Investigation of the Robustness of Neural Density Fields

J. Schuhmacher

Recent advances in modeling density distributions, so-called neural density
fields, can accurately describe the density distribution of celestial bodies
without, e.g., requiring a shape model - properties of great advantage when
designing trajectories close to these bodies. Previous work introduced this
approach, but several open questions remained. This work investigates neural
density fields and their relative errors in the context of robustness to
external factors like noise or constraints during training, like the maximal
available gravity signal strength due to a certain distance exemplified for
433 Eros and 67P/Churyumov-Gerasimenko. It is found that both models trained
on a polyhedral and mascon ground truth perform similarly, indicating that the
ground truth is not the accuracy bottleneck. The impact of solar radiation
pressure on a typical probe affects training neglectable, with the relative
error being of the same magnitude as without noise. However, limiting the
precision of measurement data by applying Gaussian noise hurts the obtainable
precision. Further, pretraining is shown as practical in order to speed up
network training. Hence, this work demonstrates that training neural networks
for the gravity inversion problem is appropriate as long as the gravity signal
is distinguishable from noise.

Code and results are available at https://github.com/gomezzz/geodesyNets

##  1 Introduction

In recent years small bodies in our solar system have been of increasing
interest as mission targets. Various past, present, and future missions
visited them and even collected samples allowing study of their composition
and properties. For example, in the past, NEAR visited the asteroid 253
Mathilde and later soft landed on 433 Eros in February 2001, or Rosetta, with
its lander Philae visiting the comet 67P/Churyumov–Gerasimenko in 2014. There
are also several upcoming missions such as Hera, ZhenHe, and Psyche [1]. Hera,
the complementary mission of ESA to NASA’s DART mission, will analyze the
latter’s impact with the smaller moon Dimorphos, which revolves around the
asteroid Didymos, with the goal of studying the process of asteroid
deflection. It is scheduled to start in 2024 and rendezvous the binary system
in 2026 together with two accompanying cubesats [2].

With these prospects, it is even more crucial for guidance, navigation and
control to have accurate and precise models of these target bodies. However,
this is especially difficult with small bodies since of their irregular shape.
To complicate matters, the density distribution of these bodies is also rarely
homogeneous but rather heterogeneous in nature [3, 4, 5].

These facts make modeling these bodies with the established three models
difficult: spherical harmonics, mascon models and polyhedral gravity models.
The former struggle with the convergence inside the Brillouin sphere, and the
irregular shapes slow down convergence, making it unsuitable for asteroids and
comets [6, 1, 7]. The minimum Brillouin sphere is the sphere centered around
the origin of the body and still enveloping all of the body’s mass. The latter
two require the knowledge of the shape model of the target body, and their
application comes with constraints like discretization [8] in the case of the
mascon model or the assumption of homogeneous density in the case of
polyhedral gravity models [9, 10].

With the recent advances in neural networks, a new approach was formulated,
requiring no previous assumptions but only existing data originating from
measurements or synthetic sources. With this data, one can train a neural
network to solve the gravity inversion problem. Multiple approaches exist for
creating such a network, as presented in Section 2. This paper focuses on
geodesyNets —a network learning the density distribution of a body whose
integration leads to gravity [1]. Previous work showed acceptable error
boundaries for models trained with a synthetic ground truth based on a mascon
model. In addition, this approach was effectively used to explain the density
of asteroid (101955) Bennu, with evidence from the OSIRIS-REx mission.
However, the results were affected negatively due to noise in the input signal
[11].

Thus, this work aims to determine if this approach is still practical given
that multiple sources of perturbations can pollute the gravity signal used for
training, ranging from non-gravitational forces like solar radiation pressure
to measurement inaccuracies. Further, the utilized gravity signal might be
weaker due to safety distance when on a trajectory far from the celestial
body, further hindering the network from learning correctly. Last, it is
investigated if pretraining on a prior shape model could improve the network
performance under these constraints and if it reduces the amount of training
iterations needed, thus reducing computational cost in a hypothetical onboard
scenario.

This work uses a polyhedral gravity model and a mascon model for conducting
these experiments. It demonstrates the applicability of geodesyNets trained
with a noisy input signal, as long as the induced error is reasonably sized
compared to the gravity signal. This means that an error relative to the
magnitude of the gravity signal in the training data set still leads to
acceptable errors. On the other hand, a large absolute limit on measurement
accuracy can render any training useless. This circumstance exemplifies in the
conducted study about the sampling distance utilized for training the network.
Finally, it is shown how pretraining can reduce the number of iterations
required for training.

All results and code are made publicly available via GitHub.

##  2 Related Work

###  2.1 Polyhedral Gravity Models

Polyhedral gravity models can calculate the full gravity tensor, including
potential, acceleration, and second derivatives for an arbitrary given point
$P$ around a polyhedral source. They provide an analytical solution given
homogeneous density distributions and given a shape model. One of the
approaches to implementing a polyhedral gravity model is given by Tsoulis et
al. [9, 10]. They use a line integral approach to convert the triple integral
into a nested summation, which, thanks to the introduction of dedicated
singularity terms avoids potential singularities that can affect these models
[12, 13]. In this work, we use Tsoulis model to compare the performance of a
geodesyNet trained with a mascon model to a network trained with a polyhedral
model, but also to determine how the mesh granularity affects the achievable
precision. Furthermore, we perform a detailed study of the achievable
precision at close range since the high accuracy, even within the Brillouin
sphere, and the analytical formulation does not require numerical integration
methods.

###  2.2 Mascon Models

Mascon models are the second approach for modeling gravity utilized in this
work. Here, the body is represented as a combination of multiple point masses,
so-called mascons (short for “mass concentrations” [14]) filling up its
volume. The mascon elements do not need to be of uniform size. Instead,
various approaches exist, combining mascons of different weights [8]. Its
simplicity and ability to model irregular shapes and density distributions as
they appear for small bodies make the model appealing. Regardless, high
accuracy of the gravity field can only be achieved with an extensive number of
mascons for non-spherical / irregular objects. Even then, the field’s accuracy
near the body’s surface remains challenging due to the discretized mass
distribution [8].

###  2.3 Gravity Modeling with Neural Networks

Artificial intelligence emerging in almost any area in recent years provides
new opportunities in modeling traditionally expensive computational processes
in physics [15, 16]. One such domain is modeling the gravity of irregularly
shaped bodies and mapping positions to accelerations. In this context, one can
distinguish between neural networks representing the actual body, enabling an
indirect mapping to gravitational accelerations, and those directly mapping
positions around a given body to gravitation. Thus, the former can also be
referred to as neural fields since they parameterize the bodies [16]. A
geodesyNet initially described by [1] represents a body through its neural
density field.

The inspiration for geodesyNets originates from Neural Radiance Fields (NeRF)
introduced by [17]. Their network learns to represent a three-dimensional
object from two-dimensional pictures. Thus, their network maps a 5D input
vector for position and view angle to a 2D color and volume density vector.
Similarly how they solve the inverse problem of image rendering, a geodesyNet
solves the inverse problem of gravity inversion mapping a cartesian vector to
a candidate body density [1]. The neural field of a geodesyNet represents the
body’s density distribution.

The subsequent Subsection 3.2 will describe it in detail since it is the
object of study, whereas this section provides a brief look over alternatives
from the first category of networks regressing acceleration on positions.

Generally, all of the following presented networks have in common that they
directly map coordinates to potentials or accelerations and are trained on a
polyhedral ground truth. However, the concrete approaches vary a lot. Furfaro
et al. [7] use Single-Layer Feedforward Networks using Extreme Learning,
reducing the number of required iterations to fine-tune their model to map the
relationship between spacecraft points and gravitational acceleration. They
report relative errors (relRMSE) below $5\%$ for asteroid 25143 Itokawa and
only slightly above $5\%$ for comet 67/P Churyumov-Gerasimenko. They conducted
two experiments. One time the global gravity field was learned using target
points sampled from a near-range sphere around the body. The second time a
local field was learned using target points from a cylinder above a given
landing zone. Their approach contrasts this work as here the robustness is
examined when also only sampling from far away.

Cheng et al. [18] use deep neural networks (DNN) with the aim of generating
trajectories in a computationally cheap way for a soft landing on small
bodies. They show that the median error of their model is $0.33\%$ when taking
the mean over the three axial directions. Given that $1\%$ deviations are
usually acceptable in traditional gravity models [18], they conclude that
their DNN is a practical approach. Similarly to the later employed geodesyNet,
they normalize the input and outputs to the range $(-1,1)$. Further, they do
not train the network with sample points from the near field but instead with
points some kilometers away from the body. In their case study for 433 Eros,
points are collected from the sphere $3\,km$ to $60\,km$.

Finally, Gao et al. [19] model the gravity field of multiple sample bodies
with a Gaussian Process Regression. They report a relative mean error of
$1.27\%$ for their bodies when validating inside the sampling area. They only
train their networks with samples including the near range of the bodies.
However, in contrast to the study presented later on geodesyNet, their
approach shows strong generalization difficulties when further away from the
body. Thus, the relative error increases beyond $60\%$ in the case of 433 Eros
at $40\,km$ distance. For comparison: Training was done here with points up to
$20\,km$ distance.

Martin et al. [20, 21, 22] also present a continuously improved model that
maps Cartesian coordinates to gravity. However, their approach differs from
the usual training in the sense that the network is bound to additional
constraints beyond the usual loss to guarantee physical correctness. These
constraints are integrated into the loss function and include, for example,
the satisfaction of Laplace’s equation. The training thus penalizes not only
inaccuracy but also physical violations of the underlying differential
equations. They also train on 433 Eros with sampling up to a distance of three
times the radius. The network generalizes well to distance, but has
difficulties predicting closer to the asteroid.

##  3 Methods

###  3.1 Ground-Truth Models

Equation 1 shows the derived formalism for computing the gravitational
acceleration around a polyhedral source at the origin with $G$ as the
gravitational constant, $\rho$ as the constant density using the polyhedral
gravity model by Tsoulis et al. [9, 10]. It consists of two summations with
the outer summing over the polyhedral faces $p$ and the inner one iterating
over the segments $q$ of each face. For a detailed description, refer to
Tsoulis et al. [9, 10].

$\overrightarrow{a}=G\rho\cdot\sum_{p=1}^{n}\overrightarrow{N_{p}}\cdot\left[\sum_{q=1}^{m}\sigma_{pq}h_{pq}LN_{pq}+h_{p}\sum_{q=1}^{m}\sigma_{pq}AN_{pq}+\text{sing}_{\mathcal{A}_{p}}\right]$
(1)

This polyhedral gravity model was recently implemented with modern C++ in a
work preceding this one [23].

This model will be contrasted with the mascon model, whose formula is
displayed in Equation 2 for a generic point $\overrightarrow{r_{i}}$ and a set
of mascons $\mathcal{M}=\\{(x_{i},y_{i},z_{i})\;i=1..n\\}$, in the following
experiments. The aim is to determine the achievable precision with the
knowledge that the mascon model, used in the original paper, has difficulties
in the immediate proximity to the surface due to discretization. Next to the
calculation of the ground truth is the evaluation of the model. For this
purpose, the neuronal density field is integrated using Equation 4. Since the
models were always trained with vectorial accelerations and not scalar
potentials, demonstratively, only these formulas are displayed.

$\overrightarrow{a}(\overrightarrow{r_{i}})=-G\sum_{j=1}^{n}\frac{m_{j}}{r^{3}_{ij}}r_{ij}$
(2)

###  3.2 GeodesyNets - Neural Density Fields

A neural density field, also referred to as geodesyNet [1, 15], is a fully-
connected neural network trained with a gravity signal from an arbitrarily
shaped body with the aim of learning the body’s density distribution. Thus, it
solves the gravity inversion problem and provides a fully differentiable
expression for mapping a cartesian point to a candidate body density (see
Equation 3) compatible with the observed gravity signal.

$f(x,y,z)\rightarrow\rho$ (3)

The appeal of this method is that it does not necessarily require a shape
model, as is the case with the aforementioned gravitational models, but that
the gravitational signal can come from any source. Further, it converges even
inside the Brillouin sphere and can learn heterogeneous density distributions.

Using the trained model, a numerical integration over the neural density field
can be performed to calculate the potential or the acceleration. This
procedure is displayed in Equation 4.

$\overrightarrow{a}=G\int_{x\in
V}\frac{\rho(x)}{|\overrightarrow{r}-\overrightarrow{x}|^{3}}(\overrightarrow{r}-\overrightarrow{x})\;dV$
(4)

Previous studies utilized this training approach in combination with the
mascon model as ground truth, demonstrating its theoretical applicability [1].
The measured relative error was always less than $1\%$ even near bodies.
There, however, it was not considered how robust the network would be if the
sample points were at a distance of several kilometers as they have been in
previous missions to minimize the risk of collision.

GeodesyNets have also been looked at from a practical point of view with
measured gravity signals obtained from the OSIRIS-REx mission, demonstrating
the practical applicability of the given approach while also giving first
insights above limitations of the approach given the noise in a measured
gravity signal [11].

This work now provides a detailed study of the disruptive factors which could
lead to wrong model behavior, including noise in the ground-truth measurements
or errors in the numerical integration. As well as the influence that the
distance has on the possible precision.

Figure 1: Training and validation toolchain: Sampling with noise and from
different distances; Training for several thousand iterations; Comparing to
the actual ground truth and calculation of relative error.

###  3.3 Sampling Points for Training

To begin with, all meshes are normalized versions so that the bodies fit in
the hypercube $[-1,1]^{3}$ The spherical envelopes from which has been sampled
for training were $(0,1)$, $(1,3)$, and $(3,5)$. The former covers the close
range around the bodies, which the original study used for sampling. Here now,
two more sampling distances are included. The mid-range $(1,3)$ and the far-
range $(3,5)$ represent realistic distances an orbiter could reach around the
body. The latter distances are the more realistic scenarios with respect to
onboard training, as they maintain a certain safety distance for the
spacecraft. In contrast, the close distance $(0,1)$ would only be practical in
a gravity-model-based (pre-)training. In the following, it is refered to
$(0,1)$, $(1,3)$, and $(3,5)$ as close-, mid- and far-range.

The utilized mesh for Eros comes from Gaskell [24], and the one for Churyumov-
Gerasimenko from the European Space Agency [25]. Eros consists of 7374
vertices and 14744 triangular faces, whereas Churyumov-Gerasimenko comprises
9149 vertices and 18294 triangular faces. Both are based on the measurements
of the probes that visited the bodies. These original meshes are now referred
to as $100\%$ resolution meshes. The mascon models used for both models are
from Izzo & Gómez [1] and are derived from the $100\%$ resolution meshes. For
this purpose, a centroid with mass $m_{j}$ is placed in each tetrahedron in
the delaunay tetrahedralized version. In the course of this study, additional
meshes were constructed. These are downsampled versions with $10\%$, $1\%$, or
$0.1\%$ of the vertices and faces. The purpose of this is to investigate to
what extent a model can be (pre-)trained, even with a low resolution model as
it may be available from astronomical observations. Figure 2 illustrates the
employed input meshes for the polyhedral gravity model.

(a) Churyumov-Gerasimenko

(b) Eros

Figure 2: The original high resolution mesh denoted with $100\%$ and
downscaled to meshes with respectively only $10\%$, $1\%$ or $0.1\%$ of
vertices and faces.

###  3.4 Addition of Noise to the Ground Truth

In addition to the ground-truth model, mesh resolution, and sampling distance,
this study has one more parameter: noise. This characteristic is applied to
the ground truth during the training. It thus represents that the
measurements, the accelerations used for training, are subject to
perturbations and measurement noise.

Three different types are investigated in the course of this study: a constant
bias, additive Gaussian noise, and multiplicative Gaussian noise.

The constant bias represents the possible effects of solar radiation pressure
(SRP) on the measurement results. In a previous study [11, 3], this was a
significant factor influencing the quality of the results. Although to
subtract the SRP, it is necessary to know the mass and size of the object,
along with its surroundings, such as the area facing towards the sun. Here, we
will determine to what extent SRP influences the training. Equation 5 [26]
shows the employed approach for calculating the acceleration affecting the
spacecraft with $P$ as the solar radiation pressure, $c$ as the speed of light
and $G_{SC}$ as the solar constant, and $R$ as the distance from the sun in
astronomical units $[AU]$. Equation 6 calculates the acceleration $a$ given
the area-to-mass ration of the spacecraft $A/m$.

$\displaystyle P$ $\displaystyle=\frac{G_{SC}}{c\cdot R^{2}}$ (5)
$\displaystyle a$ $\displaystyle=P\cdot\frac{A}{m}$ (6)

These equations are a simplified version of the calculation. Theoretically,
the reflectivity, as given in Yousef et al. [27], could also be included in
the calculation. For this work, however, having only a baseline of the
magnitude is sufficient. The calculated acceleration $a$ is then applied in
one cartesian direction, assuming that the sun always shines from this
direction. In the following experiments, a value was used for the SRP as it
could realistically affect a probe analogous to Rosetta. The mass was assumed
to be $1422\,kg$ (roughly Rosetta Dry Mass and Payload) and the area
$69.88\,m^{2}$. For the distance, the semi-major axis of the bodies was used
in each case. It should be added that the SRP for, e.g., a cubesat, would have
a value of the same magnitude due to its smaller mass but equally smaller
area.

Further, an additive Gaussian noise $X\sim\mathcal{N}(0,\sigma^{2})$ is used
during the training to simulate an absolute error in the measurement of the
gravitational signal. The standard deviation $\sigma$ is chosen in a way that
it simulates an absolute error of $10^{-4}\frac{m}{s^{2}}$ or
$10^{-5}\frac{m}{s^{2}}$. After the meshes are normalized, this standard
deviation has also been normalized. The value $10^{-5}$ is the accuracy of
GOCE (Gravity field and steady-state ocean circulation explorer) [28]. In
addition to this value, $10^{-4}$ was chosen as the absolute precision to get
an insight into how strongly one additional error magnitude affects the result
accounting for even lower precision on spacecraft.

Further, a multiplicative Gaussian noise $X\sim\mathcal{N}(1,\sigma^{2})$ is
used during the training to simulate a relative error in the measurement of
the gravitational signal. The assumption is that the magnitude of the
gravitational signal is known, but the value can only be determined to a
certain point. For example that the precision is limited relatively to the
magnitude of the accelerations to $10^{-x}$ with $x\in\\{1,2,3\\}$.

###  3.5 Training

The toolchain used for training and performance evaluation is shown in Figure
1.

This summarizes the characteristics of the training iterations. Overall, the
same configuration is used for all other parameters as in the original paper
[1]. In particular, this includes the Mean Absolute Error (MAE) calculated
from the ground truth $y$ and the model predictions $\hat{y}$ shown in
Equation 7. Here, $\kappa$ (see Equation 8) is a scaling parameter to
normalize the mass and restrict training to learning the volume rather than
finding the absolute mass [1].

$\displaystyle\mathcal{L}_{\kappa\,MAE}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}|y_{i}-\kappa\hat{y_{i}}|$ (7)
$\displaystyle\text{with}\;\kappa$
$\displaystyle=\frac{\sum_{i=1}^{n}\hat{y_{i}}y_{i}}{\sum_{i=1}^{n}y_{i}^{2}}$
(8)

###  3.6 Validation

For validation, the relative root mean squared error is used, which is
calculated as shown in Equation 9. It is calculated for each model at the end
of training for a range of distances from the body’s surface. Comparisons are
made against the polyhedral ground truth using the $100\%$ mesh resolution.
The altitude sampler employed for this purpose uses the outward normal of the
mesh faces to sample points at the appropriate altitude. So the distance in
the validation plot’s x-axis is scaled by the altitudes from the surface, not
the altitudes from the mathematical origin of the bodies.

$relRMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(\frac{|y_{i}-\kappa\hat{y_{i}}|}{||\hat{y_{i}}||_{2}}\right)^{2}}$
(9)

##  4 Results

For each scenario presented here, ten training runs have been performed with
different seeds. The graphs show the mean value of the validation results and
the standard deviation in a slightly transparent way. The batch size has been
set to $1000$ points, and every ten training iterations new point for training
are sampled. For this purpose, a spherical point sampler randomly generates
points in a spherical shell that do not lie within the sample body.

###  4.1 Polyhedral vs. Mascon Model

Figure 3 compares the relative errors of the models trained with a mascon
model and a polyhedral model for the two studied bodies. Spherical sampling
was performed in a shell $\in(0,1)$, but the results are similar for sampling
with more distant shells. No qualitative difference can be found between
models trained with the mascon or polyhedral models, not even in the close
range around the body, as one would have expected due to the polyhedral
model’s improved accuracy in the close range.

For Churyumov-Gerasimenko, the relative errors are of similar size. E.g. for
normalized distance $0.01$, the model’s errors are close at $2.2\%$, whereas
for distance $0.001$, the mascon models has a slightly larger mean error of
$6.1\%$ compared to the polyhedral trained models’ relative mean error of
$5.8\%$.

For Eros, although the average relative error is slightly lower for the
polyhedral model. The model’s relRMSE trained with the polyhedral model is
around $50\%$ to $70\%$ compared to the relRMSE of the mascon trained models
through all validation distances.

Figure 3: Relative errors of mascon-trained models and polyhedral-trained
models on the $100\%$ resolution mesh

###  4.2 Sampling Distance

Note that the blue lines in Figure 3 and Figure 5 are the same as in Figure 4
in the left column. They show the average relative error of ten trained models
with the polyhedral ground truth, respectively, based on the $100\%$
resolution mesh and with sampling in the range $(0,1)$ during training.

In Figure 4, the lines show how the relative error changes for a given height
above the target body when sampling only in the normalized range $(1,3)$ or
$(3,5)$ during training.

Immediately, it is noticeable that if the network only gets points from mid or
far away during the training, the relative error in the close range increases
to $100\%$ and above. However, the network performs well for the mid to far
range, and the mean relative error remains below $4.3\%$ until a normalized
distance of $1$ or further away for models trained in mid or far range. Hence,
the network always performs well in the sampling range or further away in the
observed cases. It generalizes to the ranges more distant than used during
training. It is also interesting to note that in the range where geometry does
not play such a large role, the network is able to generalize and scale the
predictions when getting closer to the body. This is reflected in the plots on
the right side of Figure 4. The network has been trained with points from the
region $(3,5)$, but the average relative error always remains below the
$4.3\%$ mark, even for the $(1,3)$ region.

Figure 4: Effects of Noise and Sampling Shell on obtainable precision.

###  4.3 Robustness to Noise

This section refers again to Figure 4, which also displays the results when
some noise has been applied to the ground truth for learning.

Starting with the constant bias, models trained with a constant bias
approximately equal to the magnitude of the solar radiation pressure that
would act on a probe such as Rosetta or a cubesat show an almost identical
relative error in validation as models trained with the original ground truth.

In the case of additive Gaussian noise, which corresponds to an absolute error
of $10^{-5}\,\frac{m}{s^{2}}$ or $10^{-4}\,\frac{m}{s^{2}}$ in non-normalized
units, respectively, the network is most disturbed during learning. The mean
relative errors are far beyond $5\%$ for an absolute error of
$10^{-4}\,\frac{m}{s^{2}}$ even within the training range. However, the
results differ for Churyumov-Gerasimenko and Eros. The network shows smaller
relative errors for Eros than for Churyumov-Gerasimenko.

The multiplicative Gaussian noise that impacts the ground truth by
$+/-0.1\%,1\%,10\%$ respectively settles in the middle of the computed
relative errors. Training in the close range with a relative Gaussian noise of
$1\%$ still results in relative errors below $4.7\%$ even until a range of
$0.005$ for training in $(0,1)$. However, with $\sigma=10\%$, the relative
error maximally increases to $11.0\%$ for models trained in $(0,1)$.

###  4.4 A posteriori: Using Pretraining

After the previous sections have shown that noise and a long sampling distance
increase the relative error, we will now investigate whether these results can
be improved by pretraining. The deliberation is to perform the pretraining on
an imprecise low-resolution navigation model like it would be available due to
observation before launch in the near range $(0,1)$. In the second step, the
fine-tuning happens on the $100\%$ resolution mesh in the range $(3,5)$ for a
limited number of iterations, effectively simulating a redefined improved
mesh. Even though we show results without noise terms here, experiments with
those resulted in comparable results.

Figure 5 shows the achievable precision for the polyhedral model depending on
the precision of the mesh used. A model trained with a mesh with only one-
tenth of the vertices and faces achieves a comparable performance to the
entire $100\%$ resolution mesh. Also interesting is that a geodesyNet trained
with the $0.1\%$ resolution mesh still shows a solid relRMSE of only $6.3\%$
(Churyumov-Gerasimenko) and $2.5\%$ (Eros) for a normalized range of $1$. The
models displayed in Figure 5 are utilized as a pretraining base.

Figure 6 shows the results. Pretraining was performed for $10000$ iterations,
and fine-tuning (or training for the non-pretrained models) performed for $10$
or $100$ iterations. The pretrained models correspond to those in Figure 5.

Figure 6 shows that models pretrained on the near range data (green and orange
curve) overall outperform the models without any pretraining in the near
range. Further, the more precise the model utilized for pretraining, the
better the obtainable precision in the near range. As a second observation, it
is noticeable that the relRMSE increases in the near range for the pretrained
models if the fine tuning is conducted for 100 rather than 10 iterations (e.g.
$0.1\%$ pretrained resolution, distance $0.1$: $28.1\%\rightarrow 43.2\%$)
while the relRMSE decreases for the far field (e.g. $0.1\%$ pretrained
resolution, distance $5.0$: $1.3\%\rightarrow 0.09\%$). As the previous
sections have shown, the model adapts to the far range when fine-tuning.

To summarize, pretraining allows better-performing models with fewer
iterations. However, one has to be careful when fine-tuning with far-field
data to maintain the generalization capabilities learned during pretraining in
the near field.

Figure 5: The performance of the $10000$ iterations trained models depending
on the utilized mesh resolution. The validation was done based on the $100\%$
polyhedral ground truth. Figure 6: This figure compares the relative errors of
pretrained models versus non-pretrained models. The pretraining was always
conducted in the range $(0,1)$ marked in dark gray for $10000$ iterations, and
the fine-tuning (training for none) was conducted in the range $(3,5)$ marked
in light gray for $10$ or $100$ iterations. Here, no noise was applied.

##  5 Discussion

###  5.1 Polyhedral vs. Mascon Model

No clear qualitative difference could be found in the comparison between the
mascon and polyhedral models in Subsection 4.1. In both cases, the relative
error is similar even for close distances. A possible reason could be that
imprecision due to the number of vertices/edges in the mesh may be similarly
limiting as the number of mascons. Furthermore, the numerical accuracy could
be limited during training. Especially the numerical integration could be
crucial and limit the maximum achievable precision. In addition, only single
precision is used, but the polyhedral model provides a more precise result,
i.e., precision that cannot be learned in this sense - since it cannot be
represented in single precision. Mao et al. [29], for example, could improve
accuracy using double precision in their physics-informed neural network.

###  5.2 Sampling Distance

Regarding the results presented in Subsection 4.2 on sampling, one could
expect that the network could not predict the geometry and associated
accelerations only by training in the mid-range to far-range. However, a
geodesyNet is able to generalize and scale measurements as a function of
distance successfully. This is clearly shown in the right plots in Figure 4.
Conversely, a network is able to generalize successfully from the conducted
sampling ranges up to $50$ times the distance. A possible reason for that
could be the employed normalization and loss strategy involving the
continuously re-calculated mass normalization factor. Furthermore, the
geodesyNet learns the shape of the body. Hence, it automatically satisfies the
Laplace equation and thus represents a valid gravity field generated by the
learned shape - a property enabling excellent generalization capability. In
that sense, geodesyNets contrasts the in Subsection 2.3 listed approaches
using a regression.

The experiments also showed that the models could generalize from the far to
the mid-range. As soon as the shape of the body strongly impacts the measured
gravity signal, the generalization approaching closer to the body does not
work anymore. This feature could also be beneficial in a hypothetical onboard
scenario, as it allows a probe to get closer step by step and improve the
model accordingly. As here observed, the model makes a solid prediction a
little closer than the actual sampling range during training.

###  5.3 Noise

Subsection 4.3 shows that a slight constant bias is calculable. In other
words, future missions using a typical spacecraft will not have to worry about
accounting for it.

It also shows that training a geodesyNet when expecting a sizeable absolute
error in the measurements is a challenging task since one must find a way to
overcome the noise. Otherwise, the convergence of the network to an acceptable
solution is not achievable. Especially if the error is notably large compared
to the actual gravity; this is also why models trained for Eros perform better
with additive Gaussian noise. Churyumov-Gerasimenko is less massive and
smaller. Thus, the gravity signal is smaller than it would be in the case of
Eros, and an absolute error dominates the input signal rather than merely
perturbing it.

Suppose the magnitude of the gravity signal is known, and only the number of
determinable digits is a problem, as in the case of multiplicative Gaussian
noise in the $10\%$ deviation case. Even in that case, a robust network is
constructible.

###  5.4 Pretraining

Subsection 4.4 analyzed pretraining. A first result is that even with a ground
truth of poor quality, as was the case here during training with the $0.1\%$
resolution mesh, a low error can be achieved up to the mid-range distance - an
essential requirement for pretraining on Earth before the mission.

Further, the results show that pretraining reduces the model error and allows
for obtaining a better performance with fewer gravity measurements. For
example, Figure 6 shows that the relRMSE in the case of Churyumov-Gerasimenko
is with $11.9\%$ approximately half as large using pretraining with the $1\%$
resolution mesh than without any pretraining ($21.3\%$) in the same amount of
iterations.

In an onboard scenario, where power consumption is of critical interest,
pretraining on Earth, albeit with a lower resolution ground truth, is thus
advantageous.

Moreover, it must be added in this context that the scenario, pretraining, and
fine-tuning on range $(0,1)$ has not been considered here. Instead, the mixed
range sampling scenario was considered, as it would be conceivable for a
mission: pretraining before launch on a simple navigation model based on
remote observation and slowly improving the model when approaching the target
while re-training.

##  6 Conclusion and Future Work

In summary, this work studied the robustness of geodesyNets. The variables of
interest were: the underlying training ground truth, the sampling distance for
the training points, and the effects of different types of noise and whether
pretraining can reduce the number of training iterations.

There were only minor differences when the geodesyNet was trained with the
mascon or the polyhedral gravity model, probably due to limitations by the
numerical precision and the relation of the mascon ground truth and the
polyhedral mesh.

The geodesyNet cannot learn the geometry of an irregular body properly with
only distant measurements. However, the neural density field generalizes well
and yields robust results even in areas where no training has been performed,
as long as they are not in the close range. In order to be able to predict the
acceleration in the close-range, near the surface, training in this region
needs to be conducted.

Noise negatively impacts the training’s results. However, this depends
strongly on the magnitude compared to the input gravity signal. If there
exists an absolute boundary for the measurable precision of the acceleration
and thus the actual gravity signal is no longer distinguishable from the
measurement error, the training is unproductive. However, if the magnitude of
the gravity signal is known, even with relative measurement deviations of up
to $10\%$, solid training results are achievable.

From the point of view that the magnitude of the gravity signal is known, the
training is successful.

Pretraining allows more precise results in an onboard scenario with less
sampling and is preferable, even if the ground truth is of low resolution.

Future work could consider other forms of sampling, such as sampling with
regard to realistic trajectories. Such an experiment requires a set of
efficient trajectories maximizing the gravity signal - something currently
being conducted in a related work by Maråk et al. [30].

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