# :orange[Abstract:] 
  Hall effect thrusters are one of the most versatile and
  popular electric propulsion systems for space use. Industry trends
  towards interplanetary missions arise advances in design development
  of such propulsion systems. It is understood that correct sizing of
  discharge channel in Hall effect thruster impact performance greatly.
  Since the complete physics model of such propulsion system is not yet
  optimized for fast computations and design iterations, most thrusters
  are being designed using so-called scaling laws. But this work focuses
  on rather novel approach, which is outlined less frequently than
  ordinary scaling design approach in literature. Using deep machine
  learning it is possible to create predictive performance model, which
  can be used to effortlessly get design of required hall thruster with
  required characteristics using way less computing power than design
  from scratch and way more flexible than usual scaling approach.
:orange[author:] Korolev K.V [^1]
title: Hall effect thruster design via deep neural network for additive
  manufacturing

# Nomenclature

<div class="longtable*" markdown="1">

$U_d$ = discharge voltage  
$P$ = discharge power  
$T$ = thrust  
$\dot{m}_a$ = mass flow rate  
$I_{sp}$ = specific impulse  
$\eta_m$ = mass utilization efficiency  
$\eta_a$ = anode efficiency  
$j$ = $P/v$ \[power density\]  
$v$ = discharge channel volume  
$h, d, L$ = generic geometry parameters  
$C_*$ = set of scaling coefficients  
$g$ = free-fall acceleration  
$M$ = ion mass

</div>

# Introduction

<span class="lettrine">T</span><span class="smallcaps">he</span>
application of deep learning is extremely diverse, but in this study it
focuses on case of hall effect thruster design. Hall effect thruster
(HET) is rather simple DC plasma acceleration device, due to complex and
non linear process physics we don’t have any full analytical performance
models yet. Though there are a lot of ways these systems are designed in
industry with great efficiencies, but in cost of multi-million research
budgets and time. This problem might be solved using neural network
design approach and few hardware iteration tweaks(Plyashkov et al.
2022-10-25).

Scaled thrusters tend to have good performance but this approach isn’t
that flexible for numerous reasons: first and foremost, due to large
deviations in all of the initial experimental values accuracy can be not
that good, secondly, it is hardly possible to design thruster with
different power density or $I_{sp}$ efficiently.

On the other hand, the neural network design approach has accuracy
advantage only on domain of the dataset(Plyashkov et al. 2022-10-25),
this limitations is easily compensated by ability to create relations
between multiple discharge and geometry parameters at once. Hence this
novel approach and scaling relations together could be an ultimate
endgame design tool for HET.

Note that neither of these models do not include cathode efficiencies
and performances. So as the neutral gas thrust components. Most
correlations in previous literature were made using assumption or
physics laws(Shagayda and Gorshkov 2013-03), in this paper the new
method based on feature generation, GAN dataset augmentation and ML
feature selection is suggested.

## Dataset enlargement using GAN

As we already have discussed, the data which is available is not enough
for training NN or most ML algorithms, so I suggest using Generative
Adversarial Network to generate more similar points. Generative model
trains two different models - generator and discriminator. Generator
learns how to generate new points which are classified by discriminator
as similar to real dataset. Of course it is very understandable that
model needs to be precise enough not to overfit on data or create new
unknown correlations. Model was checked via Mean Absolute Percentage
Error (MAPE) and physical boundary conditions. After assembling most
promising architecture, the model was able to generate fake points with
MAPE of $~4.7\%$. We need to measure MAPE to be sure point lie on same
domain as original dataset, as in this work we are interested in
sub-kilowatt thrusters. After model generated new points they were check
to fit in physical boundaries of scaled values (for example thrust
couldn’t be more than 2, efficiency more than 1.4 and so on, data was
scaled on original dataset to retain quality), only 0.02% of points were
found to be outliers. The GAN architecture and dataset sample is
provided as follows.

<!-- ![GAN architecture](gen.png "GAN architecture")
![Sample of generated datagray - fake, blue - real](dT.png "Sample of generated datagray - fake, blue - real") -->

# General Relations

As we will use dataset of only low power hall thrusters, we can just
ignore derivation of any non-linear equations and relations and use
traditional approach here. Let’s define some parameters of anode:
$$\alpha = \frac{\dot{m}\beta}{{\dot{m}_a}},$$
Where $\alpha$ is anode
parameter of $\beta$ thruster parameter. This is selected because this
way cathode and other losses wont be included in the model. One of key
differences in this approach is fitting only best and most appropriate
data, thus we will eliminate some variance in scaling laws. Though due
to machine learning methods, we would need a lot of information which is
simply not available in those volumes. So some simplifications and
assumptions could be made. Firstly, as it was already said, we don’t
include neutralizer efficiency in the model. Secondly, the model would
be correct on very specific domain, defined by dataset, many parameters
like anode power and $I_{sp}$ still are using semi-empirical modelling
approach. The results we are looking for are outputs of machine learning
algorithm: specific impulse, thrust, efficiency, optimal mass flow rate,
power density. Function of input is solely dependant on power and
voltage range. For the matter of topic let’s introduce semi-empirical
equations which are used for scaling current thrusters.

<div class="longtable*" markdown="2">

$$h=C_hd$$

$$\dot{m_a} = C_m hd$$

$$P_d=C_pU_dd^2$$

$$T=C_t\dot{m_a}\sqrt{U_d}$$

$$I_{spa}=\frac{T}{\dot{m_a} g}$$

$$\eta_a=\frac{T}{2\dot{m_a}P_d}$$

</div>

Where $C_x$ is scaling coefficient obtained from analytical modelling,
which makes equations linear. Generally it has 95% prediction band but
as was said earlier this linearity is what gives problems to current
thrusters designs (high mass, same power density, average performance).
The original dataset is

|          |          |        |       |       |       |              |      |           |
|:---------|:---------|:-------|:------|:------|:------|:-------------|:-----|:----------|
| Thruster | Power, W | U_d, V | d, mm | h, mm | L, mm | m_a,.g/s,    | T, N | I\_spa, s |
| SPT-20   | 52.4     | 180    | 15.0  | 5.0   | 32.0  | 0.47         | 3.9  | 839       |
| SPT-25   | 134      | 180    | 20.0  | 5.0   | 10    | 0.59         | 5.5  | 948       |
| Music-si | 140      | 288    | 18    | 2     | 6.5   | 0.44         | 4.2  | 850       |
| HET-100  | 174      | 300    | 23.5  | 5.5   | 14.5  | 0.50         | 6.8  | 1386      |
| KHT-40   | 187      | 325    | 31.0  | 9.0   | 25.5  | 0.69         | 10.3 | 1519      |
| KHT-50   | 193      | 250    | 42.0  | 8.0   | 25.0  | 0.88         | 11.6 | 1339      |
| HEPS-200 | 195      | 250    | 42.5  | 8.5   | 25.0  | 0.88         | 11.2 | 1300      |
| BHT-200  | 200      | 250    | 21.0  | 5.6   | 11.2  | 0.94         | 12.8 | 1390      |
| KM-32    | 215      | 250    | 32.0  | 7.0   | 16.0  | 1.00         | 12.2 | 1244      |
| ...      |          |        |       |       |       |              |      |           |
| HEPS-500 | 482      | 300    | 49.5  | 15.5  | 25.0  | 1.67         | 25.9 | 1587      |
| UAH-78AM | 520      | 260    | 78.0  | 20    | 40    | 2            | 30   | 1450      |
| BHT-600  | 615      | 300    | 56.0  | 16.0  | 32    | 2.60         | 39.1 | 1530      |
| SPT-70   | 660      | 300    | 56.0  | 14.0  | 25.0  | 2.56         | 40.0 | 1593      |
| MaSMi60  | 700      | 250    | 60    | 9.42  | 19    | 2.56         | 30   | 1300      |
| MaSMiDm  | 1000     | 500    | 67    | 10.5  | 21    | 3            | 53   | 1940      |
| SPT-100  | 1350     | 300    | 85.0  | 15.0  | 25.0  | 5.14         | 81.6 | 1540      |

Hosting only 24 entries in total. The references are as follows(Beal et
al. 2004-11)(Belikov et al. 2001-07-08)(Kronhaus et al. 2013-07)(Misuri
and Andrenucci 2008-07-21)(Lee et al. 2019-11)

In the next section the used neural networks architectures will be
discussed.

# Data driven HET designs

Neural networks are a type of machine learning algorithm that is often
used in the field of artificial intelligence. They are mathematical
models that can be trained to recognize patterns within large datasets.
The architecture of GAN’s generator was already shown. In this section
we will focus on fully connected networks, which are most popular for
type for these tasks. HETFit code leverages dynamic architecture
generation of these FcNN’s which is done via meta learning algorithm
Tree-structured Parzen Estimator for every data input user selects. This
code uses state-of-art implementation made by OPTUNA. The dynamically
suggested architecture has 2 to 6 layers from 4 to 128 nodes on each
with SELU, Tanh or ReLU activations and most optimal optimizer. The code
user interface is as follows: 1. Specify working environment 2. Load or
generate data 3. Tune the architecture 4. Train and get robust scaling
models

## FNN

All of Fully connected neural networks are implemented in PyTorch as it
the most powerful ML/AI library for experiments. When the network
architecture is generated, all of networks have similar training loops
as they use gradient descend algorithm : Loss function:
$$L(w, b) \equiv \frac{1}{2 n} \sum_x\|y(x)-a\|^2$$ This one is mean
square error (MSE) error function most commonly used in FNNs. Next we
iterate while updating weights for a number of specified epochs this
way. Loop for number of epochs:

\- Get predictions: $\hat{y}$

\- Compute loss: $\mathscr{L}(w, b)$

\- Make backward pass

\- Update optimizer

It can be mentioned that dataset of electric propulsion is extremely
complex due to large deviations in data. Thanks to adavnces in data
science and ML it is possible to work with it.

This way we assembled dataset on our ROI domain of $P$\<1000 $W$ input
power and 200-500 $V$ range. Sadly one of limitations of such model is
disability to go beyond actual database limit while not sacrificing
performance and accuracy.

## Physics Informed Neural Networks

For working with unscaled data PINN’s were introduced, they are using
equations 2-7 to generate $C_x$ coefficients. Yes, it was said earlier
that this method lacks ability to generate better performing HETs, but
as we have generated larger dataset on same domain as Lee et al.
(2019-11) it is important to control that our dataset is still the same
quality as original. Using above mentioned PINN’s it was possible to fit
coefficients and they showed only slight divergence in values of few %
which is acceptable.

## ML approach notes

We already have discussed how HETFit code works and results it can
generate, the overiew is going to be given in next section. But here i
want to warn that this work is highly experimental and you should always
take ML approaches with a grain of salt, as some plasma discharge
physics in HET is yet to be understood, data driven way may have some
errors in predictions on specific bands. Few notes on design tool I have
developed in this work: it is meant to be used by people with little to
no experience in ML field but those who wants to quickly analyze their
designs or create baseline one for simulations. One can even use this
tool for general tabular data as it has mostly no limits whatsoever to
input data.

## Two input variables prediction

One of main characteristics for any type of thruster is efficiency, in
this work I researched dependency of multiple input values to $\eta_t$.
Results are as follows in form of predicted matrix visualisations.
Figure 3 takes into account all previous ones in the same time, once
again it would be way harder to do without ML.



# Results discussion

Let’s compare predictions of semi empirical approach(Lee et al.
2019-11), approach in paper(Plyashkov et al. 2022-10-25), and finally
ours. Worth to mention that current approach is easiest to redesign from
scratch.

## NN architecture generation algorithm

As with 50 iterations, previously discussed meta learning model is able
to create architecture with score of 0.9+ in matter of seconds. HETFit
allows logging into neptune.ai environment for full control over
simulations. Example trail run looks like that.



## Power density and magnetic flux dependence

Neither of the models currently support taking magnetic flux in account
besides general physics relations, but we are planning on updating the
model in next follow up paper. For now $\vec{B}$ relation to power
remains unresolved to ML approach but the magnetic field distribution on
z axis is computable and looks like that for magnetically shielded
thrusters:



## Dependency of T on d,P

Following graph is describing Thrust as function of channel diameter and
width, where hue map is thrust. It is well known dependency and it has
few around 95% prediction band (Lee et al. 2019-11)



## Dependency of T on P,U



## Dependency of T on $m_a$,P

Compared to(Shagayda and Gorshkov 2013-03) The model accounts for more
parameters than linear relation. So such method proves to be more
precise on specified domain than semi empirical linear relations.



## Dependency of $I_{sp}$ on d,h



We generated many models so far, but using ML we can make single model
for all of the parameters at the same time, so these graphs tend to be
3d projection of such model inference.

## Use of pretrained model in additive manufacturing of hall effect thruster channels

The above mentioned model was used to predict geometry of channel, next
the simulation was conducted on this channel. Second one for comparison
was calculated via usual scaling laws. The initial conditions for both
are:

| Initial condition | Value             |
|:------------------|:------------------|
| $n_{e,0}$         | 1e13 \[m\^-3\]    |
| $\epsilon_0$      | 4 \[V\]           |
| V                 | 300 \[V\]         |
| T                 | 293.15 \[K\]      |
| P\_abs            | 0.5 \[torr\]      |
| $\mu_e N_n$       | 1e25 \[1/(Vm s)\] |
| dt                | 1e-8 \[s\]        |
| Body              | Ar                |

Outcomes are so that ML geometry results in higher density generation of
ions which leads to more efficient thrust generation. HETFit code
suggests HET parameters by lower estimate to compensate for not included
variables in model of HET. This is experimentally proven to be efficient
estimate since SEM predictions of thrust are always higher than real
performance. Lee et al. (2019-11)



## Code description

Main concepts: - Each observational/design session is called an
environment, for now it can be either RCI or SCI (Real or scaled
interface)

\- Most of the run parameters are specified on this object
initialization, including generation of new samples via GAN

\- Built-in feature generation (log10 Power, efficiency, $\vec{B}$,
etc.)

\- Top feature selection for each case. (Boruta algorithm)

\- Compilation of environment with model of choice, can be any torch
model or sklearn one

\- Training

\- Plot, inference, save, export to jit/onnx, measure performance

## COMSOL HET simulations

The simulations were conducted in COMSOL in plasma physics interface
which gives the ability to accurately compute Electron densities,
temperatures, energy distribution functions from initial conditions and
geometry. Here is comparison of both channels.



# Conclusion

In conclusion the another model of scaling laws was made and presented.
HETFit code is open source and free to be used by anyone. Additively
manufactured channel was printed to prove it’s manufactureability.
Hopefully this work will help developing more modern scaling relations
as current ones are far from perfect.

Method in this paper and firstly used in Plyashkov et al. (2022-10-25)
has advantages over SEM one in: ability to preidct performance more
precisely on given domain, account for experimental data. I believe with
more input data the ML method of deisgning thrusters would be more
widely used.

The code in this work could be used with other tabular experimental data
since most of cases and tasks tend to be the same: feature selection and
model optimization.


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</div>

[^1]: Founder, Pure EP