Diffusion models for inverse problems in low temperature plasmas

Thomas Marks and Alex Gorodetsky

Department of Aerospace Engineering

University of Michigan

marksta@umich.edu

Motivation: Electric spacecraft propulsion

Low-thrust, high-efficiency propulsion for station-keeping and long-duration missions

Thomas Marks (marksta@umich.edu)

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Motivation: Electric spacecraft propulsion

Low-thrust, high-efficiency propulsion for station-keeping and long-duration missions

Thomas Marks (marksta@umich.edu)

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Dawn mission (2007 – 2018)

Motivation: Electric spacecraft propulsion

Low-thrust, high-efficiency propulsion for station-keeping and long-duration missions

Thomas Marks (marksta@umich.edu)

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Earth

Vesta

Ceres

Dawn mission (2007 – 2018)

Motivation: Electric spacecraft propulsion

Low-thrust, high-efficiency propulsion for station-keeping and long-duration missions

Thomas Marks (marksta@umich.edu)

5

Dawn mission (2007 – 2018)

Earth

Vesta

Ceres

Equivalent chemical rocket system

Motivation: Hall thrusters

Thomas Marks (marksta@umich.edu)

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Widely-flown type of EP thruster, crossed electric and magnetic fields to accelerate a plasma

Motivation: Hall thrusters

Thomas Marks (marksta@umich.edu)

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Satellite station-keeping

(Starlink)

Widely-flown type of EP thruster, crossed electric and magnetic fields to accelerate a plasma

Motivation: Hall thrusters

Thomas Marks (marksta@umich.edu)

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Uncrewed exploration

(Psyche)

Satellite station-keeping

(Starlink)

Widely-flown type of EP thruster, crossed electric and magnetic fields to accelerate a plasma

Motivation: Hall thrusters

Thomas Marks (marksta@umich.edu)

9

Uncrewed exploration

(Psyche)

Satellite station-keeping

(Starlink)

Crewed Mars exploration

(future)

Widely-flown type of EP thruster, crossed electric and magnetic fields to accelerate a plasma

Motivation: Hall thrusters

Thomas Marks (marksta@umich.edu)

10

Uncrewed exploration

(Psyche)

Satellite station-keeping

(Starlink)

Crewed Mars exploration

(future)

Widely-flown type of EP thruster, crossed electric and magnetic fields to accelerate a plasma

Unable to be simulated predictively due to micro-scale plasma instabilities

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

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• Kinetic plasma instabilities lead to enhanced ”anomalous” electron transport
• Inverse energy cascade makes RANS-like closure challenging, no success to date

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Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

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• Kinetic plasma instabilities lead to enhanced ”anomalous” electron transport
• Inverse energy cascade makes RANS-like closure challenging, no success to date

​

​

​

​

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

13

• Kinetic plasma instabilities lead to enhanced ”anomalous” electron transport
• Inverse energy cascade makes RANS-like closure challenging, no success to date

​

​

​

​

Predictive modeling requires expensive whole-device kinetic simulations

These are currently infeasible (runtimes ~ years)

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

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• Instabilities manifest as enhanced collisionality in the bulk flow, like an eddy viscosity
• We prescribe an “anomalous collision frequency” field which is adjusted to match data
• This is an unknown, 1D spatial field that must be inferred.

​

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z (axial direction)

Laser velocimetry

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

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• Instabilities manifest as enhanced collisionality in the bulk flow, like an eddy viscosity
• We prescribe an “anomalous collision frequency” field which is adjusted to match data
• This is an unknown, 1D spatial field that must be inferred.

​

​

z (axial direction)

Laser velocimetry

Forward problem

Simulate plasma

given collision freq.

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

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• Instabilities manifest as enhanced collisionality in the bulk flow, like an eddy viscosity
• We prescribe an “anomalous collision frequency” field which is adjusted to match data
• This is an unknown, 1D spatial field that must be inferred.

​

​

z (axial direction)

Laser velocimetry

Forward problem

Simulate plasma

given collision freq.

Often, anomalous collision frequency profiles tuned manually by practitioners

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

17

• Instabilities manifest as enhanced collisionality in the bulk flow, like an eddy viscosity
• We prescribe an “anomalous collision frequency” field which is adjusted to match data
• This is an unknown, 1D spatial field that must be inferred.

z (axial direction)

Laser velocimetry

Forward problem

Simulate plasma

given collision freq.

Often, anomalous collision frequency profiles tuned manually by practitioners

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

18

• Instabilities manifest as enhanced collisionality in the bulk flow, like an eddy viscosity
• We prescribe an “anomalous collision frequency” field which is adjusted to match data
• This is an unknown, 1D spatial field that must be inferred.

z (axial direction)

Laser velocimetry

Forward problem

Simulate plasma

given collision freq.

Often, anomalous collision frequency profiles tuned manually by practitioners

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

19

• Instabilities manifest as enhanced collisionality in the bulk flow, like an eddy viscosity
• We prescribe an “anomalous collision frequency” field which is adjusted to match data
• This is an unknown, 1D spatial field that must be inferred.

z (axial direction)

Laser velocimetry

Forward problem

Simulate plasma

given collision freq.

Often, anomalous collision frequency profiles tuned manually by practitioners

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

20

• Instabilities manifest as enhanced collisionality in the bulk flow, like an eddy viscosity
• We prescribe an “anomalous collision frequency” field which is adjusted to match data
• This is an unknown, 1D spatial field that must be inferred.

z (axial direction)

Laser velocimetry

Forward problem

Simulate plasma

given collision freq.

The calibrated simulations that result are used to estimate other plasma properties

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

21

• Instabilities manifest as enhanced collisionality in the bulk flow, like an eddy viscosity
• We prescribe an “anomalous collision frequency” field which is adjusted to match data
• This is an unknown, 1D spatial field that must be inferred.

z (axial direction)

Laser velocimetry

The resulting fits are not unique with respect to the observables

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

22

• Instabilities manifest as enhanced collisionality in the bulk flow, like an eddy viscosity
• We prescribe an “anomalous collision frequency” field which is adjusted to match data
• This is an unknown, 1D spatial field that must be inferred.

z (axial direction)

Laser velocimetry

Forward problem

Simulate plasma�given collision freq.

The resulting fits are not unique with respect to the observables

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

23

• Instabilities manifest as enhanced collisionality in the bulk flow, like an eddy viscosity
• We prescribe an “anomalous collision frequency” field which is adjusted to match data
• This is an unknown, 1D spatial field that must be inferred.

z (axial direction)

Laser velocimetry

Forward problem

Simulate plasma�given collision freq.

This complicates calibration and makes us overly-certain in predicting other properties

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

24

• Instabilities manifest as enhanced collisionality in the bulk flow, like an eddy viscosity
• We prescribe an “anomalous collision frequency” field which is adjusted to match data
• This is an unknown, 1D spatial field that must be inferred.

z (axial direction)

Laser velocimetry

Forward problem

Simulate plasma�given collision freq.

?

Inverse problem

Obtain collision freq.

(and other states)

from ion velocity

Want rapid, automatic calibration to obtain plasma state estimates

Motivation: Anomalous transport

Thomas Marks (marksta@umich.edu)

25

• Instabilities manifest as enhanced collisionality in the bulk flow, like an eddy viscosity
• We prescribe an “anomalous collision frequency” field which is adjusted to match data
• This is an unknown, 1D spatial field that must be inferred.

z (axial direction)

Laser velocimetry

Forward problem

Simulate plasma�given collision freq.

?

Inverse problem

Obtain collision freq.

(and other states)

from ion velocity

Want rapid, automatic calibration to obtain plasma state estimates with uncertainty

Other approaches

In the past, we have used MCMC for this

Thomas Marks (marksta@umich.edu)

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Inverse problem

Obtain collision freq.

(and other states)

from ion velocity

Marks, T. A., Eckels, J. D., Mora, G. A., & Gorodetsky, A. A. (2025) Journal of Applied Physics, 138(15)

Other approaches

In the past, we have used MCMC for this

Thomas Marks (marksta@umich.edu)

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Inverse problem

Obtain collision freq.

(and other states)

from ion velocity

Marks, T. A., Eckels, J. D., Mora, G. A., & Gorodetsky, A. A. (2025) Journal of Applied Physics, 138(15)

• MCMC is parametric, we might not have an appropriate parametric model
• MCMC scales poorly with number of dimensions
• Inference is expensive online – no work saved for repeated inverse problems

Generative models

Generative models sample from empirical distributions and can solve inverse problems

Thomas Marks (marksta@umich.edu)

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?

?

Generative models

Thomas Marks (marksta@umich.edu)

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?

?

Generative models sample from empirical distributions and can solve inverse problems

Generative models

Generative models sample from empirical distributions and can solve inverse problems

Thomas Marks (marksta@umich.edu)

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Given (input, output) pairs, we can train a

model to sample from the joint distribution

Inputs

Outputs

?

?

Generative models

Generative models sample from empirical distributions and can solve inverse problems

Thomas Marks (marksta@umich.edu)

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We sample the conditional distribution to map outputs to inputs

Inputs

Outputs

Inputs

Outputs

Given (input, output) pairs, we can train a

model to sample from the joint distribution

?

?

Generative models

Generative models sample from empirical distributions and can solve inverse problems

Thomas Marks (marksta@umich.edu)

32

?

?

We sample the conditional distribution to map outputs to inputs

Inputs

Outputs

Inputs

Outputs

Given (input, output) pairs, we can train a

model to sample from the joint distribution

Bayesian posterior

Method outline

Thomas Marks (marksta@umich.edu)

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Generate library of simulations

Method outline

Thomas Marks (marksta@umich.edu)

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Generate library of simulations

Train denoising diffusion model

Method outline

Thomas Marks (marksta@umich.edu)

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Generate library of simulations

Train denoising diffusion model

Sample posterior distribution

Data generation

• Generate random samples of scalar inputs (voltage, flow rate, various model params) and random collision freq. profiles (parametric plus Gaussian random field, GRF)

Thomas Marks (marksta@umich.edu)

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Data generation

• Generate random samples of scalar inputs (voltage, flow rate, various model params) and random collision freq. profiles (parametric plus Gaussian random field, GRF)
• Use 1D fluid code to map these to outputs (plasma properties, performance)

​

Thomas Marks (marksta@umich.edu)

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Model architecture

• UNet architecture and sampling procedure based on EDM2 (Karras et al, 2024)
• Training: add noise to data tensor, attempt to predict clean version

Thomas Marks (marksta@umich.edu)

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Simulation results

put into tensor

N cells x Nc channels

Model architecture

• UNet architecture and sampling procedure based on EDM2 (Karras et al, 2024)
• Training: add noise to data tensor, attempt to predict clean version

Thomas Marks (marksta@umich.edu)

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Simulation results

put into tensor

N cells x Nc channels

Model architecture

• UNet architecture and sampling procedure based on EDM2 (Karras et al, 2024)
• Training: add noise to data tensor, attempt to predict clean version

Thomas Marks (marksta@umich.edu)

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Simulation results

put into tensor

N cells x Nc channels

Training

• Minimize mean squared difference between original tensor and denoised tensor

Thomas Marks (marksta@umich.edu)

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ℒ

Original

tensor

Denoised

tensor

2

Weight based on

noise added

Reconstruction loss

Derivative losses

Goal: ensure smooth numerical differences for future physics guidance

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1st

2nd

Training

• Minimize mean squared difference between original tensor and denoised tensor

Thomas Marks (marksta@umich.edu)

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ℒ

Original

tensor

Denoised

tensor

2

Weight based on

noise added

Reconstruction loss

Derivative losses

Goal: ensure smooth numerical differences for future physics guidance

​

​

1st

2nd

Training - results

Denoising network works well

Thomas Marks (marksta@umich.edu)

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Noisy sample

Denoised

Original

Training - results

Denoising network works well

Thomas Marks (marksta@umich.edu)

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Noisy sample

Denoised

Original

Sampling

• Use diffusion posterior sampling (DPS) to permit conditioning on data at inference time
• No physics guidance

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Thomas Marks (marksta@umich.edu)

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Sampling

• Use diffusion posterior sampling (DPS) to permit conditioning on data at inference time
• No physics guidance

​

Thomas Marks (marksta@umich.edu)

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Points from current sample to clean diffusion model prediction (high prior prob.)

Points toward better agreement with observations (high likelihood)

Prior score function

Likelihood score function

Measurement operator

Measurement variance

Sampling

• Use diffusion posterior sampling (DPS) to permit conditioning on data at inference time
• No physics guidance

​

Thomas Marks (marksta@umich.edu)

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Points from current sample to clean diffusion model prediction (high prior prob.)

Points toward better agreement with observations (high likelihood)

Prior score function

Likelihood score function

Measurement operator

Measurement variance

Forward problem

• Evaluation as a forward surrogate (condition on anomalous collision frequency)

Thomas Marks (marksta@umich.edu)

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Forward problem

• Evaluation as a forward surrogate (condition on anomalous collision frequency)

Thomas Marks (marksta@umich.edu)

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Forward problem

• Evaluation as a forward surrogate (condition on anomalous collision frequency)

Thomas Marks (marksta@umich.edu)

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• Diffusion model obeys conditioning
• Other fields recovered with high accuracy and low uncertainty
• Density variation due to high sensitivity to anomalous collision frequency

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Inverse problem

• Comparison to MCMC ground truth when conditioned on full observation of ion velocity

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Thomas Marks (marksta@umich.edu)

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Inverse problem

Thomas Marks (marksta@umich.edu)

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• Comparison to MCMC ground truth when conditioned on full observation of ion velocity

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Inverse problem

Thomas Marks (marksta@umich.edu)

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• Comparison to MCMC ground truth when conditioned on full observation of ion velocity

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Inverse problem

• Comparison to MCMC

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Thomas Marks (marksta@umich.edu)

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MCMC: 5 days to sample

Diffusion: 5 days to gen data and train, 2 minutes to sample

Comparison to experimental methods

• Test on noisy experimental data (not from model), compare to semi-analytic method of Perez-Luna et al. for obtaining electric field and ionization frequency from ion velocity distribution functions

Thomas Marks (marksta@umich.edu)

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z (axial direction)

Laser velocimetry

Comparison to experimental methods

• Test on noisy experimental data (not from model), compare to semi-analytic method of Perez-Luna et al. for obtaining electric field and ionization frequency from ion velocity distribution functions

Thomas Marks (marksta@umich.edu)

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z (axial direction)

Laser velocimetry

Comparison to experimental methods

• Test on noisy experimental data (not from model), compare to semi-analytic method of Perez-Luna et al. for obtaining electric field and ionization frequency from ion velocity distribution functions

Thomas Marks (marksta@umich.edu)

57

z (axial direction)

Laser velocimetry

Comparison to experimental methods

• Test on noisy experimental data (not from model), compare to semi-analytic method of Perez-Luna et al. for obtaining electric field and ionization frequency from ion velocity distribution functions

Thomas Marks (marksta@umich.edu)

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z (axial direction)

Laser velocimetry

• Diffusion model assimilates real data and produces field estimates that agree with semi-analytic methods
• Predicts fields not accessible from these methods (anomalous collision frequency)

• Comparison to method of Roberts and Jorns (reduce PDEs to simplified 1D ODE)
• Requires laser measurements of ion vel, plasma density, electron temp. electric field

​

Thomas Marks (marksta@umich.edu)

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Simulated data (added noise and subsampled)

Comparison to experimental methods

• Comparison to method of Roberts and Jorns (reduce PDEs to simplified 1D ODE)
• Requires laser measurements of ion vel, plasma density, electron temp. electric field

​

Thomas Marks (marksta@umich.edu)

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Comparison to experimental methods

• Comparison to method of Roberts and Jorns (reduce PDEs to simplified 1D ODE)
• Requires laser measurements of ion vel, plasma density, electron temp. electric field

​

Thomas Marks (marksta@umich.edu)

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Comparison to experimental methods

• Next, evaluate only on data obtainable from laser induced fluorescence

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Thomas Marks (marksta@umich.edu)

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Comparison to experimental methods

• Next, evaluate only on data obtainable from laser induced fluorescence
• Uncertainty increases, but median predictions still very good

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Thomas Marks (marksta@umich.edu)

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Comparison to experimental methods

Scaling evaluation

Calculate maximum mean discrepancy (MMD) between generated samples and unconditional/conditional reference datasets

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Thomas Marks (marksta@umich.edu)

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• Data size and sampling steps have largest influence on quality
• Model size improves quality, but weakly
• Limited impact of derivative scale

Applications

• Optimal experimental design
• Evaluate which data points give the most uncertainty reduction to slim down and target experimental campaigns

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• Online control
• Identify and recover from corruptions and reach operating set points in real time

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• Thruster design and experimentation
• Train diffusion model concurrently with thruster construction, use to estimate unseen properties to inform design changes

Thomas Marks (marksta@umich.edu)

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Funding Acknowledgements

Thomas Marks (marksta@umich.edu)

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This work was supported by the Laboratory Directed Research and Development program at Sandia National Laboratories, a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Conclusion

Thomas Marks (marksta@umich.edu)

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Future work

• Physics constraints during sampling
• Speed up sampling
• Online control
• Parametric inference
• New systems

• Diffusion models applied to fast approximate Bayesian inference in plasmas
• Cost (including training) competitive with MCMC, sampling much aster
• Can be conditioned on noisy experimental data, improved flexibility and accuracy vs existing methods