# Physics-Informed Neural Networks for Satellite State Estimation

Jacob Varey
MIT Lincoln Laboratory
<EMAIL_ADDRESS>Jessica D. Ruprecht
MIT Lincoln Laboratory
<EMAIL_ADDRESS>Michael Tierney
MIT Lincoln Laboratory
<EMAIL_ADDRESS>Ryan Sullenberger
MIT Lincoln Laboratory
<EMAIL_ADDRESS>979-8-3503-0462-6/24/$\$31.00$ ©2024
IEEEDISTRIBUTION STATEMENT A. Approved for public release. Distribution is
unlimited.Research was sponsored by the United States Air Force Research
Laboratory and the Department of the Air Force Artificial Intelligence
Accelerator and was accomplished under Cooperative Agreement Number
FA8750-19-2-1000. The views and conclusions contained in this document are
those of the authors and should not be interpreted as representing the
official policies, either expressed or implied, of the Department of the Air
Force or the U.S. Government. The U.S. Government is authorized to reproduce
and distribute reprints for Government purposes notwithstanding any copyright
notation herein.

###### Abstract

The Space Domain Awareness (SDA) community routinely tracks satellites in
orbit by fitting an orbital state to observations made by the Space
Surveillance Network (SSN). In order to fit such orbits, an accurate model of
the forces that are acting on the satellite is required. Over the past several
decades, highquality, physicsbased models have been developed for satellite
state estimation and propagation. These models have widely varying degrees of
fidelity: some only account for twobody Keplerian motion, while others
consider highly accurate Earth gravity models, atmospheric drag, solar
radiation pressure (SRP), perturbations from the Sun, Moon, and other
celestial bodies, etc. These models are exceedingly good at estimating and
propagating orbital states for nonmaneuvering satellites; however, there are
several classes of anomalous accelerations that a satellite might experience
which are not wellmodeled. For example, satellites using lowthrust electric
propulsion to modify their orbit, or pieces of debris which have extremely
High AreatoMass Ratios (HAMR) that experience SRP effects beyond what is
accounted for in existing models. PhysicsInformed Neural Networks (PINNs) are
a valuable tool for these classes of satellites as they combine physics models
with Deep Neural Networks (DNNs), which are highly expressive and versatile
function approximators. By combining a physics model with a DNN, the machine
learning model need not learn the fundamental physics of astrodynamics, which
results in more efficient and effective utilization of machine learning
resources to solve for only the unmodeled dynamics. This paper details the
application of PINNs to estimate the orbital state and a continuous,
lowamplitude anomalous acceleration profile for satellites. Anglesonly
observation data is simulated for satellites near Geosynchronous Orbit (GEO).
This simulation first propagates the satellite in time using a physics model
coupled with an arbitrary acceleration profile, and then simulates realistic,
anglesonly observations of the satellite as would be measured by a groundbased
optical telescope. This arbitrary acceleration profile could represent a
lowthrust orbit maneuver or a difficulttomodel SRP effect on a HAMR object,
and is used for generating truth data. The PINN is trained to learn the
unknown acceleration by minimizing the mean square error of the observations.
We evaluate the performance of pure physics models with PINNs in terms of
their observation residuals and their propagation accuracy beyond the fit span
of the observations. For a twoday simulation of a GEO satellite using an
unmodeled acceleration profile on the order of $\bm{10^{-8}\>km/s^{2}}$, the
bestfit physics model resulted in observation residuals with a rootmeansquare
error of 123 arcsec, while the bestfit PINN had an error of 1.00 arcsec,
comparable to the measurement noise. Similarly, after propagating the bestfit
physics model for five days beyond the fit span of the observations, the
propagated position of the satellite using the physicsonly model was wrong by
3860 km, compared to the PINN which had an error of only 164 km.

###### Contents

  1. 1 Introduction
    1. 1.1 Orbit State Fitting and Propagation
  2. 2 Simulating Observation Data
  3. 3 Technical Approach
    1. 3.1 Model Training
    2. 3.2 DNN Architecture and Hyper-Parameter Tuning
  4. 4 Results
    1. 4.1 Observation Residuals
    2. 4.2 Extrapolation Error
    3. 4.3 DNN Evaluation
  5. 5 Summary

##  1 Introduction

For decades the 18th Space Defense Squadron (18th SDS), a component of the
United States Department of Defense (DoD), has maintained a catalog of space
objects [1]. The Department of Commerce (DoC) has recently been tasked with
providing Space Traffic Management (STM) services to commercial space
operators [2]. While satellite owners and operators usually have exquisite
telemetry from their satellites, these government organizations generally do
not have access to that information. Instead, the Space Domain Awareness (SDA)
community relies on observations of each Resident Space Object (RSO) from the
Space Surveillance Network (SSN)— a suite of optical and radar sensors
dedicated to space surveillance [3]. The 18th SDS maintains a satellite’s
orbit by fitting an orbital state— either a Two Line Element (TLE) set or
position and velocity vectors— to observations correlated with the object.
That orbital state is used to predict where the satellite will be in the
future so that new observations can be correlated with the satellite.

For the first several decades after sending the first satellites into Earth
orbit, the challenge of maintaining knowledge of where each satellite was in
space was driven by our lack of understanding of the natural forces acting on
the satellite. More recently, high-fidelity physical models have been
developed which can be used to quickly and accurately predict where a
satellite will be at any time, assuming the satellite is not actively using
thrusters to change its orbit. For example, the Special General Perturbations
4 (SGP4) propagator, which is used to predict a satellite’s position and
velocity from a TLE, models Earth gravity, atmospheric drag, and gravitational
perturbations due to the Sun and Moon, but does not account for satellite
thrust [4]. Since the early days of human utilization of space, the assumption
that a satellite is not actively thrusting has been violated exceedingly
often. The vast majority of space missions today maintain a precise orbit
using some form of thrust. In particular, electric propulsion (ion propulsion,
Hall effect thrusters, pulsed plasma thrusters, etc.) are often utilized due
to their increased fuel efficiency; the mass savings from requiring less fuel
translates directly into more affordable satellite systems.

One important difference between electric propulsion systems and chemical
thrusters is that chemical thrusters can achieve a large acceleration in a
matter of seconds, while electric propulsion systems apply low thrust
continuously for minutes to days. This difference is depicted in Figure 1. Any
amount of thrust violates the assumptions made by the physical model used to
propagate the satellite’s state from one time to another; however, the
satellite’s new orbit can be fit from post-maneuver observations. For
satellites that use chemical thrusters to apply a large acceleration over a
short period of time, the maneuvers are nearly instantaneous, or impulsive,
and the satellite quickly changes from one orbit to another. As long as there
are a sufficient number of new observations of the satellite after the
maneuver, the satellite’s new orbit can be determined because it is once again
not thrusting in violation of the assumptions of the physical model. Because
the change in orbital state happens rapidly for impulsive maneuvers, there is
a window of opportunity to task sensors, correlate new observations of the
satellite, and estimate a new, post-maneuver state; this is difficult for
satellites that are continuously thrusting for long periods of time. Because
continuously thrusting satellites spend long periods of time violating the
assumptions of the physical model, by the time the maneuver has finished and a
new orbit can be fit, the satellite may be be very far from where it was when
the maneuver started, making the process of tasking and positively correlating
new observations to the satellite’s previous orbital state difficult.

Figure 1: A diagram illustrating orbital state changes using impulsive and
continuous thrust maneuvers. The regions shown by straight blue lines
represent times in which the satellite is not thrusting, and is thus on a
constant-energy orbit that is well-modeled by current propagators. The regions
shown in red are those in which the satellite is thrusting and the assumption
of the propagators is violated.

Satellites employing electric propulsion for slow orbital maneuvers are
difficult to maintain using pure physics models such as those built into SGP4.
Because these models do not account for the RSO actively thrusting, the best-
fit orbital state does not fit the observation data well and cannot be used to
extrapolate far into the future. In practice, these satellite states are
maintained with significant manual intervention. The process is the same as
for any satellite: a new state is fit to a span of observations, which is then
used to task sensors and correlate new observations. But because the state is
a poor fit to the data, and because it does not extrapolate well into the
future, this process must be repeated often to keep up with the ever-changing
orbital state. If new observations of the satellite are not received quickly
(e.g. due to a weather outage at an optical sensor site), it is possible that
the RSO will be lost entirely. Sometimes new observations of the satellite are
obtained, but they fail to correlate with the satellite’s stale orbital state,
and the observations will instead become an uncorrelated track (UCT). This
challenge can be mitigated by estimating the satellite thrust as part of the
orbit fitting process, making it possible to recover maneuvering objects
despite a gap in observations.

Electric propulsion systems are now used for both small, station-keeping
maneuvers, as well as large, orbital transfer maneuvers. Eutelsat 115 West B
is an example of a satellite which used electric propulsion to raise its orbit
from LEO to GEO over the course of nearly eight months using four Xenon Ion
Propulsion System (XIPS) engines and now uses the same propulsion system for
station-keeping in GEO [5]. Maintaining an orbital state on such a satellite
during the orbital transfer period is more difficult than during the station-
keeping period due to the magnitude and duration of the acceleration exerted
by the thrusters. The goal of this work is to enhance the predictive power of
an orbital state such that it can be used to task sensors and correlate new
observations further into the future as compared to current models. That is,
it is not necessary for the model to accurately predict where the satellite is
for the entire eight month orbital transfer. Instead, the model needs to be
able to accurately predict where the satellite will be a small number of days
into the future– long enough to task sensors to collect new observations, or
to positively identify serendipitous observations to the satellite.

Let the thrust profile be the magnitude and direction of the satellite’s
thrust vector as a function of time. The goal of this paper is to demonstrate
the utility of Physics-Informed Neural Networks (PINNs) for describing the
totality of the dynamics for a thrusting satellite.

A PINN in the context of this paper refers to a model that combines physics
and machine learning (ML), often a Deep Neural Network (DNN). This area of
research is referred to as Knowledge-Informed Machine Learning [6]. Physical
models are mathematical equations which have been developed and refined over
time to accurately describe the physical world. In contrast, DNNs are highly
expressive and versatile function approximators. By utilizing a PINN, the ML
model need not relearn the fundamental physics, and can focus on learning just
the deviations from the physical model present in the data, which is a more
efficient use of ML resources. PINNs are an attractive solution for modeling
the dynamics of a thrusting satellite because they combine the known physics
(i.e. astrodynamics), which describes the natural forces acting on the
satellite, with an ML model for describing the satellite thrust profile.

###  1.1 Orbit State Fitting and Propagation

Orbit fitting is the process of tuning the state parameters (i.e. the three-
dimensional position and velocity vectors) to minimize the difference between
the predicted and actual observations of the satellite. Predicted observations
are generated by propagating the initial state of the satellite to the set of
time epochs at which observations of the satellite have been made and using
geometry to compute the expected observation based on the sensor site location
and the sensing modality. For satellite tracking, the most common sensing
modalities are active RF and passive optics. Radar sensors usually produce
precise range and range-rate information, as well as less-precise angle and
angle-rate information. Conversely, passive optical telescopes usually produce
precise angles-only observations, but cannot determine range or range-rate.
This paper focuses on orbit fitting using simulated observations from a single
optical sensor observing a satellite in Geosynchronous (GEO) orbit, though the
methodology is applicable across sensing modalities and orbit regimes.

Orbit propagation broadly falls into two types of techniques: semi-analytic
methods and numerical integration methods. The former predict how a
satellite’s state will evolve based on in-depth analysis of the perturbing
forces acting on the satellite and how those forces affect the satellite’s
state. Numerical integration methods differ in that they only estimate the
total force acting upon the satellite as a function of the satellite state and
time, and then numerically integrate the acceleration using an ordinary
differential equation (ODE) solver to determine position and velocity as a
function of time. For the task of propagating a satellite state in time, let
the state vector be the concatenation of the three-dimensional position and
velocity vectors of the satellite. The derivative of the state, then, is
simply the velocity and acceleration vectors:

$\displaystyle\bm{X}$ $\displaystyle=\begin{bmatrix}\bm{r}\\\
\bm{v}\end{bmatrix}$ (1) $\displaystyle\dot{\bm{X}}$
$\displaystyle=\begin{bmatrix}\dot{\bm{r}}\\\
\dot{\bm{v}}\end{bmatrix}=\begin{bmatrix}\bm{v}\\\ \bm{a}\end{bmatrix}$ (2)

These equations define an ODE with initial conditions (ICs):

$\bm{X_{0}}=\begin{bmatrix}\bm{r}\\\ \bm{v}\end{bmatrix}_{0}$ (3)

The satellite acceleration can be described with Cowell’s formulation [7]:

$\bm{a}=-\frac{GM}{r^{3}}\bm{r}+\bm{a_{P}}+\bm{a_{T}}$ (4)

where the first term is the two-body equation of motion [8], $G$ is the
gravitational constant, $M$ is the mass of the primary body, $\bm{a_{P}}$ is
the acceleration due to natural perturbing forces (e.g. non-uniformities in
Earth’s gravitational field, SRP, Sun and Moon gravity, etc.), and
$\bm{a_{T}}$ is any acceleration the satellite is itself exerting via its
thrusters. Precise physical models exist for the first two terms of equation
4. The thrust component is often ignored, but modeling the satellite’s thrust
profile is required to accurately integrate the dynamics for a maneuvering
object. The goal of this work is to demonstrate a model architecture that is
flexible enough to learn a satellite’s thrust profile directly from
observation data.

##  2 Simulating Observation Data

The first challenge in developing a model that accurately estimates the
dynamics of a continuously thrusting satellite is acquiring observation data
of such a satellite along with true ephemeris information describing where the
satellite is as a function of time well into the future. Observation data is
needed because that is all that is typically available when fitting an orbit,
and the ephemeris data is needed for evaluating how well a model is able to
extrapolate into the future, which is a key measure of performance. Due to the
need for truth data, the demonstration of the technique described in this
paper is performed using simulated data.

The simulation creates angles-only observations of a thrusting satellite in
GEO as viewed by a ground-based optical telescope. In this demonstration, the
data amounted to thirty measurements of right ascension (RA) and declination
over the course of two days, randomly sampled in time. Sensor measurement
error was added to the observations by randomly sampling a Gaussian
distribution with zero mean and 0.5 arcsec standard deviation in both
dimensions. The resulting simulated observations in RA and declination are
shown in Figure 2. Note that daytime outages due to solar exclusion have not
been included in the simulated data for simplicity.

The acceleration profile is chosen to be periodic, with periodicity equal to
the satellite’s orbital period, as shown in Figure 3. This acceleration
profile would not be useful for a satellite to achieve a specific change in
orbit; however, the goal is to demonstrate that the DNN is capable of learning
arbitrary acceleration profiles, and thus the form of the acceleration does
not matter. Integrating the magnitude of the thrust vector over the two-day
simulation amounts to a total $\Delta$V $\approx$ 10 m/s.

To understand the feasibility of such a thrust, consider a standard Evolved
Expendable Launch Vehicle (EELV) Secondary Payload Adapter (ESPA) satellite
with mass of 180 kg [9]. For such a satellite, a $\Delta$V of 10 m/s with a
burn duration of two days would require only 10 mN of thrust, which is well
within the capability of most electric propulsion systems. For example, the
XIPS engines that are employed on several Boeing 702 class satellites are
capable of maximum thrusts of up to 165 mN [10].

Figure 2: Simulated RA and declination observations of the satellite over the
48 hour time span used for training both the PINN and the physics-only models.
Figure 3: A random periodic acceleration is generated and applied to the
satellite in inertial coordinates to simulate an arbitrary thrust profile. The
magnitude of the integrated thrust is chosen to lie within the capability of
modern electric propulsion systems.

A simple two-body physics model was used along with the acceleration profile
described above in order to generate truth data. While this physics model is
not accurate enough for most real applications, it is sufficient for a
demonstration of the methodology. One of the future goals for this research is
to incorporate a higher-fidelity physics model and use the model to fit a
state to real observations of a thrusting satellite.

##  3 Technical Approach

Ultimately, the goal is to build a model that computes the total acceleration
of a satellite in orbit, including both the natural forces and active thrust
acting on the satellite. This could be done with a pure physics model, a pure
machine learning model, or by combining physics models with machine learning
models. This section describes how a physical model— astrodynamics— can be
coupled with an ML model to describe the total dynamics of a thrusting
satellite.

By including physical models in a machine learning framework, the machine
learned model need not learn the behavior described by the physics model. For
this application, that means we can leverage the astrodynamics models that
have been developed and refined over the course of many decades without
needing to relearn those governing equations from scratch. The machine learned
model need only solve for the deviations from the physical model that best
describe the data. As discussed in Section 1, DNNs are utilized for this work
because they can be utilized as parameterized, expressive function
approximators.

To apply knowledge-informed machine learning techniques to the problem of
estimating the total dynamics of a thrusting satellite, consider that Equation
4 can be broken apart into two functions:

$\bm{a}=f(t,\bm{X})+g(t,\bm{X})$ (5)

where $f$ is a physical model incorporating astrodynamics, and $g$ is the
instantaneous thrust applied by the satellite, which will be estimated by a
DNN.

As shown in the pseudocode in Figure 4, the equations of motion, which combine
astrodynamics with the output of a DNN, are integrated using an ODE solver,
starting with an initial estimate of the satellite’s position and velocity
vectors. These initial conditions are also tuned as part of the parameter
optimization. The output of the ODE solver is the satellite position and
velocity, which can be evaluated at any time within the fit domain.

Figure 4: PyTorch- and TorchDiffEq-based pseudocode for the coupled model
architecture, orbit propagation, and loss computation. The low-level details
of computing observations and the loss function are abstracted.

###  3.1 Model Training

The loss function used for training the coupled model is the mean squared
error (MSE) of the observation residuals. The satellite position and velocity
vectors are determined by propagating the initial conditions as described
above. From those position and velocity vectors, along with the sensor
position and velocity at the same times, the expected observations can be
computed geometrically. These expected observations are then compared to the
true observations and the total loss is computed.

The loss function is back-propagated through the ODE solver in order to
determine the gradient with respect to the model parameters. This technique
was first demonstrated with the publication of Neural ODEs [11]. A number of
open-source software packages exist for solving neural ODEs, including
TorchDiffEq [12] and SciML [13], [14]. The model developed for this effort is
implemented in PyTorch [15] and uses TorchDiffEq to back-propagate through the
ODE solver.

The initial position and velocity vectors of the satellite are also tuned in
addition to the DNN parameters as a part of the model training process. This
can be achieved by either a) including the initial conditions as tunable
parameters within the model, or b) excluding these parameters from the model
and solving for them separately. Both approaches were evaluated as part of
this research, and the latter was found to converge more quickly and
consistently to the true values of those parameters. Specifically, the initial
conditions are estimated using a simple batch least-squares fit to the
observation data once every $N$ iterations through the training loop. It was
found that fitting the initial conditions every 100 training steps resulted in
good performance.

###  3.2 DNN Architecture and Hyper-Parameter Tuning

The acceleration profile to be estimated by the DNN is shown in Figure 3. The
accelerations are not particularly complicated and should be well-estimated by
a model with relatively few parameters. The results shown in this paper were
achieved using a simple Multi-Layer Perceptron (MLP) [16] with two hidden
layers of 100 nodes each and a hyperbolic tangent activation between each
layer.

As with most machine learning research, the hyper-parameters governing
training must be tuned to achieve best performance. For this work, the Adam
optimizer [17] is used to tune the ML model parameters and the learning rate
(LR) was gradually decreased throughout training using a fixed-step schedule.
Table 1 shows the values of the hyper-parameters that were used for training.

Table 1: Hyper-parameters used during training. Parameter | Value
---|---
Training Epochs | 20,000
Initial Learning Rate | 0.003
LR Scheduler Step Size | 100
LR Scheduler Scale Factor | 0.98
IC Tuning Frequency | 100
Figure 5: Training loss as a function of training epoch.

The loss curve obtained during training of the PINN is shown in Figure 5. The
MSE of the observation residuals decreases by several orders of magnitude
before asymptoting to the noise floor of the observations.

The number of training epochs required for this particular data was determined
after much experimentation. For this methodology to be used in practice, a
more robust training schedule, including early termination, should be
implemented to ensure that the model training is sufficient and achieves the
expected performance. Additional research is required to evaluate the extent
to which the model overfits the data, and how such overfitting can be
mitigated. One challenge in this domain is the general lack of data–
observations of slowly maneuvering satellites are sparse as explained in
Section 1– and thus splitting the data into train and test batches would
further reduce the amount of data used for training.

##  4 Results

Two models were developed and optimized with the loss function described
above: 1) a physics-only model that includes two-body equations of motion but
does not attempt to model thrust, and 2) the knowledge informed model
architecture described in Section 3. The first model was optimized using
simple batch least-squares optimization, while the second was tuned using the
Adam optimizer algorithm. These models were then evaluated based on how well
they fit the data and how well they were able to predict the true ephemeris in
the future, beyond the fit span of the training data.

###  4.1 Observation Residuals

Figure 6 shows a scatter plot of RA and declination observation residuals as a
function of time, and Figure 7 combines the residuals for each model into a
cumulative distribution. The median observation residual for the physics-
informed model is two orders of magnitude smaller than the physics-only model.
The physics-informed model has many more degrees of freedom to fit the data,
which should result in much lower observation residuals. Additionally, the
pure physics model is fundamentally integrating incorrect dynamics by not
accounting for the thrust exerted by the satellite.

Figure 6: Scatter plot of the RA and declination residuals for both the PINN
and physics-only models. The PINN is able to model the arbitrary thrust
profile applied, while the physics-only model, which fails to account for
thrust, has high residuals over the fit span. Figure 7: Cumulative
distributions of the observation residuals for physics-only and PINN models.

The Root Mean Square Error (RMSE) of the PINN was 1.00 arcsec, compared to 123
arcsec for the best-fit physics-only model, an improvement of two orders of
magnitude. By simulating observations with random Gaussian noise,
characterized by a standard deviation of 0.5 arcsec, the PINN adeptly captures
the dynamics of the satellite, effectively extending its explanatory capacity
to the approximate noise floor of the observations.

###  4.2 Extrapolation Error

In order to evaluate how well the models predict the true ephemeris of the
satellite in the future, each model was used to propagate the satellite’s
state for twenty days from the beginning of the simulation and compared with
the true propagated position of the satellite. The Euclidian distance between
the predicted and true position and velocity was computed. This metric is
particularly important for this application because the utility of the model
is largely based on how well the model can be used to predict where the
satellite will be in the future in order to correlate new observations with
the satellite’s orbital state. Table 2 gives the propagated position and
velocity errors for both one day and five days after the end of the two day
fit span, while Figure 8 shows the propagation errors for both models during
the full twenty days from the start of the simulation.

Figure 8: Propagation error beyond the fit span of the observations. While the propagation error grows as a function of time for both models, the PINN model outperforms the physics-only model for the full 20 days of the simulation, sometimes outperforming the physics-only model by as much as two orders of magnitude. Table 2: Extrapolation errors of the pure physics and PINN models at 1 and 5 days after the last observation. | Physics-Only | PINN
---|---|---
Observation RMSE | 123” | 1.00”
Propagation Error - 1 day |  | 397 km
---
29.5 m/s
| 3.35 km
---
0.384 m/s
Propagation Error - 5 day |  | 3860 km
---
285 m/s
| 164 km
---
12.3 m/s

###  4.3 DNN Evaluation

One benefit of coupling a physics-based model with a DNN is increased
interpretability of the combined model. Evaluating the DNN portion of the
combined model results in the best estimate for the satellite thrust as a
function of time. For this initial demonstration, the true thrust profile is
known because the observation data is simulated, which allows for direct
evaluation of how well the model estimated the thrust profile.

Figure 9: Comparison of the true acceleration profile with the estimated
acceleration profile learned by the DNN after training.

Figure 9 shows the components of the estimated thrust as well as the true
accelerations used to generate the simulated data. While not perfect, the PINN
is clearly capable of learning the general shape and scale of the thrust
needed to describe the data. The figure indicates that the DNN estimates the
z-component of the thrust profile better than the x- and y-components. It is
not inherently clear why the model is better able to fit the z-component than
the other directions. One potential reason is that the x- and y- directions
are more coupled in the case of a nearly equatorial GEO satellite, and that
there is a larger manifold of solutions that explain the observation data
equally well.

It is anticipated that better estimating the individual thrust components
would require either more precise observations or longer fit spans, as the
ability of the model to accurately estimate the thrust profile depends heavily
on the amplitude and duration of the thrust as well as the precision of the
observations. The extremely small amplitude thrusts modeled here integrate to
a $\Delta$V of only about 10 m/s over a relatively long burn duration of two
days; however, even these very small thrusts cause the satellite to deviate
from the expected trajectory determined by a pure physics model significantly.

##  5 Summary

An increasing number of satellites are employing low-thrust propulsion systems
for their mission needs. This poses a challenge for maintaining satellite
catalogs, as orbit propagation tools do not attempt to estimate thrust as part
of a satellite’s orbital state. This paper demonstrated the use of physics-
informed neural networks for learning a satellite’s thrust profile along with
the typical state vector parameters that best fit a set of angles-only
observations of the satellite. The PINN model was shown to produce a better
fit to the observations by two orders of magnitude, due to the ability of the
DNN to correctly model the satellite thrust which was not captured in the
physics-only model. When the trained physics-only and PINN models were
extrapolated beyond the last observation, the PINN model again resulted in
consistently better performance in estimating the spacecraft’s position at
future times, sometimes by as much as two orders of magnitude. These results
suggest that physics-informed neural networks offer an alternative approach to
flexibly model forces acting upon an RSO which are not well-modeled by
existing physics-based models and that the inclusion of a DNN component can
significantly improve the quality of the resulting orbital state.

###### Acknowledgements.

Research was sponsored by the United States Air Force Research Laboratory and
the Department of the Air Force Artificial Intelligence Accelerator and was
accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views
and conclusions contained in this document are those of the authors and should
not be interpreted as representing the official policies, either expressed or
implied, of the Department of the Air Force or the U.S. Government. The U.S.
Government is authorized to reproduce and distribute reprints for Government
purposes notwithstanding any copyright notation herein.

## References

  * [1] [Online]. Available: https://www.jtf-spacedefense.mil/About-Us/Fact-Sheets/Display/Article/3155799/18th-space-defense-squadron/
  * [2] U. S. W. House, “Space policy directive-3, national space traffic management policy,” Jun 2018. [Online]. Available: https://rosap.ntl.bts.gov/view/dot/60966
  * [3] L. C. G. Shepherd and A. F. S. Command, “Space surveillance network,” in _Shared Space Situational Awareness Conference, Colorado Sprincs, CO_ , 2006\.
  * [4] D. Vallado, P. Crawford, R. Hujsak, and T. Kelso, “Revisiting spacetrack report# 3,” in _AIAA/AAS Astrodynamics Specialist Conference and Exhibit_ , 2006, p. 6753.
  * [5] D. Thomas, “A comparison of geo satellites using chemical and electric propulsion,” 2016.
  * [6] L. von Rueden, S. Mayer, K. Beckh, B. Georgiev, S. Giesselbach, R. Heese, B. Kirsch, J. Pfrommer, A. Pick, R. Ramamurthy, M. Walczak, J. Garcke, C. Bauckhage, and J. Schuecker, “Informed machine learning – a taxonomy and survey of integrating prior knowledge into learning systems,” _IEEE Transactions on Knowledge and Data Engineering_ , vol. 35, no. 1, pp. 614–633, 2023.
  * [7] D. Vallado and W. McClain, _Fundamentals of Astrodynamics and Applications_ , ser. Fundamentals of Astrodynamics and Applications. Microcosm Press, 2001. [Online]. Available: https://books.google.com/books?id=OCkGmwEACAAJ
  * [8] E. Gill and O. Montenbruck, _Satellite orbits: Models, methods and applications_. Springer, 2013.
  * [9] P. Wegner, J. Ganley, and J. Maly, “Eelv secondary payload adapter (espa): providing increased access to space,” in _2001 IEEE Aerospace Conference Proceedings (Cat. No.01TH8542)_ , vol. 5, 2001, pp. 2563–2568 vol.5.
  * [10] D. M. Goebel, J. E. Polk, I. Sandler, I. G. Mikellides, J. R. Brophy, W. G. Tighe, and K.-R. Chien, “Evaluation of 25-cm xips thruster life for deep space mission applications,” in _31st International Electric Propulsion Conference (20–24 September 2009_ , 2009.
  * [11] R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. Duvenaud, “Neural ordinary differential equations,” _Advances in Neural Information Processing Systems_ , 2018.
  * [12] R. T. Q. Chen, “torchdiffeq,” 2018. [Online]. Available: https://github.com/rtqichen/torchdiffeq
  * [13] C. Rackauckas, M. Innes, Y. Ma, J. Bettencourt, L. White, and V. Dixit, “Diffeqflux.jl-a julia library for neural differential equations,” _arXiv preprint arXiv:1902.02376_ , 2019.
  * [14] C. Rackauckas, Y. Ma, J. Martensen, C. Warner, K. Zubov, R. Supekar, D. Skinner, and A. Ramadhan, “Universal differential equations for scientific machine learning,” _arXiv preprint arXiv:2001.04385_ , 2020.
  * [15] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Köpf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, “Pytorch: An imperative style, high-performance deep learning library,” 2019.
  * [16] F. Murtagh, “Multilayer perceptrons for classification and regression,” _Neurocomputing_ , vol. 2, no. 5, pp. 183–197, 1991.
  * [17] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” _arXiv preprint arXiv:1412.6980_ , 2014.

Jacob Vareyvarey.jpg received his B.S. and M.S. degrees in Aerospace
Engineering Sciences with a focus in astrodynamics and satellite navigation
from the University of Colorado at Boulder. He is currently a researcher at
MIT Lincoln Laboratory in the Artificial Intelligence Technology group. In his
eight years with Lincoln Laboratory, his research has focused on remote
sensing for space domain awareness, algorithm development for novel approaches
to faint signal detection, and deep reinforcement learning in support of
decision making in the space domain.

Jessica D. Ruprechtruprecht_cropped.png received a B.S. in Physics and
Planetary Science and an M.S. in Planetary Science from the Massachusetts
Institute of Technology (MIT). As Technical Staff in the AI Software
Architectures and Algorithms group at MIT Lincoln Laboratory, she applies her
background in observational astronomy to programs in support of Lincoln
Laboratory’s space mission. Her current research focuses on applying a variety
of artificial intelligence techniques to space problems, including
reinforcement learning, computer vision, and natural language processing.

Michael C. Tierneytierney_cropped.jpg received his B.S. in Physics from MIT
and his M.S. in Astronomy from the University of Hawai‘i at Mānoa. He is
currently a member of the MIT Lincoln Laboratory Artificial Intelligence
Technology group, where his research interests include building high-fidelity
simulators for space domain awareness scenarios, testing the robustness of
transformer-based time series classifiers, and building tools to measure the
trustworthiness and generalizability of machine learning models.

Ryan M. Sullenbergersullenberger_cropped.png earned his M.S. in Mechanical and
Aerospace Engineering from Princeton University and his B.S. in Mechanical
Engineering from Colorado State University. Currently serving as a cross-
disciplinary research engineer at MIT Lincoln Laboratory in the Applied Space
Systems group, Ryan is involved in the development, design, simulation,
analysis and experimental validation of innovative new products and sensors
used for detecting and tracking both natural phenomena and human-made space
objects. Ryan’s areas of expertise and interests include applied physics,
remote sensing, rapid prototyping, data science and machine learning.