$\bm{x}_{\pm\delta\sigma}:=\widehat{S}^{\mathcal{X}}(\bm{q}^{*},\cdot)^{-1}(\bm{0}\pm\delta\sigma)$
where $\delta m=\Delta\mathbf{e}_{1}$, $\delta\epsilon=\Delta\mathbf{e}_{2}$
and $\delta\sigma=\Delta\mathbf{e}_{3}$ for some scalar $\Delta>0$. In our
studies we set $\Delta=2$, which corresponds to the $0.977$ quantile of a
univariate standard normal random variable.

In Tab. 2, we present the posterior quadrature points from the parameter
posterior distribution that conditioned on the energy-based summary statistics
of three different synthetic top-down microscopy characterization images. They
are generated at three different sets of model parameters, with their latent
Di-BCP patterns shown in the left column. For this example, we consider the
characterization model with the reference field of view
($\mathcal{J}=\mathrm{Id}_{H^{1}(\mathcal{D})}$), a low level of the additive
noise corruption ($\text{SNR}=5$), and an insignificant level of the blurring
corruption ($\mathcal{I}=\mathrm{Id}_{H^{1}(\mathcal{D})}$). In the top row,
the case of symmetric Di-BCPs ($m=0$) is considered, which exhibit lamella
patterns. The other two are chosen to have patterns consist of primarily spots
(_middle row_) and a mixture of lamellae and spots (_bottom row_). For each of
the synthetic image data $\bm{y}^{*}$, we consider two sets of posteriors that
are conditioned on $\bm{q}^{*}=\bm{\mathcal{Q}}(\bm{y}^{*})$ with (i)
$\bm{\mathcal{Q}}=(\mathcal{Q}_{2},\mathcal{Q}_{4})$ only (_middle column_),
and (ii)
$\bm{\mathcal{Q}}=\bm{\mathcal{Q}}_{1:4}\coloneqq(\mathcal{Q}_{1},\mathcal{Q}_{2},\mathcal{Q}_{3},\mathcal{Q}_{4})$
(_right column_). When comparing the two sets of posterior samples, we observe
a strong concentration of the posterior achieved by conditioning on the set of
four energy-based summary statistics—all seven posterior quadrature points are
essentially indistinguishable in the plots. This observation is quantified via
EIG calculations that are presented in the following subsections.

Synthetic latent pattern $u^{*}$ $\pi_{\bm{X}|\bm{q}^{*}}$
$\bm{\mathcal{Q}}=(\mathcal{Q}_{2},\mathcal{Q}_{4})$
$\bm{\mathcal{Q}}=\bm{\mathcal{Q}}_{1:4}$
$\bm{\mathcal{Q}}=(\mathcal{Q}_{2},\mathcal{Q}_{4})$
$\bm{\mathcal{Q}}=\bm{\mathcal{Q}}_{1:4}$
$\bm{\mathcal{Q}}=(\mathcal{Q}_{2},\mathcal{Q}_{4})$
$\bm{\mathcal{Q}}=\bm{\mathcal{Q}}_{1:4}$

Table 2: (_left column_) The latent material states $u^{*}$, described by
order parameters, of three synthetic microscopy image $\bm{y}^{*}$ data of Di-
BCP thin films. (_middle column_) The Gauss–Hermite quadrature points of the
reference density pulled back to the model parameter posterior, illustrating
posterior concentration conditioned on
$\bm{q}^{*}=(\mathcal{Q}_{2}(\bm{y}^{*}),\mathcal{Q}_{4}(\bm{y}^{*}))$.
(_right column_) The same as the middle column, but conditioned on
$\bm{q}^{*}=(\mathcal{Q}_{1}(\bm{y}^{*}),\dots,\mathcal{Q}_{4}(\bm{y}^{*}))$.

Synthetic latent pattern $u^{*}$ $\bm{\mathcal{Q}}$
$\mathcal{F}(\bm{x}_{\bm{0}},z^{i})$ $\mathcal{F}(\bm{x}_{\pm\delta m},z^{i})$
$\mathcal{F}(\bm{x}_{\pm\delta\epsilon},z^{i})$
$\mathcal{F}(\bm{x}_{\pm\delta\sigma},z^{i})$
$(\mathcal{Q}_{2},\mathcal{Q}_{4})$ $\bm{\mathcal{Q}}_{1:4}$
$(\mathcal{Q}_{2},\mathcal{Q}_{4})$ $\bm{\mathcal{Q}}_{1:4}$
$(\mathcal{Q}_{2},\mathcal{Q}_{4})$ $\bm{\mathcal{Q}}_{1:4}$

Table 3: Posterior predictive samples, $\mathcal{F}(\bm{x},z^{i})$, where each
image on the right column is generated by independent draws from
$z^{i}\sim\nu_{Z|M}(\cdot|m)$, The model parameter samples $\bm{x}_{0}$,
$\bm{x}_{\pm\delta m}$, $\bm{x}_{\pm\delta\epsilon}$, and
$\bm{x}_{\pm\delta\sigma}$ are transformed quadrature points shown in Table 2.

Next, we present posterior predictive samples in Tab. 3 in order to examine
the downstream effects of Bayesian calibration on model predictions. That is,
we generate samples from the posterior predictive distribution

$\nu_{U^{\textrm{pred}}|\bm{Q}}(\cdot|\bm{q}^{*})=\int\nu_{U^{\textrm{pred}}|\bm{X}}(\cdot|\bm{x})\pi_{\bm{X}|\bm{q}^{*}}(\bm{x})\textrm{d}\bm{x}.$
(42)

The random variable
$U^{\textrm{pred}}=\mathcal{F}(\bm{X}|\bm{q}^{*},Z)\sim\nu_{U^{\textrm{pred}}|\bm{Q}}(\cdot|\bm{q}^{*})$
represents the Di-BCP patterns predicted by the forward operator
$\mathcal{F}(\cdot,Z)$ given samples of the calibrated parameters. These
posterior predictive samples help us to visually understand the gain in
prediction confidence from model calibration.

Generating these samples is accomplished by simulating from the forward
operator $\bm{x}\mapsto\mathcal{F}(\bm{x},z^{i})$ where
$z^{i}\sim\nu_{Z|M}(\cdot|m)$ and $\bm{x}$ are the quadrature points in Tab.
2. The Di-BCP pattern predictions of models calibrated according to the
summary statistics $\bm{Q}=(Q_{2},Q_{4})$ (_top row_ for each $u^{*}$) are
much more visually dissimilar than the predictions of models calibrated
according to the summary statistics $\bm{Q}=\bm{Q}_{1:4}$ (_bottom row_ for
each $u^{*}$), regardless of the phases of latent Di-BCP patterns.

Overall, we see that the energy-based summary statistics $\bm{Q}_{1:4}$ are
highly informative of the OK model parameters, despite having a much lower
dimension than the image data. We note that sampling from all three posterior
distributions in Tab. 2 and 3 is easily accomplished after finding the map
$\widehat{S}^{\mathcal{X}}$. Furthermore, evaluating the posterior densities
can also be easily accomplished using the map, as discussed in Section 5.

###  8.2 Expected information gain: Energy-based summary statistics

In this subsection, we present results on the estimation of EIGs for inferring
the model parameters from the energy-based summary statistics introduced in
Section 4.4.

In Fig. 14, we plot the estimation of EIGs for inferring each marginal
parameter random variable in $\bm{X}=(M,E,\Sigma)$ from the energy-based
summary statistics when observing _raw images_ in the reference field of view,
i.e., assuming no noise or blurring corruption in the characterization
process. The EIG calculations imply that the empirical spatial average summary
statistics $\widehat{M}$ strongly informs the spatial average parameter $M$,
as expected (_left_). Furthermore, other energy-based summary statistics also
jointly inform the spatial average parameter. This corroborates what we learn
from the phase diagram in Fig. 1: the spatial average parameter, or the volume
ratio parameter $f$, can be learned from the morphological phase of the Di-BCP
material. The interfacial energy $Q_{2}$ and/or the total variation energy
$Q_{4}$ strongly inform the interfacial parameter $E$ (_center_), and,
finally, the double well energy $Q_{1}$ and nonlocal energy $Q_{3}$ jointly
strongly inform the nonlocal parameter $\Sigma$ (_right_). The EIG
calculations also reveal that even though, marginally, the double well energy
$Q_{1}$ does not strongly inform $\Sigma$, $Q_{1}$ and $Q_{3}$ are jointly
very informative of $\Sigma$. In fact, the estimated EIG from these two
summary statistics, $\widehat{I}(\Sigma;(Q_{1},Q_{3}))$, is larger than the
sum of the estimated EIGs from each summary statistics marginally,
$\widehat{I}(\Sigma;Q_{1})+\widehat{I}(\Sigma;Q_{3})$282828From information
theory, we know that extracting multiple summary statistics jointly is always
at least as informative as extracting fewer summary statistics [108]. However,
the EIG from multiple summary statistics is not generally the sum of the EIGs
from each summary statistics. In fact, the former is sometimes much more that
the latter, as is the case here with $Q_{1}$ and $Q_{3}$ for informing
$\Sigma$.. Similarly, we observe that $Q_{1}$ and $Q_{3}$ jointly inform the
interfacial parameter $E$ ‘almost’ as much as $Q_{2}$ or $Q_{4}$ each informs
$E$, even though $Q_{1}$ and $Q_{3}$ marginally each provide very little
information for learning $E$.

Figure 14: The estimated EIGs for inferring each of the three parameters
$\bm{X}=(M,E,\Sigma)$ (_left_ to _right_) from (i) each of the energy-based
summary statistics $Q_{i}$, $i=1,\dots,4$, and $\widehat{M}$, and (ii) pairs
of summary statistics, $(Q_{1},Q_{3})$ and $(Q_{2},Q_{4})$. We assume no
additive noise or blurring corruption in the characterization process.

Next, to investigate the effects of additive noise and blurring corruptions on
the informativeness of the energy-based summary statistics, we plot in Fig. 15
the estimated EIGs for inferring each model parameter from its most
informative summary statistics according to Fig. 14, with different levels of
additive noise and blurring corruptions specified in Section 7.3. First, we
observe that while the empirical spatial average summary statistics
$\widehat{M}$ is highly informative of the spatial average parameter in the
absence of the noise and blurring corruptions, its informativeness
deteriorates by half with the presence of additive noise (_left_), despite the
noise being zero mean. Thus, we may have difficulty directly pinpointing the
spatial average parameter from the empirical spatial average statistics
$\widehat{M}$ of noisy characterization images. On the other hand, the
informativeness of $(Q_{1},Q_{3})$ for inferring $\Sigma$ is not significantly
degraded by the various degrees of noise and blurring corruptions (_right_),
while the informativeness of $(Q_{2},Q_{4})$ for inferring $E$ is more
significantly degraded by the blurring corruption than by the additive noise
corruption (_center_).

Figure 15: The estimated EIGs for inferring each of the three parameters
$\bm{X}=(M,E,\Sigma)$ from its most informative summary statistics according
to Fig. 14, with varying levels of additive noise and blurring corruptions. In
particular, we have $\widehat{I}(M;\widehat{M})$ (_left_),
$\widehat{I}(E;(Q_{2},Q_{4}))$ (_center_), and
$\widehat{I}(\Sigma;(Q_{1},Q_{3}))$ (_right_) .

The variables of these grids conceptually represent the specifications of the
characterization instrument one might be able to choose or control.
Ultimately, the EIG computation can help one determine the design of
characterization experiments, trading-off the cost of experiments and model
calibration accuracy. While some of these quantitative results can sometimes
be theoretically deduced without numerical simulations, often these
relationships are complex and intractable. Thus it is highly useful to be able
to _empirically_ extract these relationships with numerical simulations
instead. Using the framework for EIG computation introduced in Section 6, each
EIG value in the entries of the $3\times 4$ grids is estimated by a map
estimate $\widehat{S}^{\mathcal{X}}$, given the parameter and summary
statistics joint samples $(\bm{x}^{i},\bm{q}^{i})\sim\nu_{\bm{X},\bm{Q}}$.

Lastly, we note that the EIGs for learning model parameters are examined
marginally, although the same computations could have been conducted to
estimate the EIG for the three parameters jointly. However, since we were
mostly interested in the impact of various summary statistics on learning each
individual parameter, we leave EIG studies involving the joint parameter
$\bm{X}=(M,E,\Sigma)$ for future study.

###  8.3 Expected information gain: Fourier-based summary statistics

Next, we computed the EIG from the Fourier-based summary statistics for
learning each of the three parameters. In particular, we consider
$\bm{Q}_{a}=\tilde{\mathcal{P}}_{a}(\bm{Y})\in\mathbb{R}^{40}$, as defined in
(37), where each entry corresponds to the averaged magnitude spectrum value in
a unit interval of the radial frequency. Let $Q_{a,i}$ denote the $i$th
component of $\bm{Q}_{a}$. In Fig. 16 and 17, we show the estimated EIGs for
inferring marginally each of the three parameters from each component of
$\bm{Q}_{a}$, with various degrees of additive noise and blurring corruptions.

Figure 16: The estimated EIGs for inferring each of the three parameters
$\bm{X}=(M,E,\Sigma)$ (from _left_ to _right_) from $\mathcal{Q}_{a,i}$,
$i=1,\dots,40$, with increasing levels of the blurring corruption and a fixed
level of the additive noise corruption at $\text{SNR}=5.0$.

With the presence of the additive noise corruption, the informativeness of the
averaged magnitude spectrum decreases, while the blurring corruption primarily
reduces the information contained in the averaged magnitude spectrum at higher
radial frequencies. Both the blurring and noise corruptions tend to more
strongly impact the EIG of the spectrum at frequencies that are _most
informative_ for each parameter, with the exception of the EIG for $\Sigma$
subjects to blurring corruptions (Fig. 16, _right_). Moreover, the spectrum at
higher frequencies generally tend to inform $M$ and $E$ more while the
spectrum at lower frequencies ($\sim 10$) inform $\Sigma$ more, so that the
spectrum at both higher and lower frequencies play a non-negligible role in
informing the model parameters.

Figure 17: The estimated EIGs for inferring each of the three parameters
$\bm{X}=(M,E,\Sigma)$ (from _left_ to _right_) from $\mathcal{Q}_{a,i}$,
$i=1,\dots,40$, with decreasing levels of the additive noise corruption and no
blurring corruption.

For the remainder of this subsection, we address the difference in EIG
approximation quality when using the parametric transport framework laid out
in Section 6, compared to a traditional Gaussian approximation based EIG
approximation. In Fig. 18, the EIG computed with our non-Gaussian parametric
approach is compared to the EIG computed via a Gaussian approximation of the
posterior densities. As expected from expression (22), the EIG estimator with
a variational approximation to the posterior density is negatively biased and
this bias increases prediction for an approximate density that has a larger
discrepancy with respect to the true posterior (see Section 6.2). For some
components of the averaged magnitude spectrum, this results in an estimated
EIG based on the Gaussian approximation (_red_) that is near zero, while the
actual information gain in those components (_blue_) is quite larger as seen
from the non-Gaussian approximation. This kind of bias can hamper experimental
design decision-making that relies on EIG estimations.

Figure 18: The estimated EIGs for inference of $E$ (_left_) and $\Sigma$
(_right_) from each entry of the averaged magnitude spectrum $Q_{a,i}$,
$i=0,\dots,40$. The Gaussian approximations to the joint density lead to
under-predictions of informativeness.

We further demonstrate how poor the approximation can be by visualizing the
Gaussian approximation of the joint density that underrepresents the mutual
information between two random variables. Specifically, we consider the joint
density of $\Sigma$ and $Q_{a,17}$. In Fig. 19, we plot the joint samples from
this distribution on the left along with a Gaussian and non-Gaussian
approximation to the joint density.

Figure 19: Gaussian vs. non-Gaussian joint densities for $(\Sigma,Q_{a,17})$.
The Gaussian approximation incorrectly predicts near independence between the
random variables resulting in a poor estimate of
$\widehat{I}(\Sigma;Q_{a,17})\approx 0$.

While we considered the EIG from the individual components of the averaged
magnitude spectrum for inferring each parameter, interesting future work would
be to determine the subset of spectrum data that jointly inform the parameters
most. Techniques for doing so are found within the field of active learning or
optimal experimental design—these fields can benefit from the framework laid
out in the paper, since they often rely on EIG estimates in order to evaluate
optimization objectives.

###  8.4 Expected information gain: Levels of magnification

In this subsection, we consider the effects of varying levels of
magnification, specified by the restriction operator $\mathcal{R}$ defined in
Section 7.3, on the informativeness of energy-based summary statistics,
$\widehat{M}$, $(Q_{1},Q_{3})$ or $(Q_{2},Q_{4})$, about $\bm{X}=M$, $E$, or
$\Sigma$, respectively. In particular, we plot the estimated EIG as a function
of magnification levels for a fixed level of blurring corruption and varying
levels of noise corruptions in Fig. 20. Similarly, we plot the same quantities
in Fig. 21 for a fixed level of noise corruption and varying levels of
blurring corruptions.

Figure 20: The estimated EIG for inferring each of the three parameters
$\bm{X}=(M,E,\Sigma)$ (_left_ to _right_) from the observables
$(\widehat{M},(Q_{1},Q_{3}),(Q_{2},Q_{4}))$ with different magnification
levels, i.e., the specimen subdomain size, varying noise corruption and a
fixed blurring corruption of strength $\sigma_{blur}=5\times 10^{-3}$.

Figure 21: The estimated EIG in Fig. 20 with a fixed noise corruption of SNR =
5 and varying blurring corruption.

For learning the spatial average parameter $M$ (_left_ in Fig. 20 and 21), the
estimated EIG from the empirical mean summary statistics $\widehat{M}$
decreases as the magnification level increases. Moreover, the estimated EIG is
insensitive to changes in noise and blurring corruptions, caused by the
spatial average–preserving property of the Gaussian blur and the zero-mean
property of the noise. We expect this behaviour to change for more complex
characterization models that does not satisfy these properties. These
numerical results suggest that, under the assumptions of the data model and
parameter prior distribution, the action of zooming in will not lead to a
decrease in uncertainty in $M$ on average. These result may be interpreted by
the loss of phase information related to the material parameter $f$ (see Fig.
1) when we only observe a small portion of the top surface of a Di-BCP thin
film. For example, one might only observe several spots in a pattern that
contains a mixture of spots and strips.

For learning the interfacial parameter $E$ (_middle_ in Fig. 20 and 21), the
estimated EIG from the energy-based summary statistics $(Q_{2},Q_{4})$
decreases as the level of magnification increases in the presence of noise and
blurring corruptions. This also suggests that the action of zooming in will
not lead to a decrease in uncertainty in the $E$ parameter on average. We see
that in an idealistic noise-free setting, however, the estimated EIG increases
slightly when the level of magnification increases. We hypothesize that the
increase in image pixel density via magnification leads to better resolved
phase interfaces only in the presence of the blurring corruption, and thus
leads to a small increase in the estimated EIG for learning $E$.

For learning the nonlocal parameter $\Sigma$ (_right_ in Fig. 20 and 21), we
observe an interesting response of the estimated EIG from the energy-based
summary statistics $(Q_{1},Q_{3})$ with respect to changes in the levels of
magnification and image corruptions. In the high SNR setting, the estimated
EIG decreases as the magnification level increases. On the contrary, we see an
increase in the estimated EIG in the low SNR setting. Similarly, in the low
blurring setting, the estimated EIGs decreases as the magnification level
increases. We also observe an increase in the estimated EIG in the high
blurring setting. These results suggest that whether the action of zooming in
is beneficial for inferring $\Sigma$ depends on the level of image
corruptions.

Overall, while some of the figures behave as one might expect, the unexpected
behavior of the EIG for certain summary statistics and model parameters would
otherwise be difficult to determine quantitatively without these numerical
experiments. These empirical findings could aid and inspire further
theoretical investigations about the effects of various image corruptions and
experimental designs on model calibration and prediction.

###  8.5 Expected information gain: Known volume ratio of monomer blocks

In many experimental settings, the volume ratio of the two monomer blocks in
Di-BCPs can be determined with high certainty. In particular, we have shown in
Section 8.2 that the spatial average summary statistics is exceptional in
learning the OK parameter $m$. Therefore, in this subsection, we consider the
inference problem for two length scale related parameters in the OK model
$(E,\Sigma)$ given image data of Di-BCP patterns with a fixed, or known,
spatial average parameter.

Figures 22, 23, and 24 plot the estimated EIGs for learning the parameters
$(E,\Sigma)$ jointly from $(Q_{1},Q_{3})$, $(Q_{2},Q_{4})$, and
$(Q_{1},\dots,Q_{4})$, respectively, conditioned on three different spatial
averages $m=(0,0.3,1/\sqrt{3})$. In general, we observe a decrease in the
estimated EIGs with with increasing level of noise and blurring corruptions.
We also note that the EIGs from $(Q_{2},Q_{4})$ are more sensitive to noise
than those from $(Q_{1},Q_{3})$ or conditioning on all observables,
particularly for the spatial averages $m=0$ and $m=0.3$.

Figure 22: The estimated EIGs for inferring the joint posterior of the two
length-scale dependent parameters $(E,\Sigma)$ from energy-based summary
statistics $(Q_{1},Q_{3})$ for $m=0$ (left), $m=0.3$ (center), and
$m=1/\sqrt{3}$ (right).

Figure 23: The estimated EIGs for inferring the joint posterior of the two
length-scale dependent parameters $(E,\Sigma)$ from energy-based summary
statistics $(Q_{2},Q_{4})$ for $m=0$ (left), $m=0.3$ (center), and
$m=1/\sqrt{3}$ (right).

Figure 24: The estimated EIGs for inferring the joint posterior of the two
length-scale dependent parameters $(E,\Sigma)$ from energy-based summary
statistics $\bm{Q}_{1:4}$ for $m=0$ (left), $m=0.3$ (center), and
$m=1/\sqrt{3}$ (right).

##  9 Conclusion

In this work, we explored the Bayesian model calibration problem for block
copolymer self-assembly using imaging data. To set up the problem, we
introduced a generic probabilistic data model that includes both the
measurement uncertainty in the characterization process and the aleatoric
uncertainty in the self-assembly procedure. To calibrate the model parameters
we characterize their posterior distribution, where the likelihood function
accounts for the aleatoric uncertainty by marginalizing over high-dimensional
auxiliary variables. Moreover, we quantify the expected information gain to
measure the informativeness of data for calibrating the parameters.

We argue and demonstrate that, as a result of the generally intractable
integrated likelihood evaluations and the high dimensional data, many
conventional Bayesian inference methodologies are unsuitable for performing
the proposed calibration task. To overcome these challenges, we propose to (i)
employ a likelihood-free inference method that only requires joint samples of
data and model parameters, and (ii) construct informative low-dimensional
summary statistics for the data. In particular, we use an inference
methodology based on triangular transport maps, which allows us to (i)
consistently characterize the joint distribution of model parameters and
summary statistics with limited samples, (ii) perform amortized inference for
any realization of the data with a single collection of joint samples, and
(iii) consistently compute expected information gain with no significant
additional cost, which enables us to study the effects of different summary
statistics and corruptions in the data on the calibration results.

Lastly, we demonstrated the power of the proposed inference approach and the
computation of expected information gain on a numerical case study of
calibrating the Ohta–Kawasaki model parameters using top-down microscopy
characterization images. We introduced several energy- and Fourier-based
summary statistics to reduce the dimension of the image data. A hierarchical
uniform prior distribution for the model parameters is constructed by
analyzing the forward operator. First, we observe the concentration of the
posterior distribution in the parameter space and visualize posterior
predictive samples, which show that the proposed energy-based summary
statistics are effective for calibrating the parameters with low uncertainty.
We then estimate the expected information gains from different summary
statistics for learning the model parameters, under various experimental
scenarios with different levels of (i) noise and blurring corruptions, and
(ii) magnifications of the microscope. These numerical results allow us to
investigate the effects of experimental settings on the calibration results
for this representative model, and enable further optimal experimental design
studies of interest.

## Acknowledgement

The work of the authors is supported by the AEOLUS center, funded by the U.S.
Department of Energy, Office of Science, Office of Advanced Scientific
Computing Research, Mathematical Multifaceted Integrated Capability Centers
(MMICCS), under Award DE-SC0019303.

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##  Appendix A The auxiliary variable: a transformed Gaussian random field

In this appendix section, we define the auxiliary random field $Z$ used in the
forward operator $\mathcal{F}(\cdot,Z)$ (29) to define a random initial state
$z$ in state space when minimizing the OK energy functional. Rather than
employing the typically used i.i.d. uniform random fields for $Z$, which are
infinitely non-smooth and not amenable to discretization of the PDE equations,
we use a transformation $Z=\mathcal{Z}(\bm{X},A)$ of a Gaussian random field
$A$, that is also known as a _Gaussian related random field_ [129, 130].

We follow an approach similar to that in in [114] by taking $A\sim\nu_{A}$ to
be a Gaussian random field with zero-mean and covariance operator
$\mathcal{C}_{A}:L^{2}(\mathbb{R}^{2})\to
L^{2}(\mathbb{R}^{2}),\mathcal{C}_{A}=(\kappa\mathrm{Id}-\triangle)^{-2}$,
which possesses the covariance function

$c_{\kappa}(|\bm{s}_{1}-\bm{s}_{2}|)=\frac{\sigma^{2}}{\Gamma(1)}\kappa|\bm{s}_{1}-\bm{s}_{2}|K_{1}(\kappa|\bm{s}_{1}-\bm{s}_{2}|),$
(43)

where $K_{1}$ is the order $1$ modified Bessel function of the second kind and
$\sigma^{2}=\frac{\Gamma(1)}{\Gamma(2)4\pi\kappa}$ [131]. In this work,
$\kappa=200$ is chosen such that the spatial correlation is below the _length
scales_ of the features of all equilibrium states in our experimental scenario
modeled by the OK model (25). $A$ is isotropic and stationary since the
function $\bm{c}$ depends on Euclidean distance $|\bm{s}_{1}-\bm{s}_{2}|$
alone. For this choice of covariance, we have almost surely continuous
realizations [57, Sec. 6.5]. Realizations $A^{i}\sim\nu_{A}$ can be
numerically approximated given a sufficiently fine spatial discretization.
After discretization, these fields become Gauss–Markov random fields [132],
which are efficiently implemented using the hIPPylib python package [133],
where finite element approximations of these PDEs are solved in the FEniCS
library [134, 135].. A description of finite element discretizations for
similar random fields can be found in [47].

Figure 25: The Whittle–Matérn covariance function describing the correlation
structure of the random fields. For $\kappa=200$, the correlation decays to
near zero for distances $|\bm{s}_{1}-\bm{s}_{2}|>0.05$. Recall that the
computational domain is $\mathcal{D}_{c}=[0,40]^{2}$.

Finally, we define the transformation $Z=\mathcal{Z}(\bm{x},A)$ such that the
random field $Z$ is a pointwise uniform a.s. continuous field that satisfies
certain physical constraints. Specifically, given the point–wise variance
$\sigma^{2}$, the pointwise transformation

$\mathcal{Z}(A;\bm{x})=m+(1-m)\textrm{erf}\left(\frac{A}{\sigma\sqrt{2}}\right)$
(44)

ensures that the pointwise values of the auxiliary random field realizations
$z^{i}=\mathcal{Z}(\bm{X},A^{i})$ for $A^{i}\sim\nu_{A}$ are within the
physical order parameter bounds $(-1,1)$ and have the pointwise mean $m$.
Furthermore, its pointwise marginal densities are
$\pi_{Z(\bm{s})|M}(\cdot|m)=\mathcal{U}(-1+2m,1)\;\forall\bm{s}\in\mathbb{R}^{2}.$
Thus, they are physically reasonable random fields to employ as starting
points $Z$ for the variational procedure that defines $\mathcal{F}(\bm{x},Z)$.

To exactly match the specified spatial average $m$ for the order parameter
(27), an additional transformation is employed, in which we choose the closest
point satisfying the correct spatial average within the chosen finite element
function space. Since this projection does not account for the pointwise
constraint $z^{i}(\bm{s})\in(-1,1)$, transformed random field samples may
actually violate the constraint in regions of the domain—therefore, our
samples only _approximately satisfy_ this physical constraint. However, this
has a negligible effect on forward solutions, since the double well potential
$\mathcal{D}^{\textrm{OK}}$ heavily penalizes deviation from non-physical
regions in state space, so that any variational procedure will correct for
this within a few iterations.

Figure 26: Realizations of $\nu_{Z|M}(\cdot|m)$ for $m=0,0.3.0.5$, which have
pointwise marginals $\mathcal{U}(-1+2m,1)$.

##  Appendix B The discretization of the state space, the restriction
operators, and image interpolation

The state space $V^{u}$ for th OK model is
$\mathring{H}_{m}^{1}\big{(}\mathcal{D}\big{)}$, as in (30), with
$\mathcal{D}$ corresponds to the reference field of view, as presented in
Section 7.3, and $m$ corresponds to the spatial average model parameter.
However, when evaluating the forward operator, we solve the equivalent mixed
problem as in (31), thus the state space is transformed into the product space
$(H^{1}(\mathcal{D}))^{2}$. Furthermore, we transform the physical domain
$\mathcal{D}$ into a computational domain $\mathcal{D}_{c}$, with a side
length of $40$, thus equivalently rescale the model parameters according to
(28). The purpose of the domain scaling is to improve the numerical
conditioning of the forward operator by balancing the relative magnitude of
the three terms in the OK free energy (25).

The computational domain $\mathcal{D}_{c}$ is discretized with uniform
triangular mesh, denoted by $\mathcal{P}^{h}$, such that
$\mathcal{D}_{c}=\bigcup_{K\in\mathcal{P}^{h}}\bar{K}$. Each element is the
image of a scaling map $F_{K}$ of the reference triangular element $\hat{K}$.
We consider a linear finite element basis: let $P^{1}(\hat{K})$ denote the
space of complete linear polynomials, then the finite-dimensional spaces
$S^{h,1}\coloneqq S^{h,1}(\mathcal{P}^{h})$ by:

$S^{h,1}\coloneqq\left\\{v^{h}\in
H^{1}(\mathcal{D}_{c}):v^{h}|_{K}=\hat{v}\circ F_{K}^{-1},\hat{v}\in
P^{1}(\hat{K})\right\\}.$ (45)

In our case study, we consider $\mathcal{P}^{h}$ with same number of cells as
the number of sensors, which makes the evaluation of the sensor signal
discretization operator $\mathcal{D}$ (33) relatively straight forward.

The restriction operators $\mathcal{R}_{j}$, as introduced in Section 7.3, are
applied by projecting the space $S^{h,1}$ to spaces defined with denser
uniform triangular meshes $\mathcal{P}^{h_{j}}$. The denser meshes have the
same cells arrangement as $\mathcal{P}^{h}$ in the subdomain domains
$R_{j}(\mathcal{D}_{c})$. We then apply the imaging operator $\mathcal{I}$ and
the statistical sensor operator $\bm{\mathcal{S}}$ to the order parameter
restricted to $R_{j}(\mathcal{D}_{c})$.

The interpolation of images into $H^{1}(\mathcal{D}_{u}$) is implemented
through the similar finite element basis as define in (45). The DoFs of the
finite element basis match the the dimension of the image and the spatial
arrangement of the pixels is preserved.

The finite element discretization, the restriction operators, and image
interpolation are implemented through the FEniCS library.

##  Appendix C Parameterization and computation of transport maps

In this section, we discuss our parametrization of triangular and monotone
transport maps and an adaptive algorithm for approximating the map given
samples from the target density $\pi$. As shown in Section 5.2, the
optimization problem for the map $S$ decouples into separate optimization
problems for each map component. Thus, this section focuses on the individual
optimization problem for each component.

To learn a monotone function for approximating $S^{k}$, we consider maps that
are parameterized to be monotone by construction. Following [96], we represent
each map component $S^{k}:\mathbb{R}^{k}\to\mathbb{R}$ as

$S_{k}(\bm{w}_{1:k})=\mathcal{M}_{k}(f)(\bm{w}_{1:k})\coloneqq
f(\bm{w}_{1:k-1},0)+\int_{0}^{w_{k}}g(\partial_{k}f(\bm{w}_{1:k-1},t))\textrm{d}t,$
(46)

where $g\colon\mathbb{R}\rightarrow\mathbb{R}_{>0}$ is a positive function so
that $\partial_{k}S_{k}(\bm{w}_{1:k})\geq 0$ for all
$\bm{w}_{1:k}\in\mathbb{R}^{k}$. Different choices for $g$ include the squared
function $g(\bm{w})=\bm{w}^{2}$ and the soft-plus function
$g(\bm{w})=\log(1+\exp(\bm{w}))$. The operator
$\mathcal{M}_{k}:f\mapsto\mathcal{M}_{k}(f)$ in (46) maps the unconstrained
function $f\colon\mathbb{R}^{k}\rightarrow\mathbb{R}$ into a monotone
function. Thus, using this representation for $S_{k}$, we can focus on
approximating $f$. Similarly to the numerical experiments in [96], we consider
linear expansions for $f$ given by
$f(\bm{w}_{1:k})=\sum_{j}\alpha_{j}\psi_{j}(\bm{w}_{1:k})$, where $\psi_{j}$
are prescribed features and $\alpha_{j}\in\mathbb{R}$ are unknown
coefficients. In this work, we consider tensor products of normalized Hermite
functions with respect to each variable, i.e.,
$\psi_{j}(\bm{w}_{1:k})=\prod_{i=1}^{k}\psi_{j,i}(\bm{w}_{i})$, but we note
that $f$ can also be parameterized using other features or non-linear
expansions (e.g., neural networks).

Given a set of summary statistics, the coefficients $\bm{\alpha}=(\alpha_{j})$
can be found using a gradient-based optimization technique, such as the BFGS
or Newton–CG algorithms [136]. To identify the set of features in $f$, we use
the Adaptive Transport Map algorithm introduced in [96]. This procedure begins
with an empty set of features $\Lambda_{0}$, i.e., the zero function for $f$,
and incrementally adds feature to at a time that most reduces the value of the
objective function. To determine the stopping point, i.e., the maximum number
of features, we use a validation set. In our numerical experiments, we
partition the data by using 80% of the samples to learn the map and the
remaining 20% to evaluate the negative log-likelihood of the approximate
conditional density. If the minimum negative log-likelihood doesn’t improve
for at least 20 iterations, we stop the adaptive procedure and return the
approximation for the basis size with the lowest validation error. In
practice, we also limit the number of features to $p=100$ features for
2-dimensional maps and $p=250$ features for higher-dimensional maps. During
the iteration, we evaluated the integrals in (46) numerically using 16
Gaussian quadrature points for each sample. We observed that a larger number
of points did not affect the features selected by the adaptive procedure.