $I(\bm{q},\mathcal{T})\in\mathbb{R}^{d\times M}$:

$I_{\bm{\Omega},\alpha}(\bm{q},\mathcal{T})=U\Sigma V^{T}$ (3)

where $U\in\mathbb{R}^{d\times d}$ is the square orthogonal matrix of PCA
basis vectors, $\Sigma\in\mathbb{R}^{d\times M}$ is a diagonal matrix of
singular values, and $V\in\mathbb{R}^{M\times M}$ is the orthogonal matrix of
right singular vectors. Dimension reduction is performed by projecting the
data onto the leading $d_{a}$ PCA basis vectors as
$a=\mathcal{P}_{d_{a}}UI_{\bm{\Omega},\alpha}(\bm{q})$. The original data are
reconstructed by $\tilde{I}_{\bm{\Omega},\alpha}(\bm{q})=aU^{T}$. We set a
criteria to retain 99 % of the energy, which reduces the dimension to
$d_{a}=50$. The PCA modes of the data
$a(\mathcal{T})\in\mathbb{R}^{d_{a}\times M}$ are then input to a deep
autoencoder, which encodes the dimension to a latent space representation
$h=\chi(a;\theta_{E})\in\mathbb{R}^{d_{h}}$ and decodes to the input
$\tilde{a}=\check{\chi}(h;\theta_{D})$, where $\theta_{E}$ and $\theta_{D}$
are the encoder and decoder weights respectively. Before the autoencoder, we
subtract the mean of each PCA mode and normalize by the largest standard
deviation, $\bar{a}_{i}=(a_{i}-|a_{i}|)/\textrm{max(std(}a_{i}))$. The
autoencoder minimizes the mean squared error (MSE) of reconstruction

$\mathcal{L}_{AE}=||\bar{a}-\bar{\tilde{a}}||^{2}_{2}$ (4)

by learning the encoder and decoder weights via stochastic gradient descent as
implemented in Keras [12]. We vary the latent space dimension in the range
$d_{h}=3-20$ [43]. The MSE decreases with $d_{h}$ until a plateau value
$d_{h}\approx 5$, which we select as the latent dimension. In Fig. 4 we show
the dimension reduction process for a sample FFoRM streamline (see Fig. 5 for
flow kinematics). We also visualize the leading 4 PCA modes.

Figure 4: Dimension reduction of the phase-aligned data
$I_{\bm{\Omega},\alpha}(\bm{q},\mathcal{T})$ for an FFoRM streamline from the
test data (same as in Fig. 5) (a) Snapshots of the data at
$\mathcal{T}=0.01,0.24,0.52$. Each snapshot is projected onto the leading
$d_{a}=50$ PCA modes $U$ (b) and then encoded to the latent space with
dimension $d_{h}=5$ (c) using a deep autoencoder. The PCA modes are
reconstructed from the latent space using the decoder (d), and finally the
scattering intensity is reconstructed (e). The first 4 PCA modes are
visualized in (f).

###  2.4 Time evolution of microstructure

Next we learn the time evolution of the reduced microstructural variable $h$
and the corotating phase under flow as

$\begin{split}\frac{dh}{d\mathcal{T}}&=g_{1}(h,\sin\alpha,\cos\alpha,\textbf{Q}^{\alpha
T}\cdot\textbf{Q}^{T}\cdot\textbf{E}\cdot\textbf{Q}^{\alpha}\cdot\textbf{Q};\theta_{g1})\\\
\frac{d\alpha}{d\mathcal{T}}&=g_{2}(h,\sin\alpha,\cos\alpha,\textbf{Q}^{T}\cdot\textbf{E}\cdot\textbf{Q};\theta_{g2})\end{split}$
(5)

where $g_{1}$ and $g_{2}$ are neural network representations of the vector
fields governing the latent space and phase evolutions respectively. An
evolution equation with this structure is sometimes referred to as a neural
ODE (NODE) [10]. We use two NNs because the autoencoder latent space $h$
evolves in the phase-aligned corotating frame, whereas the phase evolves in
the corotating frame. We input only the rate of strain tensor rather than the
velocity gradient tensor because rotation of fluid elements is accounted for
by the corotating reference frame. Our choice of inputs for the corotating
phase evolution is physically informed by the evolution equation of a material
line in the corotating frame [28], as we illustrate in the Appendix.
Similarly, we transform the rate of strain tensor from the corotating to the
phase-aligned frame by $\textbf{Q}^{\alpha}$ for pattern evolution (see
Appendix for details).

Finally, we input $\sin\alpha$ and $\cos\alpha$ rather than $\alpha$ to allow
for continuous phase evolution. The phase is bounded by $\alpha\in[-\pi,\pi]$,
but there is a discontinuous change when $\alpha$ passes over the boundary at
$\pm\pi$. Whenever this occurs in the training data, we add or subtract $\pi$.
This resolves the discontuity and the resulting rotation is the same, but it
also causes a problem at long times when the phase can become arbitrarily
large. The predicted $\alpha$ is then outside the range of the training data,
and NNs perform significantly worse. Using $\sin\alpha$ and $\cos\alpha$ maps
these values back to the range $[-1,1]$, allowing time evolution for unlimited
duration.

To train the NODEs, we use a Runge-Kutta 45 integrator to forecast the latent
space and phase

$\begin{split}\hat{h}(\mathcal{T}+\tau)&=h(\mathcal{T})+\int_{\mathcal{T}}^{\mathcal{T}+\tau}g_{1}(h,\sin\alpha,\cos\alpha,\textbf{Q}^{\alpha
T}\textbf{Q}^{T}\cdot\textbf{E}\cdot\textbf{Q}\cdot\textbf{Q}^{\alpha};\theta_{g1})d\tau\\\
\hat{\alpha}(\mathcal{T}+\tau)&=\alpha(\mathcal{T})+\int_{\mathcal{T}}^{\mathcal{T}+\tau}g_{2}(h,\sin\alpha,\cos\alpha,\textbf{Q}^{T}\cdot\textbf{E}\cdot\textbf{Q}\cdot;\theta_{g2})d\tau\end{split}$
(6)

The model is trained to minimize the loss
$\mathcal{L}_{g}=||h(\mathcal{T}+\tau)-\hat{h}(\mathcal{T}+\tau)||^{2}_{2}+||\alpha(\mathcal{T}+\tau)-\hat{\alpha}(\mathcal{T}+\tau)||^{2}_{2}$
by stochastic gradient descent using automatic differentation to perform
backpropagation [10, 44]. We use the training forecast horizon
$\tau=0.1/D_{r}$, where $D_{r}$ is the rod rotational diffusion constant.
Finally, predictions can be decoded to the full state scattering intensity
$\tilde{\hat{I}}_{\bm{\Omega},\alpha}(\bm{q},\mathcal{T})=\check{\chi}(\hat{h}(\mathcal{T}))U^{T}$
and rotated to the laboratory frame by the predicted phase and the vorticity
tensor as
$\tilde{\hat{I}}(\bm{q},\mathcal{T})=\mathcal{R}(\mathcal{R}(\tilde{\hat{I}}_{\bm{\Omega},\alpha}(\bm{q},\mathcal{T}),-\hat{\alpha}(\mathcal{T})),-\theta_{\bm{\Omega}}(\mathcal{T}))$.

###  2.5 Stress

Given stress data, an additional neural network can be trained to map the
latent representation of the microstructure to the phase-aligned stress field:

$\tilde{\bm{\sigma}}_{p}^{\alpha}=F(h,\textbf{Q}^{\alpha
T}\cdot\textbf{Q}^{T}\cdot\textbf{E}\cdot\textbf{Q}\cdot\textbf{Q}^{\alpha};\theta_{F})$
(7)

We subtract the mean from each stress component and normalize the largest
standard deviation
$\bar{\sigma}_{p,ij}^{\alpha}=(\sigma_{p,ij}^{\alpha}-|\sigma_{p,ij}^{\alpha}|)/\textrm{max(std(}\sigma_{p,ij}^{\alpha}))$
and use stochastic gradient descent to minimize the loss
$L_{F}=||\bar{\bm{\sigma}}_{p}^{\alpha}-\bar{\tilde{\bm{\sigma}}}_{p}^{\alpha}||^{2}_{2}$.
The phase-aligned stress can then be transformed back to the lab frame by
$\bm{\sigma}_{p}=\textbf{Q}^{\alpha}\cdot\textbf{Q}\cdot\bm{\sigma}^{\alpha}_{p}\cdot\textbf{Q}^{T}\cdot\textbf{Q}^{\alpha
T}$. This function and Eq. 5 for the microstructure gives the evolution of the
stress – a tensorial frame-indifferent constitutive model determined from
FFoRM-sSAXS data.

We will not tackle the problem of inferring stress from velocity measurements
in the present work. Nevertheless, we provide here a sketch of how this might
be done. For a solution of particles or polymers in a Newtonian solvent with
viscosity $\eta_{s}$, and neglecting inertia, the momentum equation is
$\bm{\nabla}\cdot\bm{\sigma}_{p}=-\eta_{s}\nabla^{2}\bm{v}$, where
$\bm{\sigma}_{p}(\bm{x})$ is the desired polymer stress (inertia can be
included if necessary) [2]. In an incompressible fluid the isotropic part of
the stress is indeterminate, so we can take $\bm{\sigma}_{p}$ to be traceless.
To deal with noise, we can use the fact that the flow is steady so we can use
velocimetry data from multiple time instants. To determine $\bm{\sigma}_{p}$
from measurements of $\bm{v}$, a variational data assimilation framework,
which requires no derivatives of velocity data, can be used [1]. Here we find
the solution (stress and velocity) to
$\bm{\nabla}\cdot\bm{\sigma}_{p}=-\eta_{s}\nabla^{2}\bm{v}$ and
$\bm{\nabla}\cdot\bm{v}=0$ that minimizes
$\mathbb{E}\\{||\bm{v}-\bm{v}_{e}||^{2}\\}$, where $\bm{v}_{e}$ is the
experimental velocity, and $\mathbb{E}\\{\cdot\\}$ is expected value over the
data. Alternately, physics-informed neural networks [35, 73] have been
recently advocated for solving related problems – this approach may also be
useful for the present problem.

###  2.6 Synthetic training data

In the present work we develop and validate SIMPLE on a synthetic data set
intended to represent the dispersions of rigid, rod-like cellulose
nanocrystals (CNCs) used in the FFoRM-sSAXS experiments of [16], although
simplifications have been made. For further details of the device geometry, we
refer to this previous work. We first generate Lagrangian flow trajectories in
a model FFoRM by OpenFOAM simulations as performed in [14] assuming the fluid
is Newtonian. We vary the ratio of the inlet flow rate $Q_{1}$ and outlet flow
rates $Q_{2}$ over a range $Q_{1}/Q_{2}=(-0.7,-0.6,0.5,...,1.0)$, yielding
flows that vary from shear to extension at the stagnation point as
characterized by the nominal flow type parameter

$\Lambda=\frac{|\textbf{E}|-|\bm{\Omega}|}{|\textbf{E}|+|\bm{\Omega}|}$ (8)

where $|\bm{A}|=(\bm{A}:\bm{A}^{T})^{1/2}$. The velocity gradient tensor for a
planar flow can be expressed in terms of the flow type parameter and flow
strength $G=|\bm{\nabla v}|/\sqrt{1+\Lambda^{2}}$ with cartesian axes
coincident with the principle rate of strain axes

$\bm{\nabla v}=\frac{G}{2}\begin{bmatrix}1+\Lambda&1-\Lambda&0\\\
-(1-\Lambda)&-(1+\Lambda)&0\\\ 0&0&0\end{bmatrix}$ (9)

Streamlines are collected in the central stagnation region of the device, and
velocity gradient components in the $z$-plane are set to 0. In experiments the
scattering intensity is averaged over the depth of the device, although for
sufficient channel height these effects are small [14].

We vary the inlet flow rate $Q_{1}\approx 0.025-0.5$mL/min, corresponding to a
dimensionless rotational Peclet number $\textrm{Pe}=|\textbf{E}|/D_{r}\approx
1-20$ for the CNC suspensions, where $D_{r}=2s^{-1}$ is the rotational
diffusion constant, and $\eta_{s}\approx 1$ Pa s is the solvent viscosity. We
nondimensionalize the velocity gradient and time scales from OpenFOAM
simulations as $\bm{\nabla v}^{*}=\bm{\nabla v}/D_{r}$ and $t^{*}=tD_{r}$. For
notational convenience the $*$ superscripts are dropped hereafter. For each
combination of flow rate and inlet/outlet ratio we sample 20 streamlines
passing through the crossflow region, yielding $\sim 2000$ streamlines in
total. Compared to the resolution of streamlines available in the experimental
FFoRM-sSAXS data set, this represents a significant undersampling. Future work
will investigate the influence of the range of flow type, flow strength, and
size of the training data upon SIMPLE performance.

The velocity gradient tensor is then used to perform Brownian dynamics
simulations of dilute, high-aspect ratio cylindrical rods as in [71, 70]. We
simulate $10^{5}$ trajectories to provide nearly noise-free measurements of
the second and fourth orientation tensor moments, $\langle\textbf{uu}\rangle$
and $\langle\textbf{uuuu}\rangle$, the OPDF in spherical coordinates
$N(\theta_{rod},\phi_{rod})$, and the scattering intensity $I(\bm{q})$.

The scattering intensity calculation assumes a dilute cylindrical rigid rod
suspension

$\overline{I(\bm{q})}=\phi(\Delta\rho)^{2}V_{p}\overline{P(\bm{q})S(\bm{q})}+b$
(10)

Overbars indicate an ensemble average over the BD trajectories, or
equivalently over time for a steady state flow. For a dilute suspension
interparticle scattering effects are insignificant,
$\overline{P(\bm{q})S(\bm{q})}\approx P(\bm{q})$. The model parameters are
selected based on a fit to the CNCs and solvent in [16], yielding the particle
length $L=1680$ nm, radius $R=34$ nm, particle volume fraction $\phi=0.1$,
scattering length density difference between particle and solvent
$\Delta\rho=3.2\times 10^{-6}$ Å2, particle volume $V_{p}=\pi R^{2}L$, and
incoherent scattering cross-section $b=1\times 10^{-4}$ cm-1. For calculation
of the form factor we refer to [15], which integrates the single-particle
scattering over the OPDF, $N(\theta_{rod},\phi_{rod})$. The particle
contribution to the stress is calculated from the orientation tensor moments
using Batchelor’s expression for noninteracting particles [2, 15]. Finally,
all quantities are sampled every $0.01$ dimensionless time units, yielding
$\approx 2\times 10^{5}$ total data points. Of these, half of the flow rates
are retained for test data, so the total training data size is $M\approx
1\times 10^{5}$.

##  3 Results and Discussion

We now evaluate SIMPLE on the data described above, starting with training of
data-driven operations and interpolation within the training data. We then
extrapolate the model to startup steady shear and extensional flow protocols.

###  3.1 Training and Interpolation

In Table 1 we report the error of data-driven operations in SIMPLE evaluated
on 720 FFoRM streamlines. These data are within the range of flow rates
$Q_{1}+Q_{2}$ and flow types $\Lambda$ used in training, but were not
explicitly used to train the NNs or perform the singular value decomposition.
We report the normalized reconstruction error of the phase-aligned scattering
intensity from the truncated PCA basis,
$|I_{\bm{\Omega},\alpha}-\tilde{I}_{\bm{\Omega},\alpha}|/|I_{\bm{\Omega},\alpha}|$,
as well as the NN losses for the autoencoder, NODE, and the stress function.
The NN architectures are reported in the Appendix.

$|I_{\bm{\Omega},\alpha}-\tilde{I}_{\bm{\Omega},\alpha}|/|I_{\bm{\Omega},\alpha}|$ | $\mathcal{L}_{AE}$ | $\mathcal{L}_{g}$ | $\mathcal{L}_{F}$
---|---|---|---
$8.2\times 10^{-4}$ | $1.2\times 10^{-6}$ | $9.1\times 10^{-4}$ | $7.1\times 10^{-4}$
Table 1: Test error of data-driven operations.

As a visual example, we demonstrate the predictive capabilities of the trained
SIMPLE model for a specific streamline from the FFoRM test data. In
particular, we choose the operating conditions $Q_{1}/Q_{2}=-0.7$ and
$\textrm{Pe}\approx 20$, in which case the flow field exhibits significant
spatial fluctuations and large Lagrangian excursions in $\Lambda$ and $G$. We
time evolve the microstructure from $I(\mathcal{T}=0)$ corresponding to the
start of the beam region (Fig. 5a, star symbol) for a specified deformation
$\bm{\nabla v}(\mathcal{T})$. Specifically, we rotate the initial condition to
the phase-aligned reference frame
$I_{\bm{\Omega},\alpha}(\mathcal{T}=0)=\mathcal{R}(\mathcal{R}(I(\mathcal{T}=0),\theta_{\bm{\Omega}}(\mathcal{T}=0)),\alpha(\mathcal{T}=0))$,
project onto the truncated PCA basis, and encode to the latent space
$h(\mathcal{T}=0)=\chi(\mathcal{P}_{d_{a}}UI_{\bm{\Omega},\alpha}(\mathcal{T}=0);\theta_{E})$.
We then use the trained NNs in Eq. 5 to time evolve the latent space and
phase, yielding $\hat{h}(\mathcal{T})$ and $\hat{\alpha}(\mathcal{T})$. We
decode the latent space predictions to the scattering intensity
$\tilde{\hat{I}}_{\bm{\Omega},\alpha}(\mathcal{T})=\check{\chi}(\hat{h}(\mathcal{T});\theta_{D})U^{T}$,
and finally rotate to the lab frame using the predicted phase
$\tilde{\hat{I}}(\mathcal{T})=\mathcal{R}(\mathcal{R}(\tilde{\hat{I}}_{\bm{\Omega},\alpha}(\mathcal{T}),-\hat{\alpha}(\mathcal{T})),-\theta_{\bm{\Omega}}(\mathcal{T}))$.

We find the SIMPLE model tracks the scattering pattern and phase successfully,
and the autoencoder decodes to the full space with low error – typically
several percent. The error is largest at the inlet due to the high flow
strength $\textrm{Pe}\approx 15-20$. These are among strongest flows in the
training data, which are the most difficult to predict, as we discuss further
below. This error might be reduced by choosing the initial condition for the
microstructure to be the steady state at the inlet shear rate. The error
increases again at $\mathcal{T}\approx 0.25$, where the scattering intensity
undergoes a rapid reorientation due to the change from shear to rotational
flow and back to shear flow. The primary NODE error comes in predicting the
phase, but after this large flow gradient, the NODE tracking successfully
reproduces the true trajectory again. SIMPLE error in predicting the stress
from the time evolved latent space for this streamline is also low, as
discussed in Sec. 3.3 (Fig. 9).

Figure 5: Microstructural evolution prediction along a Lagrangian streamline
in the FFoRM. a) Streamlines in the FFoRM in black (white line is the selected
streamline) and snapshot locations of the sample predictions in (h-j) (green
symbols) b) Flow type parameter $\Lambda$ c) Flow strength $G$ d) Prediction
error versus Lagrangian time $\mathcal{T}$. e-g) True data from BD
simulations. h-j) SIMPLE predictions evolved from initial condition
$I(\mathcal{T}=0)$ with input $\bm{\nabla v}(\mathcal{T})$ as described in the
text. Snapshot time $\mathcal{T}=0.01$ (star), $\mathcal{T}=0.24$ (square),
and $\mathcal{T}=0.52$ (diamond).

Next, we quantify ensemble error of predictions on test FFoRM streamlines.
Generally the streamline time $\mathcal{T}_{max}$ varies with the starting
position $\textbf{X}_{0}$ and operating conditions, so we normalize each
streamline time by its maximum duration, $\mathcal{T}/\mathcal{T}_{max}$, and
report the ensemble average error. We show the error in the prediction of the
latent space variables $h$ and co-rotating phase $\alpha$ in Fig. 6a. We also
show the error in the lab frame scattering intensity in Fig. 6b after decoding
the latent space forecast to the phase-aligned intensity and rotating back to
the lab frame using the predicted phase. The forecasting error grows quickly
at $\mathcal{T}/\mathcal{T}_{max}\approx 0$ and
$\mathcal{T}/\mathcal{T}_{max}\approx 1$ due to large flow gradients at the
beginning and end of the stagnation region. In the center of the device near
the stagnation point the flow is relatively homogeneous, and the error is
constant.

Figure 6: Ensemble average error of SIMPLE predictions evaluated on test FFoRM
streamlines as a function of residence time a) Forecast error of the latent
variables and phase predicted by the NODE b) Combined error of forecasting,
decoding, and rotating to the lab frame.

###  3.2 Extrapolation

We have demonstrated the SIMPLE approach successfully predicts microstructural
evolution under diverse deformation histories for FFoRM test data. A greater
challenge is to extend model predictions to deformation inputs and histories
that are substantially different than those used for training. We will
demonstrate the extrapolation capabilities of the SIMPLE approach in three
ways: 1) startup of steady shear flow, which is not explicitly in the FFoRM
data; 2) startup of steady planar extensional flow along an extension axis not
seen in the FFoRM; 3) Varying the flow strength Pe beyond the training data
range.

We generate scattering intensity patterns from BD simulations using a constant
velocity gradient tensor given by Eq. 9 with $\Lambda=0$ for shear flow and
$\Lambda=1$ for extensional flow. The initial condition is an isotropic
suspension. In the FFoRM training data set, there are no streamlines in which
the fluid undergoes simple shear deformation for an extended duration. There
are streamlines comparable to startup to startup planar extensional flow, as
seen in Fig. 1b,c (left column), but the axis of extension is at an angle of
$45^{\circ}$ with respect to the $x$-axis, as compared to the current case
where the extension is $x$-axis aligned. We vary the flow strength
$G/D_{r}=0.1-50.0$, as compared to the training data range $\textrm{Pe}\approx
1-20$.

Using the same velocity gradient tensor $\bm{\nabla v}(\mathcal{T})$ as the BD
simulations, we make SIMPLE predictions from an initially isotropic scattering
intensity as detailed in Sec. 3.1. We qualitatively visualize these
predictions in comparison the the BD simulations in Fig. 7. Generally, SIMPLE
accurately predicts the qualitative scattering pattern and long-time steady
state behavior. For increasing flow strength, the prediction error grows. We
primarily attribute this to the range of training data, $\textrm{Pe}\approx
1-20$. For a purely data-driven framework, we expect the model to perform
poorly where it has not been trained. This could be addressed by physically
informed approaches to time evolution [54], or by multi-fidelity modeling
where high Pe data from an imperfect model is used to supplement the FFoRM
data [52, 53]. The white regions for SIMPLE predictions in extensional and
$\textrm{Pe}>20$ correspond to negative values of scattering intensity, which
is clearly an unphysical result. These emerge from errors in the reconstructed
predictions of the PCA modes, which can be negative valued. However, this only
occurs for scattering patterns outside the training data range. This could be
addressed by constraining the reconstruction of PCA modes to be positive
valued.

Figure 7: Steady state scattering intensity from BD simulations and SIMPLE
predictions under shear and extensional flow for increasing
$\textrm{Pe}=\dot{\gamma}/D_{r}$ and $Pe=\dot{\epsilon}/D_{r}$ respectively
from left to right. First row: BD simulation shear flow. Second Row: SIMPLE
prediction shear flow. Third row: BD simulation extensional flow. Fourth row:
SIMPLE prediction extensional flow. The dashed box indicates flow conditions
which are above the training data range $\textrm{Pe}=1-20$.The white regions
on the rightmost two scattering patterns on the bottom row indicate spurious
negative intensity predictions.

We quantify the prediction error in these extrapolation flow protocols in Fig.
8. We observe oscillations in the shear flow predictions and the extensional
flow predictions at low Pe. These emerge due to a lag between the predicted
and true phase $\alpha$ (Appendix Fig. 11). Because the phase NN evolves under
a corotational rate of strain tensor input, any phase lag in a constant
vorticity flow will emerge as an oscillation, which is then propagated through
the latent space $h$ prediction via the rotation of the rate of strain tensor
to the phase-aligned reference frame. Additionally, the equilibrium phase is
undefined, which causes challenges in data-driven modeling [43] and leads to
the low-Pe oscillations seen in the extensional flow predictions. In future
work, we plan to address this issue by physically informed NNs, with the form
$d\alpha/d\mathcal{T}=f(\bm{\Omega}(\mathcal{T}))+g_{2}(h,\sin\alpha,\cos\alpha,\textbf{E};\theta_{g2})$,
where the first term is known via the co-rotating frame evolution with the
vorticity tensor.

Figure 8: SIMPLE prediction error versus accumulated strain for increasing
flow strength under startup of a) shear flow b) extensional flow.

###  3.3 Rheology predictions

Finally, we demonstrate the ability of NNs to learn the particle stress from
the latent space scattering data in Fig. 10. As discussed above, we consider
an ideal case in which training data for the stress is available from BD
simulations. To obtain stress predictions, we take the time evolved latent
space and phase for a given initial condition and deformation (as described in
Sec. 3.1) and input them to the stress function
$\tilde{\hat{\bm{\sigma}}}_{p}^{\alpha}=F(\hat{h},\textbf{Q}^{\hat{\alpha}T}\cdot\textbf{Q}^{T}\cdot\textbf{E}\cdot\textbf{Q}\cdot\textbf{Q}^{\hat{\alpha}};\theta_{F})$.
In Fig. 9 we consider SIMPLE predictions for the FFoRM test streamline shown
in Fig. 5. We find SIMPLE reproduces the true stress accurately, even at
$\mathcal{T}\approx 0.25$ where the scattering intensity prediction error is
relatively high.

Figure 9: SIMPLE predictions of the stress for the FFoRM test streamline shown
in Fig. 5.

We then predict the stress in steady shear and extensional flow using the
SIMPLE results from Sec. 3.2. We exclude the first 10 strain units of data
before computing the steady state shear viscosity and extensional viscosity.
In Fig. 10 we compare these predictions (green x’s) to BD simulation results
(blue circles). The model accurately predicts the shear viscosity within the
training data range $\textrm{Pe}=1-20$. With respect to the poor prediction at
$\textrm{Pe}=0.1$, calculating low Pe shear viscosity is challenging even from
BD simulations because the signal-to-noise ratio is unfavorable in that
regime. SIMPLE predicts the extensional viscosity accurately up to
$\textrm{Pe}\approx 20$. For stronger flows, SIMPLE predicts a negative
extensional viscosity due to time evolution errors which lead to spurious
alignment in the direction of compression, which is clearly visible in the
bottom right panel of Fig. 7. Methods to combine data-driven modeling with
asymptotic approaches for limiting cases such as $\textrm{Pe}\rightarrow 0$
and $\textrm{Pe}\rightarrow\infty$ will be the topic of future work.

Figure 10: BD simulation results (blue circles), SIMPLE predictions (green
x’s), and stress function predictions given the true scattering intensity
$F(h,\alpha)$ (orange squares) for the particle contribution to the viscosity
in a) steady shear flow b) steady extensional flow.

To evaluate the accuracy of the stress function without time evolution errors,
we also consider predictions of the stress given the true scattering intensity
pattern (orange squares in Fig. 10). We rotate the true scattering intensity
to the phase-aligned reference frame
$I_{\bm{\Omega},\alpha}(\mathcal{T})=\mathcal{R}(\mathcal{R}(I(\mathcal{T}),\theta_{\bm{\Omega}}(\mathcal{T})),\alpha(\mathcal{T}))$,
encode to the latent space
$h(\mathcal{T})=\chi(\mathcal{P}_{d_{a}}UI_{\bm{\Omega},\alpha}(\mathcal{T});\theta_{E})$,
and make stress predictions
$\tilde{\bm{\sigma}}_{p}^{\alpha}=F(h,\textbf{Q}^{\alpha
T}\cdot\textbf{Q}^{T}\cdot\textbf{E}\cdot\textbf{Q}\cdot\textbf{Q}^{\alpha};\theta_{F})$.
We find the stress function accurately predicts the shear and extensional
viscosity at low Pe, suggesting that errors in the full SIMPLE model arise
primarily from the time evolution aspect of the model, not the dimension
reduction.

##  4 Conclusions

The FFoRM-sSAXS experiment originally demonstrated by [16] provides remarkable
access to microstructural information in a flowing complex fluid. The present
work demonstrates how to use this information in Lagrangian reference frames
attached to a large number of streamlines in the flow to develop efficient and
accurate low-dimensional models for the evolution of the microstructure. The
approach we develop, called “Scattering-Informed Microstructure Prediction
during Lagrangian Evolution (SIMPLE)”, provides a data-driven workflow to
rapidly develop general frame-indifferent evolution equations that can then be
used to simulate that fluid in an arbitrary flow. In this paper we have
introduced the basic data-driven framework, using efficient data-augmentation
methods to make the best use of the available data, as well as a corotating
reference frame that enables the microstructural evolution model to obey the
key physical principle of material frame indifference. Results thus far are
very promising, both for “interpolating” predictions for evolution along
streamlines in the FFoRM that were not included in the training data, as well
as for “extrapolating” predictions for flow histories quite different than
those found in the training data. In particular, from the Lagrangian data, we
can reconstruct the correct evolution for startup of simple shear and planar
extension.

Further methodological improvements to the SIMPLE frameworks are necessary.
The small phase errors need to be reduced, and of course the larger issue of
extracting stress from velocity fields must be addressed; we have described
potential approaches for doing so. A potential concern regarding the current
SIMPLE framework may be that the construction of the microstructural NN model
is through reduced-order scattering information rather than direct
microstructural descriptors, resulting in some lack of interpretability of the
reduced description. We note that accurate general approaches exist to convert
scattering information directly to real-space information such as the OPDF and
its moments ([15, 17]), so an alternative to the ML-based dimension reduction
pursued here would be to explicitly specify physically-motivated descriptors
to be used in the dynamic modeling aspect of the algorithm. This approach
would likely trade efficiency for interpretability, since the ML-based
approach uses an optimization procedure to determine the latent
representation.

More interestingly from the soft matter physics perspective, the SIMPLE
framework can also be integrated with physics-based evolution equations for
microstructure, rather than the data-driven ones considered here, whose only
physical input is the constraint of frame indifference. For example, the
SIMPLE formalism also provides an opportunity to combine physics- and data-
driven modeling by advancing the development of closure models. For a
suspension of rigid rods with orientation u, the evolution equation for the
second moment of the OPDF, $\langle\textbf{uu}\rangle$, depends on the fourth
moment $\langle\textbf{uuuu}\rangle$ and so on. No fully satisfactory closure
model, even for this simplest complex fluid, exists – for example none get the
scaling for shear-thinning of viscosity correct [17]. It may be possible to
use data and symmetry, in a manner analogous to the turbulence closure work of
Ling et al. [41] noted above, to find a closure model from data. For nondilute
systems, a key challenge is to incorporate particle-particle interactions into
the evolution equation for the orientational distribution function. In
principle, a general theoretical framework for describing these effects
already exists [19, 36], but the functional forms of the interaction terms are
not well understood. One potential approach is a hybrid of ML and data
assimilation, where we take the form of the evolution equation to be known,
but learn various functional contributions to it by representing them as
neural networks (or some other functional representation) whose parameters are
learned from the FFoRM-sSAXS data. More broadly, the most effective model in
practice may be one that takes a physics-based model as a baseline, with a
data-driven model learning the difference between the physics-based model and
the data.

Finally, real processing flows are almost never viscometric, whereas most
rheological data come from viscometric flows. Therefore the general importance
of a combined experimental and modeling framework that enables direct
development of fully tensorial constitutive models from transient,
nonviscometric data sets cannot be overstated. The combination of FFoRM-sSAXS
and SIMPLE provides an important step toward this aim.

## Declarations

### Funding

MDG and CY acknowledge support from ONR N00014-18-1-2865 (Vannevar Bush
Faculty Fellowship). MEH, PTC and AD acknowledge support from the Department
of Energy, Award Number DE-SC0020988.

### Conflicts of interest/competing interests

All authors certify that they have no affiliations with or involvement in any
organization or entity with any financial interest or non-financial interest
in the subject matter or materials discussed in this manuscript.

## Appendix

### Neural network architectures

The neural network architectures and activation functions are given in Table
2. The depth and width were increased empirically until improvement in the
test loss plateaued. All models are trained using the Adam optimizer in
PyTorch. We use the PyTorch package torchdiffeq [10] to train the NODEs.

Function | Shape | Activation
---|---|---
$\chi$ | $d_{a}:64:128:64:d_{h}$ | ReLU:ReLU:ReLU:Linear
$\check{\chi}$ | $d_{h}:64:128:64:d_{a}$ | ReLU:ReLU:ReLU:Linear
$g_{1}$ | $d_{h}+11:128:256:256:128:d_{h}$ | ReLU:ReLU:ReLU:ReLU:Linear
$g_{2}$ | $d_{h}+11:32:64:32:1$ | ReLU:ReLU:ReLU:Linear
$F$ | $d_{h}+9:128:128:9$ | ReLU:ReLU:Linear
Table 2: NN architectures. The additional 11 inputs to the time evolution
models $g_{1}$ and $g_{2}$ correspond to the rate of strain tensor E and the
phase $(\sin\alpha,\cos\alpha)$. The additional 9 inputs to the stress model
$F$ correspond to the rate of strain tensor. In the current application to
dilute rigid rod suspensions, $d_{a}=50$, and $d_{h}=5$.

### Evolution in the co-rotating frame

In principle, the lab frame rate of strain tensor E and the rotation matrix Q
should be sufficient for the SIMPLE to perform time evolution in the co-
rotating frame. However, it improves accuracy and reduces the amount of
necessary training data to input the rate of strain tensor in the appropriate
reference frame, as we have done in Eqs. 5-7. To illustrate the proper
transformation of E for the corotating frame, we consider here a simple case
of evolution of an infinitesimal material line. In the Lagrangian, but non-
rotating frame,

$\frac{d\textbf{u}}{d\mathcal{T}}=(\bm{\nabla v}^{T})\cdot\textbf{u}$ (11)

The co-rotating frame orientation is related to the orientation in the
nonrotating frame by $\textbf{u}=\textbf{Q}\cdot\textbf{u}^{R}$ where
$\textbf{Q}=[\bm{e}_{1}^{R},\bm{e}_{2}^{R}]$ is the unitary rotation matrix,
which evolves according to Eq. 1, and the elements of $\textbf{u}^{R}$ are the
components of u in the corotating basis given by $\bm{e}_{1}^{R}$ and
$\bm{e}_{2}^{R}$. This leads to

$\frac{d\textbf{u}}{d\mathcal{T}}=\frac{d\textbf{Q}\cdot\textbf{u}^{R}}{d\mathcal{T}}=\frac{d\textbf{Q}}{d\mathcal{T}}\cdot\textbf{u}^{R}+\textbf{Q}\cdot\frac{d\textbf{u}^{R}}{d\mathcal{T}}=(\textbf{E}+\bm{\Omega}^{T})\cdot\textbf{Q}\cdot\textbf{u}^{R}$
(12)

Applying Eq. 1, left multiplying by $\textbf{Q}^{T}$, and noting
$\textbf{Q}^{T}\cdot\textbf{Q}=\bm{\delta}$, the terms containing
$\bm{\Omega}^{T}$ cancel out and we obtain

$\frac{d\textbf{u}^{R}}{d\mathcal{T}}=\textbf{Q}^{T}\cdot\textbf{E}\cdot\textbf{Q}\cdot\textbf{u}^{R}$
(13)

Thus, $\textbf{Q}^{T}\cdot\textbf{E}\cdot\textbf{Q}$ is the appropriately
transformed representation of the velocity gradient.  Similarly, for the
transform from the co-rotating frame to the phase-aligned frame we have

$\textbf{u}^{R}=\textbf{Q}^{\alpha}\cdot\textbf{u}^{\alpha}$ (14)
$\textbf{Q}^{\alpha}=\begin{bmatrix}\cos(\alpha)&-\sin(\alpha)\\\
\sin(\alpha)&\cos(\alpha)\\\ \end{bmatrix}$ (15)

Following a similar process, after simplification we find

$\frac{d\textbf{u}^{\alpha}}{d\mathcal{T}}=\left(\textbf{Q}^{\alpha
T}\cdot\textbf{Q}^{T}\cdot\textbf{E}\cdot\textbf{Q}\cdot\textbf{Q}^{\alpha}-\begin{bmatrix}0&-\alpha_{t}\\\
\alpha_{t}&0\\\ \end{bmatrix}\right)\cdot\textbf{u}^{\alpha}$ (16)

where $\alpha_{t}=d\alpha/d\mathcal{T}$. This informs the input to the NODE
evolution of the phase-aligned scattering pattern.

### Phase prediction lag

In Fig. 11, we show an example of the time evolution of the co-rotating phase
in steady shear flow at $\textrm{Pe}=21.0$. We observe a lag between the true
and predicted phase which leads to the oscillations in error in Fig. 8. This
error emerges shortly after the start of prediction, and is constant in time
thereafter. This is consistent across the range of $\textrm{Pe}=1-50$.

Figure 11: True and predicted $\sin\alpha$ and $\cos\alpha$ for steady shear
flow $\textrm{Pe}=21.0$.

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