# Understanding the Efficacy of U-Net & Vision Transformer for Groundwater
Numerical Modelling

Maria Luisa Taccari1, Oded Ovadia2, He Wang3, Adar Kahana4, Xiaohui Chen1,
Peter K. Jimack3

###### Abstract.

This paper presents a comprehensive comparison of various machine learning
models, namely U-Net [12], U-Net integrated with Vision Transformers (ViT)
[11], and Fourier Neural Operator (FNO) [4], for time-dependent forward
modelling in groundwater systems. Through testing on synthetic datasets, it is
demonstrated that U-Net and U-Net + ViT models outperform FNO in accuracy and
efficiency, especially in sparse data scenarios. These findings underscore the
potential of U-Net-based models for groundwater modelling in real-world
applications where data scarcity is prevalent.

11footnotetext: 1School of Civil Engineering, University of Leeds, Leeds, UK,
Email<EMAIL_ADDRESS>2Department of Applied Mathematics,
Tel-Aviv University, Tel-Aviv, Israel.33footnotetext: 3School of Computing,
University of Leeds, Leeds, UK.44footnotetext: 4Department of Applied
Mathematics, Tel-Aviv University, Tel-Aviv, Israel.11footnotetext: 1School of
Civil Engineering, University of Leeds, Leeds, UK.33footnotetext: 3School of
Computing, University of Leeds, Leeds, UK.

## Introduction

Groundwater numerical models, such as MODFLOW [2], are crucial for water
resource management, although they are computationally demanding. To alleviate
this, surrogate modelling through data-driven methods offers efficient
approximations of these complex numerical techniques.

Neural Operators [19, 18], particularly the Fourier Neural Operator (FNO) [4],
have been at the forefront of recent advances, having shown potential to
approximate arbitrary continuous functions. However, the computational demand
of FNO is particularly high during training phase while these neural operators
require architectural enhancements to deliver promising results in subsurface
problems [13, 16]. This is evident in the work of Wen et al. [13], where the
integration of FNO with U-Net architecture showed improved accuracy, speed,
and data efficiency in multiphase flow problems. However, Gupta and
Brandstetter’s work [3], showing that U-Net outperforms FNOs across various
fluid mechanics problems, raises a question about the necessity of neural
operators when the vanilla U-Net architecture already exhibits remarkable
performance.

Recently, transformers [20] have seen considerable success in various fields,
including physical systems [10, 9], for which the datasets are typically
smaller compared to other domains. Only one study explores the use of
transformers in groundwater modeling [14], demonstrating that the models were
outperformed by both GRU and LSTM models to predict groundwater levels across
various stations in France with meteorological and hydrological data.

Finally, the integration of U-Net with Transformers, as exemplified in studies
like TransUNet [6] and ViTO [5], has demonstrated their utility across a broad
range of applications, particularly in the field of medical image segmentation
and operator learning for inverse PDE problems. Yet, the applicability of
these combinations in addressing time-dependent forward problems, real-world
data scenarios, and in situations with sparse data, remain areas yet to be
fully explored.

Several studies, such as the one by Brakenhoff et al. [1, 14], primarily focus
on individual time series when analysing the impact of various hydrological
stressors, including pumping rates, precipitation excess, and river stage
variations, on groundwater levels of individual monitoring wells. While this
approach provides valuable insights, it does not account for spatial
correlations, thereby limiting its use to existing time series or monitoring
wells. Similarly, previous comparisons have been predominantly limited to
specific models like LSTM, CNNs and NARX in the context of groundwater level
forecasting [15], leaving room for broader explorations.

In this paper, we present a comprehensive comparison among models—specifically
U-Net, U-Net integrated with Vision Transformers (U-Net+ViT), and Fourier
Neural Operator (FNO)—for their efficacy in modeling time-dependent forward
and inverse problems in groundwater systems. We test our model extensively on
synthetic datasets, simulating conditions from the Overbetuwe region in the
Netherlands, including sparse data scenarios. We show that both U-Net and
U-Net+ViT are particularly well-suited to these important sparse data
scenarios, with the addition of the Transformer providing enhanced predictive
capability in many cases.

## Methodology

### Example of study site and data

This subsection provides context and rationale for our study via an example
case study based upon the polder region of Overbetuwe in the Netherlands
(Figure 1). This region showcases the characteristic Dutch system of water
management where the area is divided into several polders in a mix of
agriculture, nature, and urban environments. Alongside its sparse data and
heterogeneous soil, these unique characteristics underscore the inherent
complexities of water management in similar settings, making this dataset a
suitable choice for our research. The subsoil is primarily composed of clay
and sandy clay, with soil properties being determined via borehole and cone
penetration tests. The study area features numerous observation wells for
monitoring groundwater heads while well fields (indicated as groundwater usage
facilities in the figure) are utilized for the extraction of drinking water.
The work of Brakenhoff et al. [1] considers a dataset consisting of 250 head
time series, with daily recordings starting from the year 1990 and drawdown
attributed to the extraction from up to four well fields.

Figure 1. A representation of the target study area located in the polder
region of Overbetuwe, Netherlands. [8].

For the purposes of this study, we employ synthetic data to validate the
proposed methodology, with the intention to subsequently apply the validated
method to the real-world data of the Overbetuwe region. Figure 2 represents a
sample of the high-fidelity labeled dataset, which is constructed using the
U.S. Geological Survey (USGS) finite-difference flow model, MODFLOW. The model
is composed of a single-layer representation of a confined aquifer with a
$128\times 128$ grid. The aquifer’s heterogeneity is reflected through varying
horizontal hydraulic conductivity within the bounds $k\in[0.1,0.5]$ m/d. The
hydraulic conductivity fields in our study are created using random fields
which are then thresholded to delineate different classes. A maximum of ten
pumping wells are extracting water with variable rates in the range
$Q\in[0,30]$ m3/d over a simulation period of $T=10$ days. The pumping wells
are located in random locations which vary for each sample. The boundary
conditions are delineated as Dirichlet, with the head equal to zero, mimicking
a polder encircled by ditches where a stable water level is maintained through
a comprehensive network of pumping stations.

The datasets consist of $N_{train}=5000$ training instances and
$N_{test}=1000$ testing instances. To mirror the inherent sparsity of real-
world data, a data selection strategy is adopted for the test dataset. The
locations of the boreholes for estimating the hydraulic conductivity are
chosen following a radial distribution pattern, and a helical pattern is used
for the wells monitoring hydraulic head (Figure 2).

Figure 2. An test sample of the high-fidelity dataset constructed using
MODFLOW. The first column showcases the heterogeneous hydraulic conductivity,
the second column presents the randomly positioned pumping wells, and the
third column depicts the resulting hydraulic head. The red dots (column 1 and
3) represent a selection of data points—following radial and helical patterns,
respectively—emulating the sparse observations in real-world scenarios.

### Architectures

The architectures of the three models under comparison in this study encompass
the U-Net structure, a U-Net with attention mechanism in the bottleneck, and
the Fourier neural operator (FNO).

The U-Net architecture is designed with an encoder-decoder structure where the
decoder receives the upsampled feature map, which is then concatenated with
the corresponding feature map from the encoder through a skip connection.
Detailed diagrams of the the U-Net encoder and decoder can be found in Figures
8 and 8 in Appendix A. The encoder consists of three bottleneck blocks, where
each block utilizes three layers of Conv2d, Instance Normalization, and GELU
activation to extract spatial features. These blocks increase the number of
channels by a factor of 2 and perform downsampling with a stride of 2. The
decoder is composed of a series of upsampling blocks, where each block
consists of a bilinear upsampling operation (Upsample), followed by a double
convolution operation. Each convolution within the decoderis followed by
Instance Normalization and GELU activation function. The bottleneck consists
on a single convolutional layer. In the time-dependent scenario, the time
series data of the historical pumping rates is processed through two layers of
feed-forward neural network (FNN) prior to being concatenated to the input for
the latent space representation (Figure 8).

The second model, here called UNet+ViT, employs the Vision Transformer (ViT)
[11], in the latent space representation of the U-Net, as per implementation
of TransUNet [6] and ViTO [5]. The input is tokenized into a sequence of
flattened 2D patches, each of size 1×1. Positional information is retained by
employing trainable convolutional projection to learn and add specific
position embeddings to the patch embeddings. The structure of the Transformer
includes $L$ blocks, with each block comprising Multi-Head Attention (MSA) and
FNN. This configuration involves the use of 2 blocks, each with 2 Multihead
Self-Attentions, and a FNN composed of 128 neurons. For a more detailed
visualization of the Vision Transformer, attention block, and multihead
attention, please refer to Appendix A, Figure 8.

The Fourier neural operator (FNO) [4] model leverages the fast Fourier
Transform to parameterize the integral kernel directly in the Fourier space.
The implementation of FNO for the $2D$ Darcy Flow problem as presented in [4]
is followed in this study. The total amount of parameters of FNO corresponds
to 2.38 million, that is 15 times more than UNet+ViT (151k) and 17 times more
than UNet (137k).

## Results

### Forward problem with sparse observations

This section presents the prediction of the hydraulic head at sparse
monitoring wells after a constant 10-day pumping period under two different
training conditions. We employ distinct sampling strategies for both input and
output data in our methodology. Our training data is sampled from a regular
quadratic grid, while for testing we have explored other arrangements, such as
radial and helical, to understand their potential impact on the prediction
performance. In the first scenario, training is conducted using sparse data,
with a spacing of 20 grid points for the input hydraulic conductivity field
and a spacing of 8 for the output hydraulic head. Testing is then carried out
on sparse data points, following the radial and helical patterns delineated in
subsection Example of study site and data. The resulting root mean square
error (RMSE) is found to be $5.2\times 10^{-2}$, $3.5\times 10^{-2}$ and
$8.1\times 10^{-2}$ for the vanilla U-Net, the UNet+ViT models and FNO
respectively. These results underline the superior performance of the UNet+ViT
model in handling sparse data, exhibiting a lower RMSE compared to both the
vanilla U-Net and the FNO models.

In contrast, when training is performed using the entire field and testing on
the same sparse dataset, the error marginally escalates to $3.9\times 10^{-1}$
for FNO, $3.8\times 10^{-1}$ for UNet and $3.6\times 10^{-1}$ for UNet+ViT
model. This outcome is anticipated considering the training set exhibits
sparsity in the first scenario, but not in the latter. Additionally, Figure 3
displays the prediction over the entire domain, resulting in a lower RMSE of
$1.0\times 10^{-2}$ for FNO, $1.7\times 10^{-2}$ and $1.9\times 10^{-2}$ for
the vanilla U-Net and UNet+ViT models, respectively. The FNO model, while
superior when dealing with full data, exhibits the highest predictive error
under sparse data observations. These results highlight the practical
advantages of the U-Net and especially UNet+ViT model in real-world scenarios
for which data sparsity is common.

It should be noted that traditional simpler neural networks and other machine
learning techniques may not provide adequate solutions for this specific
problem. This assertion is backed by a comparison of the results from a fully
connected neural network, a linear regression model and a random forest,
detailed in Appendix Appendix B. Despite the substantial number of trainable
parameters, reaching 51.17 million, inherent to the fully connected neural
network and the application of linear regression and random forest, these
methods significantly underperform compared to the U-Net, the UNet+ViT models,
and FNO.

Figure 3. Predictions over the entire domain are shown, with the UNet+ViT
model exhibiting a RMSE of $1.9\times 10^{-2}$. The figure demonstrates the
model’s ability to accurately capture the spatial distribution of the
hydraulic head across the entire domain.

### Identification of pumping wells

In this section, we focus on an inverse problem: specifically the
identification of pumping wells. This task requires determining the locations
and rates of pumping wells based on the observed hydraulic heads. Throughout
these experiments, we employ a single hydraulic conductivity field, which,
while spatially varying, remains identical across all samples within the
dataset.

In evaluating the performance of our models, we use both RMSE and accuracy.
The RMSE calculates the average difference between the true and the predicted
value for each pump location in the test dataset, giving a quantitative
measure of the prediction error. Complementing this, the accuracy was
determined by counting the proportion of correct pump predictions, where a
prediction is considered correct if the predicted and actual pump locations
align. This gives a sense of how often the model correctly identifies the
location of pumps.

The U-Net model performs optimally, achieving an RMSE of $5.6\times 10^{-2}$.
Interestingly, the integration of the Vision Transformer with the U-Net model
does not confer any additional precision in this scenario, yielding a near
RMSE of $6.1\times 10^{-2}$. The FNO model exhibits a higher RMSE of
$1.1\times 10^{-1}$, indicating a somewhat lower accuracy in identifying the
pumping well locations.

To visually illustrate these results, Figure 4 presents a test sample using
the U-Net + ViT model. It demonstrates an accuracy of 93% in locating the
pumps, calculated across the entire test dataset. The figure visualizes the
model’s ability to accurately identify the positions and the pumping rate of
the wells. In comparison, the FNO model achieved a notably lower detection
accuracy of 79% in the same task.

Figure 4. Test sample results for the identification of pumping wells using
the U-Net + ViT model. The model accurately identifies both the positions of
the pumping wells with an accuracy of 93 % across the entire test dataset.

### Example results for time series data

This section unveils the results achieved from the analysis of time series
data, starting with a simplified scenario, for which the inputs are the
varying hydraulic conductivity field and the pumping rate of a single pump
which varies over a 10-day simulation period. Results are evaluated in terms
of root mean square error (RMSE) with a focus on the comparison of different
configurations of the U-Net architecture with transformers. Figure 5 presents
a comparison of results over 5 time frames for the U-Net with the Vision
Transformer under autoregressive testing conditions.

Figure 5. Depiction of autoregressive testing results from the UNet+ViT model
over the final five frames of a ten-frame test sequence. This sequence
simulates a single variable-rate pump operating over a ten-day period within a
region characterized by a diverse hydraulic conductivity field. The top row
presents the ground truth, while the lower row displays the predicted
outcomes. Contour lines in the images represent groundwater levels, which are
color-coded for enhanced visual clarity. Each successive frame is employed as
input to the U-Net-Vision Transformer system, aiding in the prediction of the
subsequent frame.

The RMSE for each method was calculated to quantify the models’ performance.
The U-Net architecture alone yielded an RMSE of $1.79\times 10^{-2}$. When
supplemented with a Vision Transformer, consisting of 2 attention blocks and 2
heads, the performance improves, registering an RMSE of $1.67\times 10^{-2}$.
However, increasing the complexity of the Vision Transformer to 8 blocks and 8
heads did not further improve the performance, instead, it led to a slight
degradation in the RMSE ($1.77\times 10^{-2}$). Adding an Axial Transformer
[17] to the U-Net architecture also did not enhance the performance, yielding
an RMSE of $1.83\times 10^{-2}$. These results suggest that while adding a
Vision Transformer to the U-Net architecture leads to performance improvement,
increasing the complexity of the latent space does not necessarily do so.

## Conclusion

This paper explores and evaluates the capabilities of different machine
learning models, with a particular focus on U-Net, U-Net integrated with
Vision Transformers (ViT), and Fourier Neural Operator (FNO), in the context
of predicting hydraulic head in groundwater studies.

Our analysis and testing, conducted on synthetic datasets designed to simulate
the conditions from the Overbetuwe region in the Netherlands and including
scenarios with sparse data, firmly establish that both U-Net and U-Net + ViT
models are particularly adept at dealing with such tasks. Importantly, these
models are also preferred due to their fewer requisite parameters.

Specifically, in the case of sparse observation scenarios, the vanilla U-Net
and the U-Net + ViT models outperformed the FNO model. In particular the
performance of the UNet+ViT model was superior when handling sparse data,
highlighting the potential of the model in real-world applications, where data
scarcity is a common issue. The U-Net model demonstrated optimal performance
in identifying pumping wells. Interestingly, the integration of the Vision
Transformer with the U-Net model did not confer any additional accuracy in
this scenario. As for the analysis of time series data, supplementing the
U-Net architecture with a Vision Transformer improved the model performance,
recording an RMSE of $1.67\times 10^{-2}$ compare to $1.79\times 10^{-2}$ of
the vanilla U-Net. However, increasing the complexity of the Vision
Transformer did not further enhance the model performance, indicating that a
more complex architecture does not necessarily yield better results.

Future research will involve applying this validated methodology to real-world
data, beginning with the Overbetuwe region in the Netherlands. This will offer
an opportunity to further validate and refine the model, accounting for the
sparsity and uncertainties inherent in real-world data.

## Broader impact

The implications of this research span a wide range of potential societal
impacts, with a primary focus on improving the efficiency and reliability of
groundwater level forecasting. Given that groundwater is a crucial resource
for approximately 2.5 billion people worldwide, fulfilling their daily water
needs, and a significant source of global irrigation water, the importance of
reliable forecasts cannot be overstated. Our work, through enhancing the
performance of groundwater numerical models, offers an opportunity to
revolutionize the management and distribution of this vital resource. By
providing more accurate and data-efficient predictions, we can aid in the
formulation of informed and sustainable water management strategies. This is
particularly crucial considering the pressing challenges of population growth
and climate change.

## Acknowledgements

This work was carried out with support of the Leeds-York-Hull Natural
Environment Research Council (NERC) Doctoral Training Partnership (DTP)
Panorama under grant NE/S007458/1. Our sincere appreciation is extended to
Professor Karniadakis of Brown University. The financial assistance provided
by the Leeds Institute of Fluid Dynamics and Deltares, which made possible the
research visit to Brown University, is also gratefully acknowledged. Lastly,
we would like to express our gratitude to the reviewers. Their critiques and
suggestions have greatly enhanced the overall clarity of our work.

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## Appendix A

This appendix provides detailed diagrams of the model structures.

Figure 6. U-Net Encoder: The encoder consists of 3 bottleneck blocks, each
comprising 3 convolutional blocks (Conv2d, Instance Normalization, and GELU).
These blocks increase the channel dimension by a factor of 2 and perform
downsampling with a stride of 2, extracting spatial features. In time-
dependent scenarios, the pumping rate time series undergoes processing through
a two-layer feed-forward neural network. The resulting processed data is then
concatenated with the input, creating the latent space representation.

Figure 7. U-Net Decoder: The decoder is composed of a series of upsampling
blocks, where each block consists of a bilinear upsampling operation followed
by a double convolution operation.

Figure 8. The Vision Transformer (ViT) and its components: The input is
tokenized into a sequence of flattened 2D patches, each of size 1×1. The
structure of the Transformer encoder includes 2 attention blocks, each
containing Multi-Head Attention block and fully connected layers.

## Appendix B

This appendix sets out to examine whether simpler machine learning models,
specifically a fully connected neural network, a linear regression model, and
a Random Forest model, can achieve the same level of accuracy as more advanced
models like the U-Net, the UNet+ViT models, and FNO in predicting groundwater
levels.

The particular Random Forest model tested here used 30 estimators. The fully
connected neural network, employed for this comparison, comprises three hidden
layers, each containing 1000 nodes and using ReLU activation functions. The
model holds an impressive count of 51.17 million trainable parameters.

Unfortunately, none of the models was able to predict accurately the
groundwater levels neither capturing the location of the wells. Specifically,
the fully connected neural network and the linear regression model yielded
high RMSEs of $1.17\times 10^{-1}$ and $1.24\times 10^{-1}$, respectively. The
Random Forest model fared slightly better, achieving a lower RMSE of
$1.02\times 10^{-1}$, but it still fell short of the U-Net, the UNet+ViT
models, and FNO.

Figure 9 visually contrasts the predictions of these simpler models gainst the
ground truth. Their significant underperformance becomes evident when compared
to more sophisticated models. For a comparison of these results with accurate
outcomes produced by the UNet+ViT model, the reader is directed to Figure 3.

Figure 9. These figures illustrate the testing results using a fully connected
neural network, a linear regression model, and a Random Forest model. The top
row of each figure presents the ground truth, while the remaining rows display
the predicted outcomes generated by each respective model. Contour lines
represent groundwater levels, color-coded for clarity.


# Morphological stability and identity in self-organizing systems

Igor Strozzi

###### Abstract

In this essay I aim to investigate and discuss the process through which
bundles of things “self-organize” into other things. In particular, I engage
in such investigation by trying to apply a framework of analysis of structural
stability of morphological forms to the context of a continuous cellular
automata known as Lenia, which displays great diversity of distinguishable,
individuated complex subsystems, who behave autonomously and in lifelike
manner (such subsystems are often known as solitons). I do so in order to
suggest a path to the development of methods to formally distinguish said
subsystems from their environment.

##  1 Introduction

It could be said that this essay aims to deliberate over the following
motivating question: what makes a creature in Lenia a creature, instead of
just another splodge?

I will not delve deeply into what is Lenia Chan (2019), and I will also
consider it only in its simplest form. But, to back up the question asked, I
will provide a sketch of its definition. For the time being, let us just
consider, rather informally, it to be a dynamical system defined by a
rectangular region $\mathcal{R}$ of $\mathbb{R}^{2}$ with toroidal boundary
conditions, to which we ascribe, for each $\mathbf{x}\in\mathcal{R}$, a real
number $\sigma_{\mathbf{x}}(t)\in[0,1]$, called the state of $\mathbf{x}$ at
the instant $t$. The dynamics are given by a local update rule $\phi$, that
maps the state $\sigma_{\mathbf{x}}(t)$ of a point $\mathbf{x}$ to its state
in the next instant, with dependence on the states found in a neighborhood of
$\mathbf{x}$. At each instant, all states are synchronously updated by the
application of $\phi$. Both the neighborhoods and the mapping $\phi$ are
considered to be constant in time.

Pretty much all of the above can – and is, see Chan (2019, 2020) – be
generalized, but it suffices (rather, it is more adequate) to the purpose of
this work to consider of all cases the simplest. Indeed, the question stated
as motivating is too hard already. I do not even faintly hope to answer it, at
least not in such a brief essay. My goal, if anything more than briefly
digress over millennial issues (such as problems concerning the metaphysics of
identity) and other elusive concepts, such as that of self-organization, is to
bring a deep and promising work Thom (2018) out of its apparent oblivion, in
thrust that it is still actual and can provide useful insight into problems
such as the question asked.

##  2 Some cautionary observations

Before properly starting the proposed discussion, I should state that I do not
believe that there exists an ontological criterion to distinguish between
entities, or objects. It thus follows that there is no purely objective way to
distinguish between an object and its environment and, as a matter of fact,
that the very notion of object is ontologically disputable. This position,
evidently, goes against experience and, in fact, is quite inconvenient for
probably every practical and most theoretical problems we as humans face. It
does not, nonetheless, render this whole discussion pointless, simply because
at an epistemological/phenomenological level, individuation is possible.

The relevance of these observations lie in the fact that this position on the
fundamental nature of concepts such as identity and individual will
necessarily permeate and underlie the development of this work. It should be
said that such skepticism, although, as far as I know, unconventional
elsewhere, falls not too faraway from the generally accepted idea that the
delimitation of a system’s – or, at least, of its description – boundaries
inherently depends on the observer’s purpose and convenience.

A perhaps stronger claim can be made while still sounding reasonable: there is
a plurality – for physical systems, we would guess that uncountably many – of
equivalent descriptions of a same system. It seems to go implicitly with
either claims that no such description is identical to the system it aims to
describe. It also seems to be the case that the existence and individuation of
such a system is presupposed. An observation to be made is that if, in the
former claim, we consider the delimitation of the system itself – instead of
its description, or a model for it – to be dependent on the observer, the
assumption that it exists and can be individuated is contradictory. That in
turn implies that we can only talk about systems in an
epistemological/phenomenological level, despite assuming – or not – that they
exist as noumena.

##  3 Is self-organization actually a well-defined concept?

Even to these days, more than 70 years after what seems to be the first
appearance of the term “self-organizing system” (SOS) Gershenson et al.
(2020); Banzhaf (2009), introduced in Ashby (1947), the concept of self-
organization still remains a somewhat elusive idea, lacking a formal
Gershenson et al. (2020), unified definition and, much for this very reason,
encompassing a variety of partial, unsystematic and sometimes even
contradictory – in the sense that two different self-organized systems might
display mutually exclusive characteristics, e.g., one being taken as
deterministic and the other as stochastic, and that can lead to contradictory
outlines of the properties inherent to SOSs – different
characteristics/characterizations.

An interesting question – among others – raised in Collier (2004) is: what is
this “self” that organizes? An answer, seemingly given by Maturana Collier
(2004), is that, as it looks evident, a system cannot organize itself: its
selfness arises as it comes to being, through the spontaneous organization of
its parts. Another answer, given by Collier at the same work, is that it is
reasonable to talk about self-organization (without that being a misnomer)
since the relevant factors to the emergence of the system, and thus the
“self”, are internal (to the system?), and not modulated by external
influence. I do not feel satisfied with either of these answers, but, to me,
the latter is clearly more problematic, although the former seems to imply
that “self-organization” is in fact an oxymoron, at best – which I also do not
think to be the case.

I do not intend to solve the above dispute or answer if there is a well-
defined self that organizes itself, but I will offer a pragmatical and
temporary suggestion. A more deeper answer, I believe, can be found
investigating in which ways does it make sense to talk about “selfs” and
“systems”, as suggested in the last section. As for now, let us just discuss
what I consider a more convenient – to the present discussion – definition of
a system.

##  4 The “selfhood” of systems

As mentioned earlier, there are some widely known facts about systems that
pose, one could say, intrinsic restrictions in the grail of finding clear,
objective methods of individuation. An immediate objection is the fact that
every known “concrete” system is open, what forces any description to be no
more than that. Hence a scientific-realist position, in the context of systems
sciences, seems inherently ill-suited.

Of higher importance still is the matter of equivalence of descriptions
between “equal” systems, which rises as a major hindrance. One “same” system
can be seem under a variety of levels of description “same” system can be seem
under a variety of levels of description Koestler (2013, 1978); Haken (2006);
Mesarovic (1970), which usually lead to distinct composing entities (parts),
and different dynamics and relations between them. If selfhood is inherent to
such systen, and not some contingent property born from our point-of-view, as
observers, and subsequent description, then, in this sense, such selfhood, or,
rather, its characterization, surelly cannot rely on pretty much anything
internal to the system – i.e., components; dynamics and relations between
them. But then, what else remains to provide such characterization?

In various references 111Most of our references discuss these issues. For more
comprehensive views see Fieguth (2017); Mobus and Kalton (2015); Takahashi and
Takahara (2010); Haken (2006) the above problems are considered and one could,
as it seems, reduce them under two kinds of equivalence categories – I am
intentionally not using the more technical term “classes” – which I will call
vertical and horizontal. The former category is related to the equivalence
between levels of description “same” system can be seem under a variety of
levels of description Haken (2006); Mesarovic (1970). A system of molecules is
also a system of atoms. The latter relate sto the fact that, at least for
real/concrete systems, what separates a system from its environment is
oftentimes unclear. What seems to be the case – and I risk saying that this is
always the case – is that systems are nested within other systems. Said system
of molecules ($M$) is also a system of atoms $A$, each of which is subsystem
$A_{i}$ relatively to $M$. Within $M$, there is an environment wherein the
$A_{i}$ interact, albeit simple such environment can possibly be. This is
nothing new, and various works concern such matters (cf. Walloth (2016);
Koestler (2013)). Yet, in a work still under development – from which I will
borrow some ideas – I consider such matters under the concept of holons and
holarchies, aiming to properly define these two modes of equivalence and, what
seems to me even more elusive, the way they relate to each other.

Indeed, it seems an ubiquitous fact that vertical and horizontal equivalences
are deeply intertwined. Considering a “spatial resolution” parameter $\alpha$,
through which variation one obtains a continuum of levels of description of a
“same” system, the horizontal equivalence problem naturally arises.

For an example, consider a metallic sphere $S$. At a macroscopic level – order
of magnitude of $10^{\alpha},\ \alpha=1$ meter –, one could reasonably say
that the ball is clearly individuated from its environment, having a well-
defined surface which separes it from its surroundings. Were one vary $\alpha$
to the characteristic length of electron interactions – roughly $\alpha=-15$,
one would find a rather distinct picture. The surface that once isolated the
$S$ so conspicuously, now (heavily?) interacts with the environment.

We could, naively, say that at the very least some electrons leave the sphere,
whereas others become part of it. Even though effects resulting from these
dynamics might be negligible at macroscale, there are other interactions at
such microscales which are not.

In any case, such electrons are themselves substructures/subsystems of $S$, at
least until… they are no more. Conversely, free electrons in the environment
are not part of the system until captured by it. One could imagine, thus, that
there is, in fact, a diffuse boundary, a gradient, governing the (in-out)flux
of electrons through $S$’s “surface”, since it is not entirely clear or
determinable when an electron becomes or stops being part of $S$. To sum it
up, how “closed” a system appears to be seems, frequently, to depend on the
level of description one is using.

##  5 Some suggestions as to how define a system

We will proceed very directly to the definitions, and try to explain them as
succintly as possible. An auxiliary concept:

######  Definition 5.1 (Valued relation).

Let $S=\\{S_{i}\\}_{i\in I}$ be a family of sets and $R$ a $n$-ary relation
over $S$, that is, $R\subseteq\prod_{i\in I}S_{i}$. Let $V$ be another set,
said the set of values. If $\mathrm{v}$ is a function such that
$\mathrm{v}:R\rightarrow V$, we say that $\mathrm{v}(R)=\\{\mathrm{v}(r)\in V\
|\ r\in R\\}$, which we shall also denote by $(R,V,\mathrm{v})$, is an $n$-ary
$\mathrm{v}$-valued relation over $S$, or a $\mathrm{v}$-valuation of $R$,
and, naturally, say that $\mathrm{v}$ is a valuation of $R$. If
$R_{\mathrm{v}}=(R,V,\mathrm{v})$, we say that $R$ is the underlying relation
of $R_{\mathrm{v}}$. $\Diamondblack$

In the above definition, we consider the graph of $\mathrm{v}$ as the valued
relation, which is very natural since, well, graphs of functions are indeed
relations. Thus a $n$-ary valued relation over a family $\\{S_{i}\\}$ is
nothing else than a $(n+1)$-ary relation with some particularities: mainly,
that it is functional in $V$, but it will be almost always the case here that
the set $V$ is very different in nature from the $S_{i}$.

We will usually talk about relations defined over the elements of a system,
which will be considered as sets themselves, and the values these relations
have will usually be numbers. The underlying relations account for the
‘structure of interactions’, whereas the valuations denote the characteristics
these interactions have, the most remarkable one probably being their
strength, or intensity. ‘Structure of interactions with values’ could also be
thought of, very naturally, as a graph with weighted edges.

As for the concept of system, there are some comments to be made before we
provide our definition. The usual “minimal” Mesarovic and Takahara (1989) set-
theoretical definition considers an abstract system to be a relation
$S\subseteq\prod_{i}V_{i}$, where the $V_{i}$ represent the objects which are
parts of $S$, in a sense that were one to talk about ‘elements’ of a system,
these would not necessarily be the $V_{i}$. We shall similarly conceive a
system in a set-theoretical manner, but starting with a single set, which we
will call the underlying set of the system. This set, though, is not to be
understood as the set of ‘objects’, or ‘elements’, of the system. Rather, such
elements will be disjoint subsets of the underlying set.

The above choices are in consonance with our inclination towards spatial
systems and the idea of “leaving some space” for the environment containing
the actual system: under such interpretation, the underlying set would
correspond to a spatial region, and the system’s elements would correspond to
subregions with “properties” – the system itself being the whole of these
elements, their properties and relations between them, not including the
region itself, although fundamentally depending on it.

We should further add that despite the possibility of underlying sets having
elements of arbitrary nature, the systems in which we are interested are those
that arise from the things that are built upon the underlying set, with the
nature of its elements being in general of little importance. This choice is
to emphasize, mainly, the mereological aspects of our understanding of systems
(and the topological ones, which in fact are going to be the most relevant in
the present work). The following definition is also intented as to provide
suitable ways of allowing one to use tools that benefits from defining
hierarchies based on set-inclusion, rather than set-membership (such as
measure-theoretical and topological tools).

######  Definition 5.2 (System).

Let $S$ be a set with cardinality $|S|\geq 2$. Consider a partition
$\mathfrak{P}$ of $S$, and a subset $\Gamma_{\mathfrak{P}}$ of $\mathfrak{P}$,
with $|\Gamma_{\mathfrak{P}}|=C$. Define $\mathbf{R}=\\{R_{k}\ |\
k\in\mathbb{N}_{\leq C}\\}$ where each $R_{k}$ is a set of $n_{k}$ $k$-ary
relations over $\Gamma_{\mathfrak{P}}$, i.e., for any $R_{ki}\in R_{k}$ we
have $R_{ki}\subseteq\Gamma_{\mathfrak{P}}^{k}$ and therefore
$R_{k}=\\{R_{ki}\subseteq\Gamma_{\mathfrak{P}}^{k}\ |\ i\in\mathbb{N}_{\leq
n_{k}}\\}$. Define, now, for each $R_{k}$ and each relation $R_{ki}\in R_{k}$,
a set $P_{ki}$ (possibly empty) of functions $\mathrm{v}_{kij}$ such that, for
each $j$, $\mathrm{v}_{kij}:R_{ki}\rightarrow V_{kij}$, where $V_{kij}$ is the
set of values for the valued relation $\mathrm{v}_{kij}(R_{ki})$. Finally,
define the set $\mathbf{V}=\\{P_{ki}\ |\ k\in\mathbb{N}_{\leq C};\
i\in\mathbb{N}_{\leq n_{k}}\\}$ of sets of valuations. Then a system
$\mathcal{S}$ is a quadruple

$\mathcal{S}=(S,\ \Gamma_{\mathfrak{P}},\ \mathbf{R},\ \mathbf{V}).$

We call the set $S$ the underlying set of $\mathcal{S}$, denoted by
$\mathfrak{u}(\mathcal{S})$. We call $\Gamma_{\mathfrak{P}}$ the set of
elements of $\mathcal{S}$, which we denote by $\mathfrak{e}(\mathcal{S})$. We
call $\mathbf{R}$ the set of relations of $\mathcal{S}$, denoted
$\mathfrak{r}(\mathcal{S})$, and $\mathbf{V}$ the set of valuations of
$\mathcal{S}$, denoted $\mathfrak{v}(\mathcal{S})$. Strictly speaking,
$\mathbf{V}$ is, as said, a set of sets of valuations, since for each relation
in $\mathbf{R}$ we might have multiple distinct valuations; with respect to a
system $\mathcal{S}$, though, we simply call it its set of valuations.

We say that a non-empty set $s\subset S$ is an element of a system
$\mathcal{S}$ iff $s\in\Gamma_{\mathfrak{P}}$, and denote the system-
membership relation by
$\mathrel{\raisebox{0.43057pt}{\ooalign{$\sqsubset$\cr$-$}}}$. Naturally, it
is thus true that:
$s\mathrel{\raisebox{0.43057pt}{\ooalign{$\sqsubset$\cr$-$}}}\mathcal{S}\iff
s\in\Gamma_{\mathfrak{P}}$. $\Diamondblack$

Here, I will consider the particular case where $S=(X,\mathcal{T})$ is a
topological space. With this, we alter definition $5.2$ so that
$\Gamma_{\mathfrak{P}}\subset\mathcal{T};\
\bigcap\Gamma_{\mathfrak{P}}=\emptyset$.

We need two more definitions.

######  Definition 5.3 (Subsystem).

Let $\mathcal{S}=(S,\Gamma_{\mathfrak{P}},\mathbf{R},\mathbf{V})$ be a system.
Consider the set
$\mathfrak{R}=(\bigcup_{k}\mathbf{R})\cup(\bigcup_{i}\bigcup_{k}\mathbf{V})$.
A subsystem
$\mathcal{S}^{\prime}=(S^{\prime},\Gamma_{\mathfrak{P^{\prime}}}^{\prime},\mathbf{R^{\prime}},\mathbf{V^{\prime}})$
of $\mathcal{S}$ is a system such that $S^{\prime}\subseteq S$,
$\bigcup\Gamma_{\mathfrak{P^{\prime}}}^{\prime}\subseteq\bigcup\Gamma_{\mathfrak{P}}$,
and for which there are families of functions $\\{f_{n}\\},\\{g_{m}\\},\
n,m\leq|\Gamma_{\mathfrak{P^{\prime}}}^{\prime}|=C^{\prime}$, such that, for
some $p_{n},q_{n}$,
$f_{n}:\mathfrak{R}^{p_{n}}\rightarrow\Gamma_{\mathfrak{P^{\prime}}}^{\prime
q_{n}}$ and for $p_{m},q_{m}$,
$g_{m}:\mathfrak{R}^{p_{m}}\rightarrow\mathcal{P}(\Gamma_{\mathfrak{P^{\prime}}}^{\prime
q_{m}}\times V_{kij}^{\prime})$, for given sets of values $V_{kij}^{\prime}$,
satisfying:

i) for any element $R^{\prime}$ of $\bigcup_{k}\mathbf{R}^{\prime}$ there is
exactly one $n$ such that for some
$\boldsymbol{\sigma}\in\mathfrak{R}^{p_{n}}$,
$f_{n}(\boldsymbol{\sigma})=R^{\prime}$;

ii) for any element $R^{\prime}$ of $\bigcup_{k}\mathbf{R}^{\prime}$, if there
is a valuation $\mathrm{v}_{kij}^{\prime}$ whose domain
$\mathrm{dom}(\mathrm{v}_{kij}^{\prime})$ is $R^{\prime}$, then there is
exactly one $m$ such that for some
$\boldsymbol{\sigma}\in\mathfrak{R}^{p_{m}}$,
$g_{m}(\boldsymbol{\sigma})=\mathrm{v}_{kij}^{\prime}$. $\Diamondblack$

In good english, what the definition above means is that a subsystem
$\mathcal{S}^{\prime}\subset\mathcal{S}$ has as its underlying set
$S^{\prime}$ a subset of the underlying set $S=\mathfrak{u}(\mathcal{S})$, and
that its relations and valuations depend on the set $\mathfrak{R}$ of all
relations/valuations of $\mathcal{S}$. It is too restrictive to
aprioristically assume basically anything about how relations on a subsystem
relate to relations on its supersystem. In principle, a relation on a
subsystem could depend in arbitrarily complicated ways on relations of its
supersystem. We cannot even guarantee that a relation $R$ of
$\mathcal{S}^{\prime}$ that is mapped from
$\boldsymbol{\sigma}\in\mathfrak{R}$ and has a valuation $\mathrm{v}$ satisfy
something like $g_{m}(\boldsymbol{\sigma})=\mathrm{v}$, i.e., if $R$ comes
from some $\boldsymbol{\sigma}$, its valuations not necessarily depend on the
same $\boldsymbol{\sigma}$.

Indeed, an unvalued relation represents a ‘structure of interactions’ and as
such could be the same for very different valued relations. For instance, two
global coupling parameters can describe very distinct features of a system,
and have very different values, but would share a same unvalued (unary)
relation. It is not unreasonable to consider, then, that, being these features
sufficiently different in quality, their dependence on relations at the
superlevel could be different enough as to render, at the very least,
inconvenient to consider them functions of a same tuple $\boldsymbol{\sigma}$.

Let us now define a supersystem.

A really useful definition of supersystem does not follow all that trivially
from definition 5.3. Conceptually, the reason is that while one can use the
knowledge about a system to infer a characterization of a subsystem, when only
the latter is known it is in general much harder to completely – or, at least,
satisfactorily – describe some of its supersystems. The more heterogeneous the
system, the harder.

More formally, the motive is that the functions that define a subsystem from a
given system do not necessarily have inverses, so that when we define a
subsystem, we not necessarily define how to obtain a corresponding supersystem
(which is not even unique). It is, though, obvious to think, for a given
system $\mathcal{S}^{\prime}$, of a supersystem of it as a system
$\mathcal{S}$ such that $\mathcal{S}^{\prime}\sqsubset\mathcal{S}$. Moreover,
it is reasonable, and more adequate to our purposes, to define a supersystem
for sets of systems, despite the fact that such sets could be singletons. We
define supersystems as follows:

######  Definition 5.4 (Supersystem).

Let $\mathfrak{S}=\\{\mathcal{S}_{i}\\}_{i\in I}$ be a family of systems.
Define, for each $\mathcal{S}_{i}$, the set
$\mathfrak{R}_{i}=(\bigcup\mathfrak{r}(\mathcal{S}_{i}))\cup(\bigcup\bigcup\mathfrak{v}(\mathcal{S}_{i}))$.
Define $\mathfrak{R}=\bigcup_{i\in I}\\{\mathfrak{R}_{i}\\}$. A system
$\mathcal{S}$ is said a supersystem of each $\mathcal{S}_{i}$ (and, by
extension, of $\mathfrak{S}$) if it satisfies the following:

i)
$\bigcup\limits_{s\in\mathfrak{S}}\mathfrak{u}(s)\subseteq\mathfrak{u}(\mathcal{S})$;

ii)
$\bigcup\limits_{s\in\mathfrak{S}}\mathfrak{e}(s)\subseteq\mathfrak{e}(\mathcal{S})$;

iii) for each $R\in\bigcup\mathfrak{r}(\mathcal{S})$, there is a natural
number $p\leq|\mathfrak{R}|$ and a function $f$ such that, for some
$\boldsymbol{\sigma}\in\mathfrak{R}^{p}$, $R=f(\boldsymbol{\sigma})$;

iv) for each $V\in\bigcup\bigcup\mathfrak{v}(\mathcal{S})$, there is a natural
number $q\leq|\mathfrak{R}|$ and a function $g$ such that, for some
$\boldsymbol{\sigma}\in\mathfrak{R}^{q}$, $V=g(\boldsymbol{\sigma})$.
$\Diamondblack$

Lastly, we define a superlevel of a system.

######  Definition 5.5 (Superlevel (of a) system).

Let $\mathcal{S}$ be a system. We say that another system, $\mathcal{S}^{+}$,
is on the (immediate) superlevel of $\mathcal{S}$ if there is a family
$\\{S_{i}\\}$ of subsystems of $\mathcal{S}$ such that
$\mathcal{S}\sqsubset\mathcal{S}^{+}$, i.e., $\mathcal{S}^{+}$ satisfies the
conditions given at definition 5.4. We denote the immediate superlevel of a
system $\mathcal{S}$ by ${\curlywedgeuparrow}{\mathcal{S}}$ or, sometimes, by
$\mathcal{S}^{+}$. $\Diamondblack$

All of that put, we can finally say that self-organization produces
individuation through the supervenience of a supersystem, in a superlevel,
over its composing subsystems, which are, themselves, selves. Thus we answer
the problem of individuation by stating that it tends to, or perhaps
necessarily do so, incur in the production of an infinitely descending chain
of “subselves”.

##  6 Forms and structural stability

In this section, I will define, exactly as in Thom (2018), the ideas of form
and structural stability.

######  Definition 6.1 (Form).

Let $E$ be a topological space and $G$ a group (or pseudogroup) acting on $E$.
A $G$-form is defined as an equivalence class of closed sets of $E$ modulo the
action of $G$.

To put the above into context, consider $E$ to be
$\mathbf{S}=\mathbb{R}^{2}\times[0,1]$ with the usual euclidean norm (which we
will denote by $\|.\|$). Consider $\tau$ to denote the set of open sets of
$\mathbb{R}^{2}$. Let $G$ be a pseudogroup with homeomorphisms
$\Gamma=\\{\gamma:s_{1}\mapsto s_{2}\ |\ s_{1},s_{2}\in\tau\\}$. Returning to
Lenia, we have that, for each configuration $A^{t}:\mathbb{R}^{2}\mapsto[0,1]$
we can consider a point in $E$ (cf. Chan (2019)). Such a point corresponds to
a configuration. I believe that Thom stated the definition considering $G$ to
be a group – instead of a pseudogroup –, since otherwise the equivalence class
appears to be ill-defined (what would be “an equivalence class of closed sets
of $E$ modulo the action of $G$” if $G$ were a pseudogroup?).

This observation notwithstanding, let us now try to understand what
intuitively this definition means. Considering that $G$ is a pseudogroup and
defines a quotient space of $E$ (the orbit space), a $G$-form can be seen as
given by the equivalence relation $\sim_{G}$ that identifies open sets in the
configuration space. What one has, roughly speaking, is that open sets of
different, but neighboring, configurations are mapped onto the same orbits
through the configuration space. That gives a way to talk about “fuzzy
boundaries” for regions/subsystems in respect to their evolution in time.

######  Definition 6.2 (Structural stability).

A $G$-form $A$ is said structurally stable if any form $B$ sufficiently close
to $A$ in $E$ is $G$-equivalent to $A$. Equivalently, a $G$-equivalence class
$F$ defines a structurally stable form if and only if the set of points in $E$
pertaining to $F$ is open.

The above definition looks a bit confusing, since a form $B$, without respect
to any group or pseudogroup, is hitherto undefined. I will consider that $B$
is defined with respect to any other suitable (pseudo)group, which seems the
most obvious thing to do. In this sense, we can talk about arbitrary forms and
define a specific $G$-form $A$, in the case we are considering, to be
structurally stable if there is some $\varepsilon_{0}\in\mathbb{R}_{>0}$ such
that for any other $\varepsilon\leq\varepsilon_{0}$ it holds true that if

$\sup_{c\in A\cup B}|\inf_{a\in A}\|c-a\|-\inf_{b\in B}\|c-b\||<\varepsilon$

then $a\sim_{G}b,\ a\in A,\ b\in B$.

##  7 Morphological stability as an identity criterion?

Finally, let us consider the following:

  1. 1.

Creatures in Lenia correspond to regions in $\mathbb{R}^{2}$, the regions they
occupy;

  2. 2.

These regions can be divided infinitely and arbitrarily. One can reasonably
assume that, in some random initialization that yields a creature, at some
instant, it is possible to define a region $\mathcal{R}\subset\mathbb{R}^{2}$
that will contain the creature and some parts of such region that will, by the
very nature of the rules that define Lenia, interact in an autopoietic
fashion, i.e., so as to dynamically sustain a set of relations invariant in
time Maturana and Varela (1980);

  3. 3.

Regions as such correspond to open sets of a point in the configuration space
$\mathbb{R}^{2}\times[0,1]$;

  4. 4.

For a suitable choice of a pseudogroup $G$, the configuration space is
partitioned precisely in the orbit space that makes said
regions/creatures/systems structurally stable, hence morphologically stable;

  5. 5.

It thus individuates such creature, despite allowing fuzzy boundaries, since
we can disturb it without breaking its stability.

##  8 Conclusions

To wrap it up, despite the inherent difficulties in defining boundaries for
objects, entities or systems, and the stated skepticism in the possibility of
an ontological criterion of discernibility, we aimed to show some tools given
at Thom (2018) that might help is in such distinctions. By their very nature –
i.e., since the choice of the (pseudo)group $G$ is arbitrary –, they actually
correspond to our view that distinctions of such nature are inherently
epistemological.

The idea of self-organization was briefly discussed and, again, problems, that
I believe related to the elusive nature of the concept of identity, concerning
the discernibility of entities. A simple modification on the definition of
system given by Mesarovic and Takahara (1989) was proposed, with intent of
suggesting a path out of the issue posed.

The concepts of form and structural stability, as found in Thom (2018), were
discussed and an attempt to apply them to Lenia’s case was made. The relations
between all previous sections were outlined, at last.

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