# Towards Comparative Physical Interpretation of Spatial Variability Aware
Neural Networks: A Summary of Results

Jayant Gupta<EMAIL_ADDRESS>University of Minnesota, Twin-
CitiesMinnesotaUSA ,  Carl Molnar<EMAIL_ADDRESS>University of Minnesota,
Twin-CitiesMinnesotaUSA ,  Gaoxiang Luo<EMAIL_ADDRESS>University of
Minnesota, Twin-CitiesMinnesotaUSA ,  Joe Knight<EMAIL_ADDRESS>University
of Minnesota, Twin-CitiesMinnesotaUSA and  Shashi Shekhar<EMAIL_ADDRESS>University of Minnesota, Twin-CitiesMinnesotaUSA

###### Abstract.

Given Spatial Variability Aware Neural Networks (SVANNs), the goal is to
investigate mathematical (or computational) models for comparative physical
interpretation towards their transparency (e.g., simulatibility,
decomposability and algorithmic transparency). This problem is important due
to important use-cases such as reusability, debugging, and explainability to a
jury in a court of law. Challenges include a large number of model parameters,
vacuous bounds on generalization performance of neural networks, risk of
overfitting, sensitivity to noise, etc., which all detract from the ability to
interpret the models. Related work on either model-specific or model-agnostic
post-hoc interpretation is limited due to a lack of consideration of physical
constraints (e.g., mass balance) and properties (e.g., second law of
geography). This work investigates physical interpretation of SVANNs using
novel comparative approaches based on geographically heterogeneous features.
The proposed approach on feature-based physical interpretation is evaluated
using a case-study on wetland mapping. The proposed physical interpretation
improves the transparency of SVANN models and the analytical results highlight
the trade-off between model transparency and model performance (e.g.,
F1-score). We also describe an interpretation based on geographically
heterogeneous processes modeled as partial differential equations (PDEs).

Spatial Variability, Neural Networks, Spatially Heterogeneous Features,
Explainable AI

††conference: Sigspatial ’21: ACM SIGSPATIAL; November 02–05, 2021;
Sigspatial, Beijing††booktitle: Sigspatial ’18: International Conference on
Advances in Geographic Information Systems, November 02–05, 2021, Beijing,
China††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Information systems
Information systems applications††ccs: Information systems Spatial-temporal
systems††ccs: Computing methodologies Neural networks††ccs: Computing
methodologies Machine learning algorithms

##  1\. Introduction

Interpretability of models changes with their complexity as simpler models
(e.g., rule-based) are easier to interpret than complex models (e.g., multi-
layer neural networks). For a model to be interpretable, it needs to be
simulatable i.e., it can be thought through by a human, decomposable i.e., the
input, parameters and output could be explained individually, and
algorithmically transparent i.e., the process to compute the output should be
clear (Arrieta et al., 2020). However, this may not be enough for
computational models built to represent physical processes which must adhere
to different physical constraints (e.g., Newton’s laws, geographic
heterogeneity, etc.). Figure 1 illustrates the relation between physical
interpretability and interpretability where comparison (i.e., comparative
physical interpretation) or transformation (i.e., transformative physical
interpretation) of a given model to other models or representations which are
sensitive to known physical constraints and can serve as a bridge between the
two. In this work, we strive towards comparison-based physical
interpretability of spatial variability aware neural networks (SVANNs) (Gupta
et al., 2020) which are sensitive to geographic heterogeneity.

Figure 1. Towards physical interpretability.

The physical interpretability of a spatial variability aware model is
important to ensure its re-usability, to allow debugging of model errors, and
to provide output determinism (i.e., enhance model explainability to a jury in
a court of law). The need for interpretability can vary depending on the needs
of the end-user (Rossi, 2019). For example, artificial intelligence (AI)
system builders need model interpretability to enhance model understanding and
improve system performance, whereas lawmakers and medical practitioners need
interpretability to enhance trust and confidence in the outputs. Regulatory
bodies want model interpretability to ensure fairness for their constituents,
while the affected users want to understand the factors responsible for an
outcome.

Table 1. User and applications of physical interpretability. User |  Applications
---|---
AI system builders |  Enhancement of model understanding, Improvement in system performance.
Lawmakers, medical practitioners |  Enhancement of trust and confidence.
Regulatory bodies |  Ensure trust and confidence in the system.
Affected users |  Understand factors responsible for an outcome.

The physical interpretation of a multi-layer neural network (MLNN) is hard due
to the following factors.

There are many reasons it is hard to interpret models built using multi-layer
neural networks (MLNNs) or convolutional neural networks (CNNs) (LeCun et al.,
1995) in physical terms. First, MLNNs have a large number of parameters which
change with every learning step at different rates. Second, the networks can
be highly unstable and small perturbations in the input (e.g., noise) and
hyper-parameters can produce substantially different results (Goodfellow et
al., 2014). Third, generalization error bounds are often vacuous with little
explanatory power because they often show trends contradictory to the true
error (Valle-Pérez and Louis, 2020). Finally, the networks are at risk of
over-fitting and may not generalize well on unseen data, leading to a strong
bias in the system.

Related work in this area includes post-hoc interpretation of MLNNs using
model-agnostic or model-specific techniques. Model-agnostic techniques can be
used for any model and rely on techniques that simplify the model to a locally
linear model (e.g., Local Interpretable Model-Agnostic Explanations (LIME)
(Ribeiro et al., 2016)); feature relevance estimation, which calculates
desirable properties to predict a given class (e.g., Shapely Additive
exPlanations (SHAP) (Lundberg and Lee, 2017)); and techniques to visualize
input and output results. On the other hand, model-specific techniques for
MLNNs split the inputs for each classification decision to highlight the
contribution of its input elements (Montavon et al., 2017). Interpretability
techniques for MLNNs can differ from those for CNNs, which generally: either
explain the effect of an input on the network output, or interpret how each
layer sees the input (i.e., what is learnt by each layer). For the first
category, Matthew et al. (Zeiler et al., 2011) output feature maps for each
layer showing strong and soft activation when an input image feeds-forward
through a CNN. For the second category, a Layer Relevance Propagation
technique (Bach et al., 2015) was proposed, which generates a heatmap
representing the contribution of a single pixel to the final prediction. One
limitation of all these methods, however, whether model-agnostic or model-
specific post-hoc methods, is that they do not consider physical constraints
and properties of geographic spaces.

This paper investigates a novel approach to model interpretation based on
geographically heterogeneous features. The features are constructed using
well-known remote sensing indices. Remote sensing indices are derived from
spectral imagery bands and owing to their simplicity many (indices) have a
high level of interpretability for a given geographic region of study. Later,
we also discuss an interpretation based on geographically heterogeneous
processes where the processes are represented using partial differential
equations (PDEs). PDEs are fundamental scientific elements which are used to
understand many real-world phenomena (e.g., sound, heat, etc.) which means
they are also highly interpretable. The two (i.e., indices and PDEs) help in
developing an independent physical interpretations of SVANN based models.

We find that our spatial heterogeneity feature based interpretation of SVANNs
improves their decomposability which are further substantiated by a case study
on wetland mapping using remote-sensing indices.

Theoretically, our process-based interpretation of SVANNs should improves
their simulatibility, decomposability, and algorithmic transparency, however,
this is yet to be validated.

Following is the summary of our key contributions:

Contributions:

  * •

We provide two physical interpretations of SVANNs as follows:

    * –

Geographically heterogeneous feature-based using remote sensing indices.

    * –

Geographically heterogeneous process-based using partial differential
equations (PDEs).

  * •

We provide a case study on wetland-mapping to validate the physical
interpretation based on geographically heterogeneous features.

    * –

Investigate image upsampling for multi-scale data fusion preprocessing to
improve model results.

Scope: In this paper we limit our discussion of geographically heterogeneous
process-based interpretation to PINNs (Raissi et al., 2019), which is one of
the many techniques to solve partial differential equations (PDEs) using
multi-layer neural networks (MLNNs). Further, experimental evaluation of
process-based interpretation is not explored in this paper and is planned in
the near future. The experimental evaluation of Geographically heterogeneous
feature-based interpretation is limited to two remote sensing indices: the
Normalized Difference Vegetation Index (NDVI) (Tucker, 1979) and the
Normalized Difference Water Index (NDWI) (McFEETERS, 1996), and two case-study
regions, which characterize their uses. Further, our approximation of rule-
based methods is kept simple (i.e., based on only one index) specifically to
serve as a highly interpretable baseline method.
Organization: The paper is organized in the following manner: Section 2 gives
a brief background on spatial heterogeneity and SVANNs. Section 3 describes
our geographically heterogeneous feature-based approach to interpreting
SVANNs. Section 4 describes the wetland-mapping based evaluation of the
feature-based interpretation and performance based evaluation. Section 5
discusses the experimental results and describe process-based model
interpretability using PDEs. Finally, we conclude and consider future work in
Section 6.

##  2\. Related Work and Background

###  2.1. Spatial Heterogeneity

Geographic properties differ across different areas giving rise to varied
geophysical and cultural phenomena. This spatial variability results in
different physical models across geographic areas. Knowledge of spatial
variability is necessary to understand the spatial patterns of events and
objects over an area that vary spatially (Turner and Chapin, 2005). A
computational model can be affected by two types of spatial variability:
variability in the object of interest itself, which may differ in shape, size,
or both; and variability in the background of the object of interest. For
example, a computational model that is trained to find residential housing in
the the US may have difficulty finding houses in other places where housing
construction is adapted to different local climates or other conditions (e.g.,
cave houses in Petra, igloos in polar regions, etc.). Here the neighbor
surroundings differ as well. Figure 2 shows the spatial variability in houses
and their background across the globe.

Figure 2. Spatial variability in houses and their surroundings.

Spatial variability has been observed in many geo-phenomena including climatic
zones, USDA plant hardiness zones (of Agriculture, 2012), and terrestrial
habitat types (e.g., forest, grasslands, wetlands, and deserts). The
differences in climatic zones affect the plant and animal life of the region.
Spatial variability is considered as the second law of geography (Leitner et
al., 2018) and has been adopted in regression models (e.g., Geographically
Weighted Regression (Brunsdon et al., 1998)) to quantify spatial variability,
the relationships among variables across a study area.

Spatially heterogeneous features: Remote sensing indices are becoming
increasingly used for identifying geophysical features such as plants, soil,
and water (Gaitán et al., 2013). Typically the index would be used as part of
subsequent analysis steps; they are indicators of information, not results by
themselves. These indices also have limitations in the extent of the area
which can be evaluated using them, due to spatial heterogeneity between
regions (Gaitán et al., 2013). For example, some coastal regions (e.g., Miami-
Dade County, Florida) consist of freshwater and intertidal regions in close
proximity, which causes high variation in vegetation density and moisture
levels. In this case, it can be beneficial to use a remote sensing index that
measures differences in vegetation density, such as NDVI, or an index which
specializes in measuring differences in vegetation moisture, such as NDMI. By
selecting either of these remote sensing indices in this region, we can see an
increase in the ability to delineate wetland features (Figure 3a-c).

However, for regions which consist of only dense freshwater forested
vegetation, we may wish to select different remote sensing indices for
identifying features. While NDVI may still contribute some useful information
in such a region, the need for NDMI is diminished, due to a similar moisture
level in the region’s vegetation (Figure 3d-f).

Figure 3. Using the same remote sensing indices across spatially heterogeneous
regions.

Current multi-layer neural network and convolution neural network based models
follow a One-Size-Fits-All (OSFA) where models are trained without
consideration to spatial variability. To address the issue, we proposed
spatial variability-aware neural networks (SVANN) (Gupta et al., 2020) whose
network weights and architecture can differ across locations to account for
the heterogeneity in a geographic space that follows Newton’s laws. The
approach was evaluated using CNN based deep neural networks.

###  2.2. Spatial Variability Aware Neural Networks

There are two types of SVANN approaches as described below.
SVANN-I (Gupta et al., 2020) is a spatially explicit neural network model
where each parameter (e.g., weight) is a function of the model’s location
$loc_{M}$. The architecture $f$ is composed of a sequence of $K$ weight
functions or layers $(w^{1}(loc_{M}),...,w^{K}(loc_{M}))$ which map a
geographic location-based training sample $x(loc_{S})$ to a geographic
location-dependent output $y(loc_{M})$ as follows:

(1) $\displaystyle y(loc_{M})$
$\displaystyle=f(x(loc_{S});w^{1}(loc_{M}),...,w^{K}(loc_{M})),$

where $w^{i}(loc_{M})$ is the weight vector for the $i^{th}$ layer. In this
approach we assume that the architecture is location invariant i.e., $K$ is
constant for all the models.

Figure 5 shows an example of SVANN-I (termed as SVANN) approach where the
geographical space has 4 zones and deep learning models are trained for each
zone separately. For prediction, each zonal model predicts the test samples in
its zone.
SVANN-E (Gupta et al., 2021) is a spatially explicit neural network model
whose architecture is a function of model location $loc_{M}$. The architecture
$f_{loc_{M}}$ is a sequence of $K$ weight functions or layers mapping a
geographical location based training sample $x(loc_{S})$ to a geographical
location dependent output $y(loc_{M})$ as follows.

(2) $\displaystyle y(loc_{M})$
$\displaystyle=f_{loc_{M}}(x(loc_{S});w^{1}(loc_{M}),...,w^{K}(loc_{M})),$

where $w^{i}(loc_{M})$ is the weight vector for the $i^{th}$ layer. Further,
as the architecture varies across locations, it implies that the weights are
also a function of location.

SVANN-I is a special case of SVANN-E; therefore, the classification accuracy
of SVANN-I can never be greater than the classification accuracy of SVANN-E
(Gupta et al., 2021).

In the following text, we will use SVANN to refer to SVANN-I models. In case
otherwise, we will use SVANN-E.

##  3\. Proposed Approach

Our approach is to interpret SVANNs using geographically heterogeneous
features. In this section, we define what we mean by simulatibility,
decomposability, and algorithmic transparency (Section 3.1). Next, we describe
an interpretation of SVANNs using geographically heterogeneous features and
compare the interpretability of different models.

###  3.1. Basic Definitions

Simulatability denotes the ability of a model to be simulated, or thought
about strictly, by a human. Hence, model complexity (e.g., number of
parameters, architecture) is relevant, where the model needs to be self-
contained enough for a human to think and reason about it as a whole. For
example, NDVI can be easily simulated given a set of Red and NIR bands
(Equation 3), in contrast, a decision tree with a depth of $100$ is hard to
simulate.
Decomposability represents the ability to explain each part of a model (input,
parameter and calculation). Decomposability requires every input to be readily
interpretable and every part of the model to be understandable by a human
without the need for additional tools. For example, a linear regression model
is decomposable as each parameter is associated with a meaningful variable,
whereas the parameters in deep learning models are hard to interpret as the
models rely on their machine based representation learning.
Algorithmic transparency deals with the ability of a user to understand the
process followed by the model to produce any given output from its input data.
For example, a linear model is considered transparent because its error
surface can be readily understood. In contrast, deep neural network
architectures are considered opaque as the loss landscape cannot be fully
observed, and instead has to be approximated through heuristic optimization
(e.g., stochastic gradient descent). Algorithmically transparent models are
constrained by means of mathematical analysis and methods (Arrieta et al.,
2020).

###  3.2. Feature-Based Interpretation

A transformation-based approach converts the given model (say model 1) to a
physically interpretable model. For physically interpretable models, the
mapping is relatively simpler. For example, if we consider a partial
differential equation as model 1 and a set of rules as model 2. Then, the
domain space of model 1 can be mapped to model 2 depending on the physical
constraints. On the other hand, model transformation can be difficult for
complex models or models which do not adhere to physical constraints. For such
models we describe the comparison-based approach which is also the focus of
this work.

A comparison-based approach compares the output of the given model to a
physically interpretable model. For example, consider a model (say Model 1)
which has 4 parameters (or weights) out of which 3 have known physical
properties, however, 1 does not. Further, we have another Model (say Model 2)
which can perform the same task and is physically interpretable. Then, the
model may be useful to identify the parameter with the unknown properties.
Figure 4 below shows the contrast between the two approaches.

Figure 4. Physical interpretability by Transformation or Comparison. We assume
that Model $2$ is a physically interpretable model.

A more general form of this example would be a model with $n$ output
parameters where $d$ features have known physical properties whereas $n-d$
parameters have unknown physical properties and the physical interpretability
of these features is established through a more physically interpretable
model.

To show how spatial heterogeneity affects model interpretability, we use the
example of two spatially heterogeneous locations, $R_{1}$ and $R_{2}$, where
$R_{1}$ is a region with a high density of forests and $R_{2}$ is a region
with a significant water and wetland coverage. We also have two types of
spatially heterogeneous features (e.g., remote sensing indices), one of which
is useful to delineate between different vegetation types (say $RSI_{1}$) and
one that delineates water-based land-cover features ( say $RSI_{2}$).

A SVANN-based approach will consider training two distinct models to classify
land cover for each region. The SVANN outputs are locally dependent weights
and lack any physical interpretation. Now, consider two rule-sets built using
each of the two heterogeneous features ($RSI_{1}$ and $RSI_{2}$) to classify
the land cover across $R_{1}$ and $R_{2}$.

A comparison of output weights from SVANN with the rule-sets would in theory
highlight the set of weights responsible for the two spatially heterogeneous
regions (i.e., $R_{1}$ and $R_{2}$). Furthermore, the prediction of spatially
heterogeneous weights should behave similar to the spatially heterogeneous
features for any input. For example, the set of SVANN weights which behave
similar to rule-sets for $RSI_{1}$ are suitable for forest-based regions and
similarly the other set for water and wetland coverage. In this work, we use
accuracy and F1 measures as a proxy to compare the model behavior.

The example above helps to separate the weights, which are otherwise
indistinguishable based on their sensitivity to heterogeneity. This clearly
shows that comparison-based approach helps to improve the decomposability of
the SVANN weights.

The above interpretation is highly useful when the nature of channels to SVANN
based models is not known. In such instances, one can attempt to compare
various SVANN models with simple rule based approach to identify the channel
used for a given SVANN model.

Comparative physical interpretation depends upon the complexity of the
physically interpretable models. For example, models such as partial
differential equations (PDEs) have a higher representative power than rule-
based models. Furthrmore, PDEs are known to be far more physically
interpretable due to their extensive usage to model physical processes (e.g.,
heat flow). We briefly describe an approach of modeling SVANNs in Section 5.5.

##  4\. Evaluation Procedure

This section describes how we evaluated an interpretation of SVANNs based on a
wetland mapping case-study for selecting remote sensing indices based on
geographic location. We present the problem statement, some background on the
remote sensing indices used in the case-study, and experiment design. We then
describe the model architecture, evaluation metrics, and the dataset used in
the case-study. Results of the evaluation are given in Section 5.

###  4.1. Problem Statement

In this case-study, our goal was to use a SVANN approach to aid in the
selection of remote sensing indices for a set of wetland mapping tasks. A
wetland refers to a flooded area of land having a distinct ecosystem based on
hydrology, hydric soils, and vegetation adapted for life in water-saturated
soils (Keddy, 2010). Wetland inventory maps are essential for wetland
management, protection and restoration. However, development of highly
accurate wetland inventories can be expensive and technically challenging.
Further, they require periodic updates due to seasonal changes, land use
change and climate change. In recent work, semi automated wetland mapping
approaches are common (e.g., (Kloiber et al., 2015)), which use a combination
of image segmentation and random forest classification, along with aerial
photo interpretation. However, these approaches are iterative and time-
consuming to derive, and heavily require expert analysis.

We defined our wetland mapping task as a pixel-level binary image segmentation
process where we label each pixel of an image as a class, consisting of
wetland or non-wetland. Figure 5 shows the input and output of the system: the
three color (i.e., red, green, and blue) channels, acquired from multispectral
satellite imagery, joined with an index channel (e.g., NDVI). Our input also
consisted of the corresponding binary wetland mask as a true label. The white
region in the binary mask represents the pixels that belong to a wetland
class. The two inputs are used to train the final semantic segmentation
models. Various remote sensing indices are used in the domain of wetland
mapping. These indices are optimized for identifying different land cover
features. Remote sensing indices must be carefully chosen for a specific
geographic location, due to differences in climate and ecosystem. In this
case-study, we do so by comparing semantic segmentation results between CNN
models which use different remote sensing indices across spatially
heterogeneous locations.

Figure 5. Example input and output for the SVANN evaluation.

###  4.2. Geographically Heterogeneous Indices

Each multispectral Landsat-8 image in the study area consisted of several
bands (e.g., Blue, Green, Red, Near Infrared (NIR), Shortwave Infrared 1).
Each band represents a particular portion of the electromagnetic spectrum or
radiation. These bands can be used to calculate remote sensing indices to
better delineate land-cover features in a geographical area. We chose to
perform our evaluations using two well-known remote sensing indices which are
relevant to wetlands in northern Minnesota.

The first index was the Normalized Difference Vegetation Index (NDVI) (Tucker,
1979). It is formulated as Equation 3 by Near-Infrared (NIR) and visible red
spectrum (Red), which corresponds to band 5 and band 4 of the Landset-8
dataset. The formula for NDVI is given as:

(3) $\text{NDVI}=\frac{\text{NIR - Red}}{\text{NIR + Red}}$

The second index was the Normalized Difference Water Index (NDWI) (McFEETERS,
1996) formulated as Equation 4, where band 3 of the Landset-8 dataset
represents the visible green spectrum (Green). The formula for NDWI is given
as:

(4) $\text{NDWI}=\frac{\text{Green - NIR}}{\text{Green + NIR}}$

While both NDVI and NDWI have a range of $[-1,1]$, NDVI is more sensitive to
chlorophyll content in leaves (Nasser Mohamed Eid et al., 2020), so it has
been found useful for distinguishing vegetated and non-vegetated areas
(Corcoran et al., 2013). In contrast, NDWI is more sensitive to water content,
and a positive NDWI value indicates water areas and a negative value indicates
a non-water surface (Lv et al., 2019).

###  4.3. Experiment Design

To evaluate the efficacy of SVANN as a method of selecting remote sensing
indices based on location, we first trained four SVANN models (SVANN Models
1-4), where SVANN Models 1 and 2 were trained and evaluated on a Freshwater
Forested/Shrub Wetland region, and SVANN Model 3 and 4 were trained and
evaluated on a Lakes Wetland region. SVANN then selects the best performing
model for each region.

SVANN Versus Rule-Based OSFA: To compare with a rough simulation of
traditional wetland mapping automation, we built two baseline rule-based OSFA
classifiers (Rule-Based Models 1 and 2), which use an established ruleset for
each remote sensing index to classify wetland features.

SVANN Versus UNet OSFA: To compare with a standard deep learning approach,
which does not consider spatial variability when selecting input features, we
trained an OSFA UNet model (OSFA Model), which used both remote sensing
indices and both geographic regions as its input. We then compared the best
performing SVANN models against the OSFA model on the same region. Figure 6
shows the experiment design used in the evaluation.

Figure 6. Experiment design used in the evaluation.

Comparative physical interpretation: To validate the comparative physical
interpretation of SVANNs we compare their F1-score to the rule-based models
for each test area. If the behavior of a SVANN model is like a particular
rule-based approach, it suggests that the SVANN weights are suitable for that
area over the other.

The validation becomes further relevant as the zonal boundary starts to
overlap i.e., the test area starts to overlap. We plan to explore the
validation on overlapping test regions in the future work.

Figure 7. Experiment design showing the comparisons made in the case-study to
distinguish SVANN as a method of feature selection.

###  4.4. Modeling

Rule-Based Methodology: We built baseline rule-based classifiers for wetland
classification using the NDVI and NDWI remote sensing indices.

NDVI is frequently used as a tool for wetland identification. We defined the
rules for our rule-based NDVI index wetland classification following
established value ranges for certain wetland and non-wetland features (e.g.,
(Brown, 2015)).

Table 2. NDVI Rule-Based Classification Classification |  Values
---|---
Water |  $-1\text{ to }-0.1$
Rocks, clouds, buildings, etc |  $-0.1\text{ to }0.1$
Sparse Wetland Vegetation |  $0.1\text{ to }0.73$
Dense Non-Wetland Vegetation |  $0.73\text{ to }1$

NDWI is commonly used for water identification, not vegetation. As a result,
common value ranges for NDWI are derived specifically for identifying water
features (Ji et al., 2009). Therefore, we manually defined the rules for our
rule-based NDWI index wetland classification by adjusting the classification
threshold to account for vegetation features as well as water features.

Table 3. NDWI Rule-Based Classification Classification |  Values
---|---
Non-Wetland |  $-1\text{ to }-0.6$
Wetland |  $-0.6\text{ to }1$

By thoroughly evaluating the generated prediction masks, we found these two
rulesets to be fairly successful on our case-study regions. These rule-based
approaches correctly classify wetland features with reasonable accuracy and
make highly interpretable predictions. Therefore, we were able to use these
rulesets in our rule-based approach as a robust, interpretable baseline to
compare against our SVANN model.

Semantic Segmentation Methodology: We built the SVANN models using the UNet
(Ronneberger et al., 2015) architecture. UNet is an established technique for
image segmentation in biomedical imagery that has been adapted for aerial
imagery (Iglovikov and Shvets, 2018). This architecture follows an encoder-
decoder framework which combines local pixel information with its context. To
achieve this, high resolution features from the contracting set of layers
(i.e., context) are concatenated with the output from the up-sampled images
(i.e., localization). We provide a brief overview of UNet in Appendix B.

###  4.5. Evaluation Metrics

Since our case-study was a binary classification task, we were able to derive
evaluation metrics by constructing a confusion matrix consisting of correct
and incorrect model predictions, compared with the ground truth labels. The
values in the confusion matrix were labeled as true positive (TP), true
negative (TN), false positive (FP), and false negative (FN). Since our study
had an imbalanced class distribution, it was necessary to select evaluation
metrics which punish false predictions and therefore measure the relevance and
completeness of the model predictions (Gonzales and Sakla, 2019). For this
reason, we used the metrics precision and recall, called respectively the
complement of commission errors and the complement of emission errors by the
remote sensing community. We also used F1-score which balances the trade-off
between precision and recall. Detailed description of precision, recall, and
F1-score can be found in Appendix A.

###  4.6. Dataset

We used multispectral imagery from Landsat-8 Operational Land Imager (OLI)
(Roy et al., 2014) and part of a set of wetland maps developed by the National
Wetlands Inventory (NWI) (Fish and Service, 2020) for the mapping task. The
multispectral imagery consisted of the Blue, Green, Red, NIR, SWIR1, and SWIR2
bands, each with a spatial resolution of 30m and acquired on Sept. 11, 2020.
The NWI wetland maps in this case-study consisted of 718,304 features, which
represent the extent, approximate location, wetland types and surface water
habitats in northern Minnesota. All the features in the NWI wetland map were
classified according to the Cowardin classification system (Cowardin, 1979).
In this work, we limited our analysis to two study areas located in northern
Minnesota. The first region predominantly consists of Freshwater
Forested/Shrub wetlands. The second region mainly consists of small Lakes
wetlands. Figure 8 (a) shows the location of the two study areas. Overall, the
study regions had 5 different types of wetlands, as shown in Figure 8b-c.

Figure 8. Location and types of wetland (Best in color).

###  4.7. Preprocessing

To align and limit the analysis to the study area, the wetland shapefile and
multispectral imagery were first re-projected to the WGS84 reference system.
Next, the wetland shapefile and imagery were cropped to the case-study
regions.

Since the Landsat-8 imagery has a spatial resolution of 30m (Roy et al.,
2014), while the NWI wetland shapefile has a positional accuracy of about 2.2m
(of Natural Resources, 2019), we then up-sampled the multispectral imagery by
a factor of 4 using bilinear interpolation, which is a nearest-neighbor
algorithm for increasing the number of pixels. Up-sampling provided a $4^{2}$,
or $16$, times total increase in imagery pixels, resulting in $8,306\times
5,434$ pixels from the Freshwater Forested/Shrub Wetland region and
$9,046\times 5,709$ pixels for the Lakes Wetland region.

We then partitioned the imagery (Figure 9) and its mask into images having the
dimensions of $256\times 256$ pixels. These steps resulted in $672$ and $770$
samples for the Freshwater Forested/Shrub Wetland and Lakes Wetland regions
respectively. Partitions at the right and bottom edges of the imagery were
removed, as a majority of the area was empty.

Figure 9. Preprocessing imagery and polygon based wetland maps.

The binary masks for training and testing were created in two steps. First, we
used the polygons in the wetland shapefile to extract the wetland from the
imagery for each region. We then used the extracted wetland imagery to create
a binary mask for model training. Figure 9 shows the initial imagery from the
Freshwater Forested/Shrub Wetland region, an example of extracted wetland
imagery for each image partition, and the corresponding binary mask where
white pixels represent wetlands. The dataset was split into training ($\sim
80\%$), validation ($\sim 10\%$), and testing ($\sim 10\%$) datasets to build,
validate and test the models.

##  5\. Results and Discussion

In our first experiment, we evaluated the effect of training SVANN with and
without up-sampled imagery. The second and third set of experiments compared
trained SVANN models against two one-size-fits-all approaches, a rule-based
approach (Rule-Based Models 1 and 2) and a deep learning UNet model (OSFA
Model). Since our overarching goal is to interpret the SVANN approach, we also
include a discussion of SVANN’s interpretability.

###  5.1. Up-Sampling Versus Non Up-Sampling

In the preprocessing stage of the case-study, we compared our results when
training SVANN models with up-sampled imagery (Up-Models 1-4 in Table 5, where
the model chosen by SVANN is shown in bold) versus training with non up-
sampled imagery (Models 1-4). Previously, we had found that the lower spatial
resolution of the Landsat-8 multispectral imagery (30m), compared with the
positional accuracy of the NWI wetland shapefile (approx. 2.2m), caused a
significant amount of fine-scale information to be lost when preprocessing the
input data for the SVANN models, which made model training difficult for the
Lakes Wetland region. As discussed earlier, we addressed this issue by first
up-sampling the multispectral imagery using bilinear interpolation before
generating the binary wetland masks. Then, we used the up-sampled imagery and
binary wetland masks as model inputs for the up-sampled models (Up-Models
1-4). The input for the non up-sampled models (Models 1-4) was the original
imagery, which was used to generate their binary wetland masks.

Table 4 shows the results of this comparison. In the Lakes Wetland region, Up-
Model 4 sees a significant improvement (+0.440) in validation accuracy and
evaluation metrics (e.g., +0.298 in F1 score) over Model 4. Our results
suggest that the up-sampling of the imagery provided a significant improvement
in the ability of the model to classify wetlands in this region. This
improvement is likely due to the prevalence of small wetland features (i.e.,
Figure 8c), which cannot be accurately represented by a low-resolution binary
wetland mask. In the Freshwater Forested/Shrub Wetland region, Up-Model 1 sees
a slight decrease in F1 score (-0.032), but an increase in validation accuracy
(+0.149), over Model 1. We speculate that this is due to the homogeneity of
the wetlands in the region, which allowed the low-resolution binary mask to
capture the wetland information with reasonable accuracy, and made evaluation
on the test data slightly more difficult. Overall, we found the up-sampling to
be useful for two reasons: firstly, model performance was overall improved,
since the up-sampled models produced more consistent and overall better
classification accuracy. Secondly, model decomposability was improved, since
the up-sampled models were trained using more detailed and contiguous binary
wetland masks, which are more clearly understood by humans than the non up-
sampled binary masks (see Figure 10).

Figure 10. Imagery up-sampling process, demonstrating the increase of wetland information retained by the binary mask when the imagery is up-sampled. Table 4. Experimental results showing the effect of up-sampling on model performance. Model | Index | TN | FP | FN | TP | Precision | Recall | F1 | Val. Acc. | Epochs
---|---|---|---|---|---|---|---|---|---|---
Test Area: Freshwater Forested/Shrub Wetland region
Up-Model 1 | NDVI | 1,114,819 | 432,164 | 763,862 | 2,145,603 | 0.832 | 0.737 | 0.782 | 0.786 | 3
Up-Model 2 | NDWI | 240,074 | 1,839,944 | 13,532 | 2,362,898 | 0.562 | 0.994 | 0.718 | 0.758 | 3
Model 1 | NDVI | 94,020 | 27,567 | 30,313 | 126,628 | 0.821 | 0.807 | 0.814 | 0.637 | 3
Model 2 | NDWI | 87,923 | 33,664 | 57,227 | 99,714 | 0.748 | 0.635 | 0.687 | 0.594 | 3
Test Area: Lakes Wetland region
Up-Model 3 | NDVI | 2,266,152 | 847,533 | 581,179 | 1,351,408 | 0.615 | 0.699 | 0.654 | 0.696 | 3
Up-Model 4 | NDWI | 2,423,985 | 689,700 | 455,228 | 1,477,359 | 0.682 | 0.764 | 0.721 | 0.723 | 3
Model 3 | NDVI | 222,181 | 63,466 | 11,497 | 18,248 | 0.223 | 0.613 | 0.327 | 0.231 | 3
Model 4 | NDWI | 237,436 | 48,211 | 8,830 | 20,915 | 0.303 | 0.703 | 0.423 | 0.283 | 3

###  5.2. SVANN Versus One-Size-Fits-All

We then compared SVANN against the two different OSFA approaches. Table 5
shows the results of the experiments.

####  5.2.1. SVANN Versus Rule-Based OSFA

To compare the SVANN models with the rule-based OSFA approach, we made two
sets of regional comparisons: In the Freshwater Forested/Shrub Wetland region,
we compared SVANN Model 1 and Rule-based Models 1 and 2. In the Lakes Wetland
region, we compared SVANN Model 4 and Rule-based Models 1 and 2. For both of
these regions, we found that the SVANN model outperformed the rule-based
approaches on all three evaluation metrics (i.e., precision, recall, and F1
score).

This comparison tells us two things: first, it suggests that deep learning is
superior to the baseline rule-based approach, since the deep learning models
in the case-study are capable of learning more complex features of wetland
delineation than the rule-based approach. Second, it suggests that SVANN is
superior to an OSFA approach, since SVANN is capable of both training model
weights and selecting remote sensing indices as a function of location,
whereas the rule-based approach consists of a static set of rules and input
data for classifying wetlands all regions.

####  5.2.2. SVANN Versus UNet OSFA

To compare the SVANN models with the OSFA UNet model, we again made two sets
of regional comparisons: In the Freshwater Forested/Shrub Wetland region, we
compared SVANN Model 1 and the OSFA Model. In the Lakes Wetland region, we
compared SVANN Model 4 and the OSFA Model. In the first comparison, we find
that the OSFA UNet model performs slightly better than the SVANN model in
precision, F1 score, and validation accuracy, with roughly equal recall. In
the second comparison, the SVANN model has a slightly better recall, F1 score,
and validation accuracy, with lower precision. Overall, the experimental
results show that the SVANN models have roughly similar performance with the
OSFA UNet model.

From these results, we find that using SVANN models to select remote sensing
indices based on location achieves similar performance to an OSFA UNet model,
even though the OSFA model uses roughly twice as much input data as well as an
additional index feature during training. The findings support the claim that
considering spatial heterogeneity using SVANNs during feature selection offers
significant value, since only the most useful remote sensing indices are
selected by the SVANN approach.

Table 5. Experimental results comparing SVANN approach with OSFA. Model | Index | TN | FP | FN | TP | Precision | Recall | F1 | Val. Acc. | Epochs
---|---|---|---|---|---|---|---|---|---|---
Test Area: Freshwater Forested/Shrub Wetland region
SVANN Model 1 | NDVI | 695,391 | 522,235 | 255,792 | 2,983,030 | 0.851 | 0.921 | 0.885 | 0.827 | 8
SVANN Model 2 | NDWI | 874,312 | 672,671 | 249,550 | 2,659,915 | 0.798 | 0.914 | 0.852 | 0.817 | 8
Rule-based Model 1 | NDVI | 1,066,717 | 480,266 | 403,777 | 2,505,688 | 0.839 | 0.861 | 0.850 | - | -
Rule-based Model 2 | NDWI | 1,115,080 | 431,903 | 518,482 | 2,390,983 | 0.847 | 0.822 | 0.834 | - | -
OSFA Model | NDVI, NDWI | 736,624 | 481,002 | 258,417 | 2,980,405 | 0.861 | 0.920 | 0.890 | 0.831 | 8
Test Area: Lakes Wetland region
SVANN Model 3 | NDVI | 2,074,210 | 1,039,475 | 405,994 | 1,526,593 | 0.595 | 0.790 | 0.679 | 0.737 | 8
SVANN Model 4 | NDWI | 2,586,145 | 527,540 | 461,125 | 1,471,462 | 0.736 | 0.761 | 0.749 | 0.787 | 8
Rule-based Model 1 | NDVI | 2,313,277 | 800,408 | 467,235 | 1,465,352 | 0.647 | 0.758 | 0.698 | - | -
Rule-based Model 2 | NDWI | 2,413,404 | 700,281 | 536,717 | 1,395,870 | 0.666 | 0.722 | 0.693 | - | -
OSFA Model | NDVI, NDWI | 2,608,688 | 504,997 | 495,945 | 1,436,642 | 0.740 | 0.743 | 0.742 | 0.780 | 8

###  5.3. Interpretation

The experimental results of our wetland mapping case-study show the
suitability of comparative physical interpretation of SVANNs using rule-based
methods to achieve decomposability.

For the test area having Forested/Shrub Wetland we find that SVANN Model 1 is
better than Model 2 and for rule-based Model 1 is better than Model 2. If we
interpret SVANN model 1 with rule-based Model 1, we can say that they have a
higher chances that the weights correspond to the $NDVI>0.1$, compared to that
of SVANN Model 2.

In contrast, if we compare the rule-based Model 1 with OSFA Model we cannot
conlude whether the Model is more suitable for Forested/Shrub Area or Wetland
Area. Similarly, for the test area having lakes Model 4 behaves similar to
that of rule base model 2, which suggests that SVANN model 2 is more suitable
for lakes.

We also discussed model interpretability in terms of simulatability,
decomposability, and algorithmic transparency in Section 3. Based on the above
interpretation we find that UNet based SVANNs are more decomposable than a
OSFA approach, however limited in their simulatibility and algorithmic
transparency due to the underlying use of UNet architecture. We believe to
achieve additional transparency, SVANNs needs to be compared with more
sophisticated yet highly interpretable models like partial differential
equation (PDEs). Below, we initiate a discussion in that direction.

###  5.4. Multi Layer Neural Networks for PDE

Representation learning allows multi-layer neural networks (MLNNs) to
automatically derive features for a given set of inputs. In addition, by the
universal approximator theorem (Cybenko, 1989) a sufficiently large network
can learn a significant range of functions. Thus, MLNNs can serve as a natural
solver for PDEs. Use of MLNNs to solve PDEs has been explored as far back as
the 1990s. For example, Lagaris et al. (Lagaris et al., 1998) and Lee et al.
(Lee and Kang, 1990) used neural networks to solve PDEs and ODEs on an a
priori fixed mesh.

Recently, there has been a renewed interest in this direction due to
significant advances in computing resources and software libraries such as
TensorFlow (Abadi et al., 2016), which can efficiently compute high-
dimensional derivatives using automatic differentiation (Baydin et al., 2018).
For example, Raissi et al. (Raissi et al., 2017) constructed an estimate of
PDEs using neural network models called Physics informed neural networks
(PINNs). These networks can be considered as a class of data-efficient
universal function approximators that naturally encode any underlying physical
laws as prior information (e.g., through loss function). The approach allows
for the estimation of physical models from limited data and leverages a priori
knowledge that physical dynamics should obey a class of PDEs. More recent
methods such as the Deep Galerkin Method (Sirignano and Spiliopoulos, 2018) do
not assume such a mesh structure and are analogous to the Galerkin method,
where the solution is approximated by neural networks, which can serve as a
linear combination of basis functions. In this work, we limit our physical
interpretation to PINNs and plan to explore additional interpretation(s) in
the future work.

For a better understanding of PINNs and automatic differentiation, we provide
an example of solving a PDE using PINNs in Appendix C and details on automatic
differentiation in Appendix D.

###  5.5. Towards process-based model interpreta- tion using partial
differential equations

Partial differential equations (PDEs) have been studied extensively and are
well-understood by various scientific communities. Therefore, representing
PDEs with multi-layer neural networks (MLNNs) allows them to be simulatable by
physicists, mathematicians, and other communities well-versed in PDEs.
Further, partial differential equations are far more decomposable compared to
MLNNs, as each term is a derivative, partial derivative, or a constant with
well-defined properties. PDEs also offer increased transparency, as they can
be dissected and studied mathematically and their error surface can be
analyzed–manually for smaller equations, and through software for larger
equations.

Next, we give an example of geographically heterogeneous processes based on
streamflow in flat and stepped rice fields. We model the process using SVANNs
and the provide a physical interpretation using PINNs.

Basic knowledge of Physics tells us that streamflow in rice paddies on the
plains should differ from streamflow of rices paddies on non-planar step
fields (Figure 11). Streamflow models therefore need different parameters
(flat farm vs hillside agriculture) that reflect the different physical
constraints across these regions.

Figure 11. Geographically heterogeneous modeling of streamflow.

Let $Q_{i}$ be the functional representation of surface water run-off for the
$i^{th}$ zone and $lat^{i}$, $long^{i}$ be the sample coordinates within the
$i^{th}$ zone. Then,

$Q_{1}=f_{flat}(lat^{1},long^{1})$ $Q_{2}=f_{stepped}(lat^{2},long^{2})$

where, let $Q_{1}$ denote the streamflow model for flat rice fields and
$Q_{2}$ denote the streamflow model for stepped rice fields. For each
functional representation, we define a partial differential equation
(PDE)-based formulation as follows,

$M_{1}=\left(\frac{\partial^{2}\hat{Q}_{1}}{\partial
x^{2}}+\frac{\partial\hat{Q}_{1}}{\partial
x}-b\right)+[\hat{Q}_{1}^{(0)}-Q_{1}^{(0)}]+[\hat{Q}_{1}^{(1)}-Q_{1}^{(1)}]$
$M_{2}=\left(\frac{\partial^{2}\hat{Q}_{2}}{\partial
x^{2}}+\frac{\partial\hat{Q}_{2}}{\partial
x}-b\right)+[\hat{Q}_{2}^{(0)}-Q_{2}^{(0)}]+[\hat{Q}_{2}^{(1)}-Q_{2}^{(1)}]$

where $Q_{i}^{(i)}$ and $Q_{i}^{(1)}$ are the initial and boundary conditions
for the function.

Let $Q_{3}$ represent a unified functional representation of the two models
such that,

$Q_{3}=f(Q_{1},Q_{2}).$

Then,

$M_{3}=\left(\frac{\partial^{2}\hat{Q}_{3}}{\partial
x^{2}}+\frac{\partial\hat{Q}_{3}}{\partial
x}-b\right)+[\hat{Q}_{3}^{(0)}-Q_{3}^{(0)}]+[\hat{Q}_{3}^{(1)}-Q_{3}^{(1)}]$

Also, let the average prediction error of a Model $M_{i}$ on a location (lat,
long) be $Error_{Avg}(M_{i},lat,long)$. Then we postulate,

$Error_{Avg}(M_{3},lat^{1},long^{1})>Error_{Avg}(M_{1},lat^{1},long^{1})$
$Error_{Avg}(M_{3},lat^{2},long^{2})>Error_{Avg}(M_{2},lat^{2},long^{2})$

Hence, using heterogeneous data to train models reduces their representational
ability locally.

Further, SVANN-I approach will have similar structure for $f_{flat}$ and
$f_{stepped}$, whereas, for SVANN-E approach the structure for $f_{flat}$ can
differ from $f_{stepped}$ as shown in Figure 11.

Overall, we find that the models considered in this work had different
physical interpretability. Table 6 summarizes the interpretability of the
models based on different interpretability metrics.

Table 6. Physical interpretability of the models considered.

Model | Interpretability Metric
---|---
Simulatibility | Decomposability | Algorithmic Transparency
OSFA (UNet) | Low | Low | Low
SVANN (UNet) | Low | Medium | Low
Rule-based | High | High | High
PINN | High | High | High

##  6\. Conclusion and future work

Given Spatial Variability Aware Neural Networks (SVANNs), the goal was to
investigate mathematical (or computational) models for physical interpretation
of SVANNs towards transparency. This problem is important due to several
important use-cases (e.g., reusability, debugging, etc.) and challenging due
to a large number of SVANN model parameters, vacuous bounds on generalization
performance of neural networks, risk of overfitting, sensitivity to noise,
etc. The related work on either model-specific or model-agnostic post-hoc
interpretation is limited due to a lack of consideration of physical
constraints (e.g., mass balance) and properties (e.g., second law of
geography). To overcome the limitations of related work, we provide a physical
interpretation of a SVANN using remote sensing indices which honor to spatial
variability. Our results on the task of wetland mapping show that the proposed
physical interpretations improve the transparency of SVANN models and the
analytical results highlight the trade-off between model transparency and
model performance (e.g., F1-score). We also describe an interpretation based
on geographically heterogeneous processes modeled as partial differential
equations (PDEs).

In the future, we plan to explore an experimental based validation of the
PINN-based interpretation of SVANNs. In addition, we plan to expand our
process-based interpretation of SVANNs to include additional neural network
based solvers for PDEs. Finally, we plan to use an extensive set of remote
sensing indices to further explore their use in the context of SVANNs and
expand the validation to overlapping test areas.

###### Acknowledgements.

This material is based on work supported by the National Science Foundation
under Grant No. 1737633. We would also like to thank Kim Koffolt and the
spatial computing research group for their helpful comments and refinements.

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##  Appendix A Precision, Recall, and F1-score

Precision is a metric for prediction relevance. It calculates the percentage
of the model’s true predictions which were correctly classified. The formula
for precision is given as:

(5) $\text{Precision}=1-\text{Commission Error}=\frac{\text{TP}}{\text{TP +
FP}}$

Recall is a metric for prediction completeness. It calculates the fraction of
total true predictions which the model correctly classified as true
predictions. The formula for recall is given as:

(6) $\text{Recall}=1-\text{Omission Error}=\frac{\text{TP}}{\text{TP + FN}}$

There is inherently a trade-off between precision (i.e., prediction relevance)
and recall (i.e., prediction completeness), which necessitates a metric which
considers this interaction. As such, we used the F1 score, which is the
harmonic mean of precision and recall, as another metric to evaluate the
pixel-level classification performance (this metric is fundamentally similar
to the ’dice coefficient’ used by the computer vision community). While the
precision and recall metrics can offer useful information on their own (i.e.,
prediction relevance and prediction completeness), the F1 score considers
these two metrics as equally important in its calculation, which is given as:

(7) $\text{F1
Score}=2\cdot\frac{\text{Precision}\cdot\text{Recall}}{\text{Precision}+\text{Recall}}$

With these three evaluation metrics, we obtain a robust baseline for
quantitatively evaluating the ability of the model to correctly classify.
Further, these metrics are commonly used when evaluating semantic segmentation
models for wetland mapping tasks, which allows us to compare our results with
rule-based approaches.

##  Appendix B Brief overview of UNet

Figure 12 shows the UNet architecture adapted from its original paper
(Ronneberger et al., 2015), presented earlier in (Gupta et al., 2021). UNet
has two paths, a contracting path and an expanding path. Each step in the
contracting path consists of two $3\times 3$ convolutions (with padding), each
followed by a rectified linear unit (ReLU) and a $2\times 2$ maxpooling
operation with a stride of $2$ for downsampling (which reduces each image
dimension by half). Each step in the expanding path consists of up-sampling
(i.e., up-convolution), which doubles the image dimensions, followed by a
$2\times 2$ convolution which reduces the number of feature channels by half.
This is followed by concatenation with the corresponding feature map from the
contracting path, and two $3\times 3$ convolutions each followed by a ReLU.
The final layer consists of a $1\times 1$ convolution layer mapping each image
pixel to the class. In addition, we use a dropout layer between each of the
convolution layers to avoid over-fitting. Overall, UNet consists of 9 steps; 5
steps in the contracting path and 4 steps in the expanding path.

Figure 12. Architecture of UNet where arrows represent the operations (i.e.,
convolution, maxpool, dropout, and copy and concatenation). Feature dimensions
are shown above the features and the number of channels in the convolution
operation are shown above the arrows. (Best in color).

##  Appendix C Example of Physics Informed Neural Networks to solve a PDE

We provide a toy example of solving PDEs using physics informed neural
networks (PINNs). We consider a PDE defined as $\frac{\partial u}{\partial
t}+3.\frac{\partial u}{\partial x}=0$ with initial condition as
$u(x,0)=x.e^{-x^{2}}$.

The PINNs model function $u(x,t)$ as $\hat{y}$, for which the loss function
can be defined as a linear combination of the PDE and the initial condition as
follows:
Loss function ($\mathcal{L}$) = $\left(\frac{\partial\hat{y}}{\partial
t}+3.\frac{\partial\hat{y}}{\partial x}\right)+(\hat{y}(x,0)-x.e^{-x^{2}})$
Multi-layer neural network (MLNN) architecture: For illustration, we use a
basic MLNN with two input, one output, and one hidden layer, with 2 sigmoid
nodes as shown in Figure 13. For simplicity, we use sigmoid nodes in the
hidden layers and linear nodes in the input and output layer. Further, for the
readability of equations we use the following two notations: First,
$f_{w_{i}w_{j}}=\sigma(w_{i}.x+w_{j}.t)$ and Second,
$f_{w_{i}}=\sigma(w_{i}.x)$. For example, the notations simplify $\hat{y}$ to
$w_{5}.f_{w_{1}w_{3}}+w_{6}.f_{w_{2}w_{4}}$.

Figure 13. Architecture of the MLNN used in the toy example.

The loss function in terms of network parameters is as follows,

(8) $\mathcal{L}=w_{5}.f_{w_{1}w_{3}}.(1+f_{w_{1}w_{3}}).(3.w_{1}+w_{3})\\\
+w_{6}.f_{w_{2}w_{4}}.(1+f_{w_{2}w_{4}}).(3.w_{2}+w_{4})\\\
+f_{w_{1}}.w_{5}+f_{w_{2}}.w_{6}-x.e^{-x^{2}}.$

The weights of a given $x$ are updated using the back-propagation rule (Table
7), where the loss function ($\mathcal{L}$) is used to update the weights as
shown in Equations 9-14. In theory, when the weights are updated, $\hat{y}$
should approach the theoretical value of the function represented by the PDE
for a given input $x$. Table 3 shows the results for 5 iterations (i.e.,
loops) of $x=0.1$. It should be noted that the proposed architecture may not
be the most appropriate for the given PDE, but simply illustrate the PINN
approach.

Table 7. Back-propagation weight update rule for the weights. $\mathbf{w_{1}}$ |  $\mathbf{w_{2}}$ |  $\mathbf{w_{3}}$ |  $\mathbf{w_{4}}$
---|---|---|---
$w_{1}+\eta.\frac{\partial\mathcal{L}}{\partial w_{1}}.x$ |  $w_{2}+\eta.\frac{\partial\mathcal{L}}{\partial w_{2}}.x$ |  $w_{3}+\eta.\frac{\partial\mathcal{L}}{\partial w_{3}}.x$ |  $w_{4}+\eta.\frac{\partial\mathcal{L}}{\partial w_{4}}.x$
$\mathbf{w_{5}}$ | $\mathbf{w_{6}}$
$w_{5}+\eta.\frac{\partial\mathcal{L}}{\partial w_{5}}.\sigma(w_{1}.x+w_{3}.t)$ | $w_{6}+\eta.\frac{\partial\mathcal{L}}{\partial w_{6}}.\sigma(w_{2}.x+w_{4}.t)$

(9) $\small\frac{\partial\mathcal{L}}{\partial
w_{1}}=x.w_{5}.(3.w_{1}+w_{3}).f_{w_{1}w_{3}}.(1+f_{w_{1}w_{3}})^{2}+w_{5}.f_{w_{1}w_{3}}.(1+f_{w_{1}w_{3}}).\\\
(x.(3.w_{1}+w_{3}).f_{w_{1}w_{3}}+3)+x.w_{5}.f_{w_{1}}.(1+f_{w_{1}})$ (10)
$\small\frac{\partial\mathcal{L}}{\partial
w_{2}}=x.w_{6}.(3.w_{2}+w_{4}).f_{w_{2}w_{4}}.(1+f_{w_{2}w_{4}})^{2}+w_{6}.f_{w_{2}w_{4}}.(1+f_{w_{2}w_{4}}).\\\
(x.(3.w_{2}+w_{4}).f_{w_{2}w_{4}}+3)+x.w_{6}.f_{w_{2}}.(1+f_{w_{2}})$ (11)
$\small\frac{\partial\mathcal{L}}{\partial
w_{3}}=t.w_{5}.(3.w_{1}+w_{3}).f_{w_{1}w_{3}}.(1+f_{w_{1}w_{3}})^{2}+w_{5}.f_{w_{1}w_{3}}.(1+f_{w_{1}w_{3}}).\\\
(t.(3.w_{1}+w_{3}).f_{w_{1}w_{3}}+1)$ (12)
$\small\frac{\partial\mathcal{L}}{\partial
w_{4}}=t.w_{6}.(3.w_{2}+w_{4}).f_{w_{2}w_{4}}.(1+f_{w_{2}w_{4}})^{2}+w_{6}.f_{w_{2}w_{4}}.(1+f_{w_{2}w_{4}}).\\\
(t.(3.w_{2}+w_{4}).f_{w_{2}w_{4}}+1)$ (13)
$\frac{\partial\mathcal{L}}{\partial
w_{5}}=f_{w_{1}w_{3}}.(1+f_{w_{1}w_{3}}).(3.w_{1}+w_{3})+f_{w_{1}}$ (14)
$\frac{\partial\mathcal{L}}{\partial
w_{6}}=f_{w_{2}w_{4}}.(1+f_{w_{2}w_{4}}).(3.w_{2}+w_{4})+f_{w_{2}}$

Table 8. Weight updates for the MLNN for $x=0.1,y=0.1$. The solution of the PDE at the sample point is $-0.208$. Loop |  $\mathbf{w_{1}}$ |  $\mathbf{w_{2}}$ |  $\mathbf{w_{3}}$ |  $\mathbf{w_{4}}$ |  $\mathbf{w_{5}}$ |  $\mathbf{w_{6}}$ |  $\hat{y}$ |  Loss
---|---|---|---|---|---|---|---|---
$0$ |  $0.5$ |  $0.5$ |  $0.5$ |  $0.5$ |  $0.5$ |  $0.5$ |  $0.525$ |  $2.01$
$1$ |  $0.487$ |  $0.487$ |  $0.495$ |  $0.495$ |  $0.389$ |  $0.389$ |  $0.408$ |  $1.52$
$2$ |  $0.476$ |  $0.476$ |  $0.491$ |  $0.491$ |  $0.28$ |  $0.28$ |  $0.294$ |  $1.05$
$3$ |  $0.469$ |  $0.469$ |  $0.488$ |  $0.488$ |  $0.173$ |  $0.173$ |  $0.182$ |  $0.6$
$4$ |  $0.464$ |  $0.464$ |  $0.486$ |  $0.487$ |  $0.067$ |  $0.067$ |  $0.07$ |  $0.17$
$5$ |  $0.462$ |  $0.462$ |  $0.486$ |  $0.486$ |  $-0.038$ |  $-0.038$ |  $-0.04$ |  $-0.25$

In this example, the equation is a transport equation whose general form is
given by $\frac{\partial u}{\partial t}+v.\frac{\partial u}{\partial x}=0$,
where $v$ is a constant. The exact solution to the equation in the example is
$u(x,t)=(x-3t).e^{(x-3t)^{2}}$. Proving exact solution is out of the scope.

##  Appendix D Automatic Differentiation

Automatic differentiation (AD) is based on the fact that all numerical
computations are compositions of a finite set of elementary operations for
which derivatives are known. Combining the derivatives of the constituent
operations using the chain rule gives the overall composition derivative. In
general, there are forward and reverse accumulating mode (Baydin et al., 2018)
of AD, but most of the mainstream machine learning libraries including
TensorFlow and PyTorch implement the reverse accumulating mode, so we limit
this discussion to the reverse accumulating mode.

Figure 14. Computational graph of the toy example of MLNN described in Appendix C. All the known variables are marked in bold. Table 9. Numerical example of automatic differentiation using reverse accumulating mode. All the weights $w_{i}$ are initialized to $0.5$ for $i=1,2,\dots,6$, and $x,t$ are initialized to 0.1. The order of computation to follow is from top to bottom for the forward pass, and bottom to top for the backward pass. Forward Pass | Backward Pass
---|---
Node | Eqa | Val | $\partial$ | Eqa | Val
$w_{1}$ | - | 0.5000 | $\frac{\partial\hat{y}}{\partial w_{1}}$ | $\frac{\partial\hat{y}}{\partial v_{1}}\frac{\partial v_{1}}{\partial w_{1}}$ | 0.0118
$x$ | - | 0.1000 | $\frac{\partial\hat{y}}{\partial x}$ | $\frac{\partial\hat{y}}{\partial v_{2}}\frac{\partial v_{2}}{\partial x}+\frac{\partial\hat{y}}{\partial v_{1}}\frac{\partial v_{1}}{\partial x}$ | 0.1175
$w_{2}$ | - | 0.5000 | $\frac{\partial\hat{y}}{\partial w_{2}}$ | $\frac{\partial\hat{y}}{\partial v_{2}}\frac{\partial v_{2}}{\partial w_{2}}$ | 0.0118
$w_{3}$ | - | 0.5000 | $\frac{\partial\hat{y}}{\partial w_{3}}$ | $\frac{\partial\hat{y}}{\partial v_{3}}\frac{\partial v_{3}}{\partial w_{3}}$ | 0.0118
$t$ | - | 0.1000 | $\frac{\partial\hat{y}}{\partial t}$ | $\frac{\partial\hat{y}}{\partial v_{4}}\frac{\partial v_{4}}{\partial t}+\frac{\partial\hat{y}}{\partial v_{3}}\frac{\partial v_{3}}{\partial t}$ | 0.1175
$w_{4}$ | - | 0.5000 | $\frac{\partial\hat{y}}{\partial w_{4}}$ | $\frac{\partial\hat{y}}{\partial v_{4}}\frac{\partial v_{4}}{\partial w_{4}}$ | 0.0118
$v_{1}$ | $w_{1}.x$ | 0.0500 | $\frac{\partial\hat{y}}{\partial v_{1}}$ | $\frac{\partial\hat{y}}{\partial v_{5}}\frac{\partial v_{5}}{\partial v_{1}}$ | 0.1175
$v_{2}$ | $w_{2}.x$ | 0.0500 | $\frac{\partial\hat{y}}{\partial v_{2}}$ | $\frac{\partial\hat{y}}{\partial v_{6}}\frac{\partial v_{6}}{\partial v_{2}}$ | 0.1175
$v_{3}$ | $w_{3}.t$ | 0.0500 | $\frac{\partial\hat{y}}{\partial v_{3}}$ | $\frac{\partial\hat{y}}{\partial v_{5}}\frac{\partial v_{5}}{\partial v_{3}}$ | 0.1175
$v_{4}$ | $w_{4}.t$ | 0.0500 | $\frac{\partial\hat{y}}{\partial v_{4}}$ | $\frac{\partial\hat{y}}{\partial v_{6}}\frac{\partial v_{6}}{\partial v_{4}}$ | 0.1175
$v_{5}$ | $v_{1}+v_{3}$ | 0.1000 | $\frac{\partial\hat{y}}{\partial v_{5}}$ | $\frac{\partial\hat{y}}{\partial v_{7}}\frac{\partial v_{7}}{\partial v_{5}}$ | 0.1175
$v_{6}$ | $v_{2}+v_{4}$ | 0.1000 | $\frac{\partial\hat{y}}{\partial v_{6}}$ | $\frac{\partial\hat{y}}{\partial v_{8}}\frac{\partial v_{8}}{\partial v_{6}}$ | 0.1175
$v_{7}$ | $\sigma(v_{5})$ | 0.5250 | $\frac{\partial\hat{y}}{\partial v_{7}}$ | $\frac{\partial\hat{y}}{\partial v_{9}}\frac{\partial v_{9}}{\partial v_{7}}$ | 0.5000
$w_{5}$ | - | 0.5000 | $\frac{\partial\hat{y}}{\partial w_{5}}$ | $\frac{\partial\hat{y}}{\partial v_{9}}\frac{\partial v_{9}}{\partial w_{5}}$ | 0.5250
$w_{6}$ | - | 0.5000 | $\frac{\partial\hat{y}}{\partial w_{6}}$ | $\frac{\partial\hat{y}}{\partial v_{10}}\frac{\partial v_{10}}{\partial w_{6}}$ | 0.5250
$v_{8}$ | $\sigma(v_{6})$ | 0.5250 | $\frac{\partial\hat{y}}{\partial v_{8}}$ | $\frac{\partial\hat{y}}{\partial v_{10}}\frac{\partial v_{10}}{\partial v_{8}}$ | 0.5000
$v_{9}$ | $v_{7}.w_{5}$ | 0.2625 | $\frac{\partial\hat{y}}{\partial v_{9}}$ | $\frac{\partial\hat{y}}{\partial\hat{y}}\frac{\partial\hat{y}}{\partial v_{9}}$ | 1.0000
$v_{10}$ | $v_{8}.w_{6}$ | 0.2625 | $\frac{\partial\hat{y}}{\partial v_{10}}$ | $\frac{\partial\hat{y}}{\partial\hat{y}}\frac{\partial\hat{y}}{\partial v_{10}}$ | 1.0000
$\hat{y}$ | $v_{9}+v_{10}$ | 0.5250 | $\frac{\partial\hat{y}}{\partial\hat{y}}$ | - | 1.0000

The reverse accumulating mode consists of a forward pass and a backward pass.
The forward pass will define the computational graph and compute the numerical
values of all the nodes starting from the input to the output in a feed-
forward manner, and these values will be stored in the memory for efficient
computation. When the forward pass is completed, the graph is evaluated in a
backward pass to compute the gradient of each node (i.e., the partial
derivative of the output node with respect to all the other nodes). Figure 14
gives an example of the computational graph derived from the toy example in
Appendix C, and Table 9 provides an numerical example of how AD works.

##  Appendix E Resources

We used Python’s geopandas, rasterio, PIL and numpy libraries for
preprocessing the imagery and shapefiles. We used Keras, a high-level deep
learning interface for TensorFlow, to implement UNet. Model training and
evaluation was done on a 3.70GHz Intel Core i7-8700K CPU, NVIDIA GeForce GTX
1080Ti GPU, and 16GB 2400MHz DDR4 RAM. Python based implementation of SVANN is
at the following link: https://github.com/jayantgupta/SVANN.