# Nonlinear System Identification of Swarm of UAVs Using Deep Learning Methods

Saman Yazdannik  MSc student
Department of Aerospace Engineering
K. N Toosi University of Technology
Email<EMAIL_ADDRESS>Morteza Tayefi  Assistant Professor
Department of Aerospace Engineering
K. N Toosi University of Technology
Email<EMAIL_ADDRESS>Mojtaba Farrokh  Associate Professor
Department of Aerospace Engineering
K. N Toosi University of Technology
Email<EMAIL_ADDRESS>
###### Abstract

This study designs and evaluates multiple nonlinear system identification
techniques for modeling the UAV swarm system in planar space. learning methods
such as RNNs, CNNs, and Neural ODE are explored and compared. The objective is
to forecast future swarm trajectories by accurately approximating the
nonlinear dynamics of the swarm model. The modeling process is performed using
both transient and steady-state data from swarm simulations. Results show that
the combination of Neural ODE with a well-trained model using transient data
is robust for varying initial conditions and outperforms other learning
methods in accurately predicting swarm stability.

##  1 Introduction

The investigation and modeling of the interactions and dynamics of animal and
robot swarms is a longstanding area of interest in academia. This area falls
under the discipline of nonlinear System Identification (SID). Swarm dynamics
are often nonlinear and governed by Ordinary Differential Equations (ODEs).
Several data-driven SID methods exist, some of which will be discussed in the
related work section. Multiple swarm models exist and there is no unified
model that governs all swarm systems. This study focuses on a specific type of
swarm model from a particular class, with the general form described as.

$\displaystyle\dot{r_{i}}=v_{i}(t)$ (1)
$\displaystyle\dot{v_{i}}=f(v_{i}(t))-\sum_{j=1,i\neq
j}^{N}g(r_{i}(t),r_{j}(t))+\eta_{i}(t)$ (2)

The swarm model under examination is governed by a second-order differential
equation, where swarm behavior arises from acceleration ($v_{i}$) as outlined
in Equation 2. It consists of three components: the intrinsic dynamics of an
agent is dependent on its velocity, the interaction dynamics is a sum of
functions of the agent’s position and the positions of other agents, and model
noise, typically assumed to be Gaussian white noise.

Models of this form can exhibit various steady-state behaviors including
flocking, milling, and aggregation. Given time series position and velocity
data obtained from observing a swarm at a fixed sampling rate and an unknown
governing equation, the goal is to find an approximate function that
accurately predicts the trajectories for any arbitrary initial conditions.

SID methods can be grouped into two categories. Given the dynamical process
$\dot{x}=f(x(t))$ and the initial condition $x_{0}$, the first class of
methods aims to directly predict future trajectory $x$, or to approximate
$x_{0}+\int_{0}^{T}f(x(t))dt$, giving $x(T)$ for any T based on the initial
state at $t=0$. The second class of SID methods approximates $f(x(t))$ and
then uses an ODE solver to integrate numerically and derive future $x(t)$
trajectories based on the initial state at $t=0$. [RNN, CNN and MLP] belong to
the first class while Neural ODE belongs to the second class, as evaluated in
this study.

Due to inherent model noise, perfect prediction of all agents’ future
trajectories in a swarm is not feasible. The objective of this study is to
identify models that accurately capture overall swarm behavior. A model is
considered successful if it demonstrates correct convergence to the actual
swarm’s steady state and is robust to varying initial conditions. Experiments
and metrics are presented to evaluate both criteria.

This study utilizes time-series data from simulations of a swarm model using a
basic ODE solver based on Euler’s method. The swarm’s trajectory is divided
into two phases: transient and steady state. With randomly initialized
positions and velocities, a swarm initially enters the transient state before
settling into the steady state. Results of training on data from both phases
are presented. Time-series data prompted the consideration of MLPs, RNNs, and
CNNs as initial models for this study.

##  2 Modeling Swarm Dynamics in Nonlinear Systems

We concentrate on a single swarm model from the previously mentioned swarm
category.

$\displaystyle\dot{r_{i}}=v_{i}$ (3)
$\displaystyle\dot{v_{i}}=(1-|v_{i}|^{2})v_{i}-\frac{a}{N}\sum_{j=1,i\neq
j}^{N}g(r_{i}(t),r_{j}(t-\tau))+\eta_{i}(t)$ (4)

In the present study, the intrinsic dynamics dictate that each agent
experiences an acceleration or deceleration, leading to a constant velocity of
1 as time approaches infinity. The interaction dynamics cause each agent to
accelerate towards the mean position of all other agents at a rate
proportional to the coupling strength (a). It is assumed that Gaussian white
noise is equivalent for all agents.

In this swarm model, three forms of stability are observed: ”milling”,
”rotation”, and ”flocking”. The first stability, ”milling”, is achieved under
conditions of zero time lag, low noise, and an optimal coupling strength. The
agents ultimately converge to a rough circular pattern around the mean field
with a constant radius, resulting in a ”milling” effect. This stability can
manifest as all agents moving in a single direction or a mixture of clockwise
and counter-clockwise movement. The second stability, ”rotation”, occurs when
a significant time delay $(\tau)$ is introduced. The agents will eventually
aggregate and rotate in a circle with a fixed radius. It is worth noting that
simulations using Euler’s method with large time steps are equivalent to
introducing time delay to the swarm system. Our experiments show that, under
specific model parameters, a swarm exhibiting ”milling” stability can
transition to ”rotating” stability as the simulation time step is increased.
The third stability, ”flocking”, emerges with the introduction of significant
noise. The agents will all flock in a single direction. This study focuses on
the learning of systems that exhibit the first two forms of stability.

The examination of model stability is a topic within the field of dynamical
systems. This paper does not delve into the mathematical proof of model
stability. Instead, we present empirical results obtained from our
simulations. The second order ODE described in equation 4 can be reformulated
as a first order ODE. As an example, consider a swarm consisting of 5 agents.
The dynamics of each agent are governed by the second order ODE in equation 4.
To simplify this ODE to a first order system, we introduce two variables for
the $i^{th}$ agent: $x_{1i}=r_{i}$ and $x_{2i}=v_{i}$. This allows us to
express the dynamics of the entire swarm as follows:

$\displaystyle\dot{x_{11}}=x_{21},$ (5)
$\displaystyle\dot{x_{21}}=(1-|x_{21}|^{2})x_{21}-\frac{a}{5}\sum_{j=1}^{5}(x_{11}(t)-x_{2j}(t))+\eta_{1}(t),$
(6) $\displaystyle\dot{x_{12}}=x_{22},$ (7)
$\displaystyle\dot{x_{22}}=(1-|x_{22}|^{2})x_{22}-\frac{a}{5}\sum_{j=1}^{5}(x_{12}(t)-x_{2j}(t))+\eta_{2}(t),$
(8) $\displaystyle\dot{x_{13}}=x_{23},$ (9)
$\displaystyle\dot{x_{23}}=(1-|x_{23}|^{2})x_{23}-\frac{a}{5}\sum_{j=1}^{5}(x_{13}(t)-x_{2j}(t))+\eta_{3}(t),$
(10) $\displaystyle\dot{x_{14}}=x_{24},$ (11)
$\displaystyle\dot{x_{24}}=(1-|x_{24}|^{2})x_{24}-\frac{a}{5}\sum_{j=1}^{5}(x_{14}(t)-x_{2j}(t))+\eta_{4}(t),$
(12) $\displaystyle\dot{x_{15}}=x_{25},$ (13)
$\displaystyle\dot{x_{25}}=(1-|x_{25}|^{2})x_{25}-\frac{a}{5}\sum_{j=1}^{5}(x_{15}(t)-x_{2j}(t))+\eta_{5}(t),$
(14)

or we can write this as vector-matrix form:

$\begin{bmatrix}\dot{x_{11}}\\\ \dot{x_{12}}\\\ \dot{x_{13}}\\\
\dot{x_{14}}\\\ \dot{x_{15}}\\\ \dot{x_{21}}\\\ \dot{x_{22}}\\\
\dot{x_{23}}\\\ \dot{x_{24}}\\\ \dot{x_{25}}\end{bmatrix}\\\
=\begin{matrix}zeroes(5,5)&I(5,5)&zeroes(5,5)\\\ \frac{a}{5}\cdot
ones(5,5)-a\cdot I(5,5)&I(5,5)&-I(5,5)\end{matrix}\begin{bmatrix}{x_{11}}\\\
{x_{12}}\\\ {x_{13}}\\\ {x_{14}}\\\ {x_{15}}\\\ {x_{21}}\\\ {x_{22}}\\\
{x_{23}}\\\ {x_{24}}\\\ {x_{25}}\\\ |x_{21}|^{2}{x_{21}}\\\
|x_{22}|^{2}{x_{22}}\\\ |x_{23}|^{2}{x_{23}}\\\ |x_{24}|^{2}{x_{24}}\\\
|x_{25}|^{2}{x_{25}}\end{bmatrix}+\begin{bmatrix}0\\\ 0\\\ 0\\\ 0\\\ 0\\\ 0\\\
0\\\ 0\\\ 0\\\ 0\\\ \eta_{1}\\\ \eta_{2}\\\ \eta_{3}\\\ \eta_{4}\\\
\eta_{5}\end{bmatrix}$ (15)

sectionLiterature Review on collective motion of a swarm

###  2.1 Literature Review on collective motion of a swarm

The collective motion of a swarm can be understood as the interplay of three
key components: short-range repulsion to prevent collisions, local
interactions to align the velocity vectors of neighboring units, and a global
positional constraint to maintain the coherence of the swarm [1]. A universal
feature observed in collective swarm motion in biological systems such as
schools of fish, bird flocks, and herds of mammals is the tendency for the
velocity vectors of individual units to be parallel with one another [2]. A
comprehensive examination of various types of swarms is presented in [3]. The
field of System Identification (SID) has generated significant academic
interest, as it encompasses the study of systems to unveil and model the
interactions and dynamics in animal or robot swarms. A deeper understanding of
these dynamics can aid in the advancement of more advanced swarm systems.
Identification algorithms have been proposed that integrate structure
determination with parameter estimation using an orthogonal least squares
approach [4]. Determining the parameters through estimation of causal entropy
can provide insight into the impact of components in multivariate time series
data [5]. Another non-parametric method, entropic regression, has been
utilized to estimate the parameters of dynamic equations [6]. Recently,
various learning-based methods have also been developed for SID. Temporal
Convolutional Neural Networks (CNNs) utilize masked convolutions to implement
a fixed-range sequence model [7]. Additionally, multi-step neural network
approaches, such as Adams-Bashforth [8], Cluster-Networks [9], and Physics-
Informed Deep Learning models [10] and [11], have been proposed. Another study
evaluates the use of machine learning models to supplement knowledge-based
mathematical models in cases where analytical models may fail, for example,
due to unknown noise models or approximations [12].
A recent development in the field of nonlinear system identification has
introduced a new method for uncovering the structure of a nonlinear dynamic
system from data. This approach utilizes symbolic regression to determine the
system’s dynamics and strikes a balance between the model’s complexity (number
of terms) and its agreement with the data. However, this regression problem
can be computationally expensive and may not scale well to large-scale
systems, and may result in over-fitting. Other techniques for discovering
emergent behavior and governing equations from time-series data include
automated statistical inference of dynamics, equation-free modeling, and
empirical dynamic modeling. One method, Sparse Identification of Nonlinear
Dynamics (SINDy), produces a sparse, nonlinear regression that automatically
identifies the relevant terms in the system from a library of functions [13].

##  3 Methods and Approaches for System Identification

Simulations were performed to generate data, utilizing equation (15) in
Python. The resulting data comprised of the position and velocity of each
swarm particle for N timesteps. A representative example of the generated
data, presented in CSV format, can be found in the appendix. The system’s
steady state is characterized by the time derivatives of its parameters being
equal to and maintaining zero values. The eventual attainment of the steady
state may vary depending on the system’s initial conditions and the path or
duration required to reach it. To comprehend the system more thoroughly, the
methodologies applied should be categorized into four distinct training
approaches.

  1. 1.

Training on Transient Dynamics with Identical Initial Conditions to Test Data

  2. 2.

Training on Transient Dynamics with Dissimilar Initial Conditions to Test Data

  3. 3.

Training on Steady State Dynamics with Identical Initial Conditions to Test
Data

  4. 4.

Training on Steady State Dynamics with Dissimilar Initial Conditions to Test
Data

The baseline techniques were executed and evaluated as described above. The
models were trained through a folding approach, where a consecutive set of
observations were utilized to predict the subsequent step. For instance, the
first five steps were utilized to forecast the sixth step, the second to sixth
steps predicted the seventh step, and so forth. This resulted in a single
batch shape, assuming a swarm of 32 particles, as follows:

$\displaystyle 5\times 32\times 4$ (16) $\displaystyle 5\times 128$ (17)

In order to assess and compare our models, we defined the Mean Field Error
(MFE) as a metric. The MFE quantifies the discrepancy between the mean
positions of the predicted swarm particles and the ground truth particles. It
is computed as the Euclidean distance between the mean positions of the swarm
particles, providing insight into the stability achieved by the model and its
ability to learn from the time-series data. The calculation of the MFE is
described as follows:

$\displaystyle
MFE=\sqrt{(x_{m,true}-x_{m,pred})^{2}+(y_{m,true}-_{m,pred})^{2}}$ (18)

##  4 Non Deep learning Models

This section outlines one method: a Regression Model-based Forecasting
approach for predicting swarm behavior for deriving the equation form of the
swarm’s dynamical system.

###  4.1 Least Square-Regression Model

The data utilized to construct models predicting the future behavior of swarms
is time-series data. For basic time-series forecasting, Linear Regression
Models are commonly used. Despite the swarm system being non-linear, linear
regression models can locally approximate the non-linear function and still be
employed for forecasting future samples, provided the local neighborhood of
the estimation is small. One of the initial models established is a Linear
Regression Model based on Least Squares. The Ordinary Least Squares (OLS)
method is employed to evaluate the various regression lines, where the
regression line that minimizes the sum of the squared differences between the
observed and predicted dependent variables is selected according to OLS.

Given a set of m samples, $x=[x_{1},x_{2},x_{3},...,x_{m}]$, the model will
learn a weight vector, w, to predict future samples,
$\tilde{x}=[x_{m+1},x_{m+2},x_{m+3},...,x_{m+n}]$, as $\tilde{x}=w^{T}x$,
where the predicted samples are n steps ahead of the current time. The
quantity m represents the number of in-samples, while n represents the number
of out-samples that the model predicts. The optimal feature set size (m) and
optimal prediction horizon (n) are generally determined through cross-
validation and serve as hyperparameters for the model.

  1. 1.

Model for Steady State Dynamics: A swarm model was generated using the defined
model from Section 2, producing data over 3000 time steps. A test-train split
was created, using 2000 samples for training and 1000 samples for testing. The
data primarily represents the steady state behavior of the swarm system. A
regression model was built, taking 10 samples as input and predicting the 11th
sample. After training, a sliding window approach was used for testing the
model. The model was initially given the last 10 samples of the training data
to predict a new sample, then the window was shifted to include the predicted
sample to predict additional samples until 1000 samples were predicted.

  2. 2.

A Regression Model for Transient Dynamics: The swarm model data consisting of
250 time steps was generated as per the procedure outlined in Section 2. A
test-train split was established, with 150 samples designated for training and
100 samples reserved for testing, to primarily capture the transient dynamics
of the swarm system. A regression model was built, which employed 10 samples
to predict the 11th sample. After training, a sliding window approach was
employed for evaluating the model’s performance. The last 10 samples of the
training data were initially fed to the model, which predicted the next
sample. This process was repeated by folding ahead and incorporating the
predicted sample until 100 samples were predicted.

Figure 1: MFE for Steady State Predictions

The study finds that a regression model with an optimal feature size and
prediction horizon of 10 and 1 respectively can accurately predict the steady
state evolution of a swarm system with minimal mean field error (MFE). As the
number of prediction samples increases, the MFE begins to rise. The model can
predict the steady state behavior of the swarm up to approximately 1000 steps
ahead with acceptable MFE. However, its performance on transient data is poor
due to its inability to capture the highly non-linear nature of the swarm
system. This linear regression model does not provide insight into the
underlying dynamics of the swarm system as it is linear in nature and the
swarm system is highly non-linear. The model’s success in predicting steady
state may be due to the linear stability of the multi-agent system in the
steady state.

Figure 2: MFE for Transient Predictions Figure 3: Prediction for Steady State
Figure 4: Prediction for Transient

##  5 Deep Learning Models

The study evaluated three deep learning baselines, including a standard feed
forward network, recurrent neural networks (RNNs), and temporal convolutional
neural networks (CNNs). The RNNs and temporal CNNs demonstrated superior
performance in extrapolating steady swarm dynamics and accurately matching the
ground truth. However, when trained on transient data, these models failed to
extrapolate and reach stability, revealing their limitations in capturing the
inherent dynamics of the system. This highlights the strengths of these models
in inferring temporal dependencies but not in learning the underlying dynamics
of the system.

###  5.1 Multi-Layer Perceptron

The state dynamics of the system is characterized by a set of non-linear
equations. To capture this non-linearity, a baseline was implemented using a
fully connected neural network. This network, with appropriate activation
functions between layers, has the potential to capture the temporally
dependent dynamics of the swarm. The input to the network is a tensor
consisting of positional and velocity data of 32 particles over five time
steps, and the objective of the model is to predict the state and velocity of
the swarm for the next time step. The following table presents the final
hyperparameters that produced the most reliable outcomes:

Hidden Layer Size | 256
---|---
Optimizer | SGD
Learning Rate | 0.001
Training Window for Dataset | 5
Table 1: Hyperparameters for MLP

###  5.2 Recurrent Neural Network

Recurrent Neural Networks (RNNs) are extensions of feedforward neural networks
with a built-in memory component. They are recurrent in nature as they perform
the same computation for every sequence of input data, with the output
dependent on the previous computation. The output is then copied and fed back
into the network, allowing it to consider both the current input and the
previous output in making a decision. This makes RNNs highly effective in
identifying patterns in time-series data or data where the samples at a
particular time are assumed to be dependent on the preceding sample.

Figure 5: RNN Unrolled Structure

A single-layer RNN was employed, followed by a linear activation function. The
loss function used was nn.MSELoss(). The hyperparameters optimized included
the hidden size, the number of RNN layers, the optimizer, and the learning
rate, as detailed below:

Hidden Layer Size | 256
---|---
Number of RNN layers | 1
Optimizer | SGD
Learning Rate | 0.005
Training Window for Dataset | 5
Table 2: Hyperparameters for RNN

###  5.3 Convolutional Neural Networks

While Recurrent Neural Networks (RNNs) and their derivatives such as Long
Short-Term Memories (LSTMs) and Gated Recurrent Units (GRUs) have
traditionally been the preferred models for time-series based data,
Convolutional Neural Networks (CNNs) have also been effectively applied to the
learning of time-series data as described in [14]. Given the presence of
multiple swarm particles with underlying interaction forces in the data,
incorporating a form of convolution may be beneficial in capturing both local
and temporal information. In this implementation, 1-dimensional convolutions
were used instead of 2-dimensional convolutions. The following presents the
final architecture of our implemented model:

Figure 6: Implemented CNN Architecture

Several architectural and filtering configurations were evaluated, with the
final configuration producing the most consistent results. The use of Max-
Pooling, Adaptive Average-Pooling, and Dropout resulted in suboptimal
performance, hence not included in the final architecture. The ReLU activation
function was applied to most layers. The chosen loss function was Mean Square
Error $(nn.MSE)$. Both Stochastic Gradient Descent $(SGD)$ and Adam
optimization algorithms were experimented with. Although Adam demonstrated
fast convergence at times, its performance was inconsistent across different
learning rates. On the other hand, SGD consistently converged quickly at a
learning rate of $5\times 10^{-3}$ and was chosen as the final optimizer and
learning rate.

Loss Function | Mean Square Error
---|---
Optimizer | SGD
Learning Rate | 0.0005
Training Window for Dataset | 5
Table 3: Hyperparamters for CNN First Lyer | Second Layer | Third Layer
---|---|---
Filter 1 | 5 | Filter 2 | 5 | Filter 3 | 3
Padding 1 | 2 | Padding 2 | 2 | Padding 3 | 1
Stride 1 | 2 | Stride 2 | 2 | Stride 3 | 1
Table 4: Parameters of Convolution Layers

The performance of the models was inadequate when extrapolating and predicting
transient swarm dynamics. These findings suggest the need for further research
in developing robust deep learning solutions capable of generalizing the
system’s dynamics and accurately forecasting the future trajectories of swarm
particles during transient states.

###  5.4 Swarm Dynamics Predictive Performance through Neural Network Models

The Multi-Layer Perceptron (MLP) model was initially evaluated as a baseline.
The MLP was capable of capturing the non-linear properties of the model.
However, its performance was highly contingent on the initial conditions of
the particles, resulting in a marked decrease in its ability to predict the
true test trajectories of the swarm when these conditions deviated from the
training data. The comparison between the performance of the model on steady-
state data and data consisting of both transient and steady-state dynamics
revealed a substantial reduction in performance during the latter case,
particularly during the transient phase, as indicated by the Mean Field
Errors. These findings suggest that the MLP is not capable of uncovering the
underlying behavior of the swarm and instead focuses solely on minimizing the
Mean Field Error. As a result, it is prone to failure when initial conditions
differ or during transient scenarios. In addition, the training time required
by the MLP was significantly longer (500 epochs) compared to the RNN and CNN
models (50 epochs), further hindering its utility. Given these limitations,
the MLP was not tested as extensively as the CNN and RNN models.

Figure 7: Training loss for model trained on Steady State for MLP

The RNN and CNN models exhibited efficient performance during the training
process, exhibiting rapid convergence across the four methodologies outlined
in the Methods section. Experiments with various epoch lengths revealed
consistent results for lengths exceeding 50 epochs. The loss plots for the two
models in one case are depicted in Figures 8 and 9.

Figure 8: Training loss for model trained on Steady State for RNN Figure 9:
Training loss for model trained on Steady State for CNN

Despite their rapid convergence to a minimal cost (within 50 epochs), the RNN
and CNN models only demonstrated exceptional performance under one
methodology, namely when they were trained on steady-state data with initial
conditions identical to the test data. The results were inconsistent for the
other training methodologies, as illustrated by the Mean Field Error plots. As
demonstrated in Figures 10 and 11, no clear better-performing model emerges,
owing to the presence of a transient state in the evaluation data. As
previously noted, the model trained on steady-state data was expected to
perform well, and it did once the transient state was removed, causing a
significant change in the Mean Field Error. This suggests that both the RNN
and CNN models can only learn the stable part of the swarm system. The plots
for all other truncated tests are included in the appendix. Although RNN and
CNN seem to have comparable performance, a closer examination of the predicted
data from each model via animation reveals that while the Mean Field Error is
similar for both models, the CNN model exhibits a noticeable phase shift.

Figure 10: trained on Steady State for RNN MFE Figure 11: trained on Steady
State for CNN MFE

The analysis of the results presented in this section indicates that the RNN
and MLP models have an advantage in learning inter-particle interactions and
individual particle behaviors compared to only learning the steady state of
the entire system, which is the case with the CNN model. The three deep-
learning-based models (MLP, RNN, and CNN) are capable of learning steady-state
models from data with the same initial conditions, but they are unable to
generalize well when dealing with the presence of transient states or states
with different initial conditions. The results suggest that the deep-learning-
based models, similar to the non-baseline models, are not able to identify the
hidden state dynamics of the swarm system and struggle with capturing the
highly non-linear properties of the transient system.

##  6 Advanced Deep Learning Model - Neural ODE

The proposed model boasts two distinctive features that enhance its efficacy.
Firstly, it adopts the concept of Neural ODE, a non-architectural but
innovative method for optimizing through the utilization of an ODE solver.
Backpropagation through an ODE solver is feasible, but it necessitates the
storage of variables at each time step, leading to substantial memory
overhead. This model circumvents this issue by defining adjoint states, the
time dynamics of which can be obtained through numerical integration with the
ODE solver. These adjoint states then facilitate the calculation of
derivatives of each parameter with respect to the loss function, thereby
enabling the ODE solver to be utilized as a ”black box”, regardless of the
specific ODE solver employed.

Our objective is to approximate the function $\dot{X}=f(X)$ as a neural
network $\hat{f}$ with parameters $\theta$ , using the form presented in
equation 15. The optimization target is to minimize the scalar-valued loss
function L, given by $L(ODESolve(X(t_{0}),\hat{f},t_{0},t_{1},\theta))$. The
definition of adjoint states and the calculation of their time dynamics are
detailed in [15]. In this model, the selection of the neural network
architecture utilized for approximating the function $\hat{f}$ is the second
aspect considered. Given that the dynamics of $\dot{X}$ is the direct target
for approximation, physical characteristics of the swarm system can be
leveraged to inform the choice of the neural network. The homogeneity of the
observed swarm implies that the dynamics of each agent must be uniform and
that the interactions between agents must also exhibit consistency. In light
of these considerations, the architecture was designed as depicted in Figure
12. The utilization of a physics-informed network significantly shrinks the
parameter space, thereby expediting the training process.

| Intrinsic | Interaction | Aggregation
---|---|---|---
Number of Layers | 3 | 3 | 3
Nonlinearity Input | Cubic | None | None
Table 5: Physics-informed model architecture ODESolver Type | Euler
---|---
ODESolver Step | 0.05
Optimizer | Adam(lr=0.01)
Loss Function | MSE
Table 6: Physics-informed model hyperparameters

Tables 5 and 6 present the architecture of the final model and the
hyperparameters employed for the training process, respectively.

Figure 12: Physics-informed model for 3 agent swarm team Figure 13: Learning
curve of NODE for 3 agent swarm team Figure 14: Predicted trajectory of agents
compare to ground truth

##  7 Conclusions

In this study, we investigated the effectiveness of various neural network
models in capturing the dynamics of nonlinear Ordinary Differential Equations
(ODEs) systems. Our aim was to evaluate the performance of existing deep
learning and non-deep learning models in learning the stability of swarm
systems through time-series data. The results showed that traditional deep
learning models and non-deep learning baselines were not robust enough to
handle the nonlinearity and transient states of the swarm systems. The models
performed well when evaluating systems with the same initial conditions but
struggled with systems with different initial conditions and transient states.

This led us to explore a new deep learning method, Neural ODE, which provides
a framework for training ODEs within larger models. Despite finding a working
model, it remains unclear how the network, in conjunction with Neural ODE,
captures the true dynamical process of the system. This is because the
presence of bifurcations and catastrophes in dynamical systems can cause
instability and hysteresis, hindering the learning of a good model. Further
research is needed to fully understand the limitations and capabilities of
Neural ODEs in capturing the nonlinear dynamics of swarm systems.

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