Transfer Learning of Multiobjective Indirect Low-Thrust Trajectories Using Diffusion Models and Markov Chain Monte Carlo

arXiv:2605.09125 · eess.SY, cs.LG, cs.SY, math.OC · Submitted 2026-05-09 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "Transfer Learning of Multiobjective Indirect Low-Thrust Trajectories Using Diffusion Models and Markov Chain Monte Carlo".

Dev: The gist:

Rosa: First, who's behind it and why it matters.

Title and authors: Rosa: So, let's talk about who wrote this and what they're calling their work. The paper is titled "Transfer Learning of Multiobjective Indirect Low-Thrust Trajectories Using Diffusion Models and Markov Chain Monte Carlo." The authors are Jannik Graebner and Ryne Beeson.

Dev: They are focusing on making the training data generation more efficient for diffusion models, specifically by combining parameter homotopy with MCMC to handle indirect trajectory optimization problems.

Taro: So, when we look at the title again, "Transfer Learning," it suggests they are building a system that can learn something from one context and apply it effectively to a new, related context.

Rosa: That’s right. It means they are using past solutions to train the diffusion model so that when you change mission parameters slightly, the model doesn't need entirely new training data for those new values.

Dev: The implication is that this bypasses the problem where generating training data for these models is usually very expensive because you have to run a gradient-based numerical solver for every single parameter value.

Taro: So instead of running that expensive solver repeatedly, they are using homotopy to keep the problems linked together so they can generate training data more cheaply.

Rosa: That’s the core mechanism, and it allows them to explore a continuous range of mission-parameter values while keeping those successive optimization problems sufficiently similar.

Dev: It’s about generating that training data more efficiently, which is what they state in the abstract when they introduce this transfer learning framework.

Taro: So we're moving from a situation where we generate data at fixed parameter values to one where we can extrapolate to new ones based on learned distributions.

The paper's summary: Rosa: Now let’s go into the main summary of this work for "Transfer Learning of Multiobjective Indirect Low-Thrust Trajectories Using Diffusion Models and Markov Chain Monte Carlo." They say the approach reformulates the multiobjective optimization problem as sampling from an unnormalized target distribution in costate space.

Dev: This means instead of solving it as a fixed problem, they're treating it like drawing samples from some underlying density function that describes all the possible optimal trajectories.

Taro: So, what does that actually mean for a mission designer? It shifts the task from finding one single point to understanding the whole landscape of solutions.

Rosa: Exactly. And this allows diffusion models to learn a conditional sampling distribution over these clusters in costate space, which are shown in Figure one <ref:2605.09125#pg1>.

Dev: Those clusters in costate space correspond to families of locally optimal trajectories, and learning a conditional sampling distribution over those enables efficient generation of new trajectory candidates across parameter values.

Taro: That’s what I mean when I say they can generate new trajectory candidates without starting from scratch for every single parameter value.

Rosa: And this whole process is accelerated by using the diffusion models to learn a conditional sampling distribution over these clusters, which is key for efficient generation of new solutions.

Dev: The limitation they mention in the text is that generating training data remains expensive, and opportunities exist to better exploit past data.

Taro: So they acknowledge that creating all that initial training data was costly before this framework existed.

The paper's improvements: Rosa: Let’s talk about what improvements the authors suggest in "Transfer Learning of Multiobjective Indirect Low-Thrust Trajectories Using Diffusion Models and Markov Chain Monte Carlo." They are focusing on using homotopy in a mission parameter with MCMC to generate training data more efficiently.

Dev: This means they're not just doing one thing, but combining these techniques into a unified framework to leverage existing training data better.

Taro: So the improvement is about making the process of creating that data less reliant on generating it anew every time you change a mission parameter.

Rosa: Precisely. It lets them generate training data across a continuous range of mission-parameter values while keeping successive problems sufficiently similar for transfer learning to work.

Dev: At each homotopy step, samples obtained for one parameter value are used to initialize the Markov chains for the next, which transfers the learned solution structure across that entire parameter space.

Taro: That means if we've already solved a problem well at one setting, we don't have to re-solve it completely when we move to a new setting.

Rosa: So they are transferring the learned solution structure across the mission-parameter space using these homotopy steps as the bridge.

Dev: This is how they improve upon existing methods by creating a data generation pipeline that is much less computationally demanding for those diffusion models.

Taro: It sounds like a big step forward in making this kind of AI applicable to real-world, high-cadence mission design scenarios where you need solutions quickly.

Conclusion: Rosa: We've covered the main points of "Transfer Learning of Multiobjective Indirect Low-Thrust Trajectories Using Diffusion Models and Markov Chain Monte Carlo." To summarize, this work proposes combining parameter homotopy with MCMC to generate training data more efficiently for diffusion models.

Dev: The key is that they are sampling from an unnormalized target distribution in costate space rather than solving a fixed problem.

Taro: And the implication is that we can use learned structures to quickly generate solutions for new mission parameters without having to start all over again every time we change the parameters.

Rosa: This leads to a denser Pareto front and higher quality results because of how they fine-tuning the diffusion model with reward-weighted data.

Dev: The overall idea is that this is a way to generate solution data across a continuous range of mission-parameter values, which is really useful for indirect trajectory optimization problems.

Taro: It shows MCMC and diffusion models are well suited for this kind of transfer learning in the field.

Princeton University

eess.SY, cs.LG, cs.SY, math.OC

Submitted: 2026-05-09

Updated: 2026-10-08

Comments: v2: Updated publication information only; manuscript content is unchanged. The version of record is available at https://doi.org/10.1007/s40295-026-00630-x

Journal ref: The Journal of the Astronautical Sciences 73 (2026) 91

DOI: 10.1007/s40295-026-00630-x

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 80/100

The gist: The gist: This work introduces a transfer-learning framework that combines homotopy in a mission parameter with Markov chain Monte Carlo (MCMC) to generate training data more efficiently for

Key concepts

Transfer Learning Framework
A method that reuses knowledge from one problem (e.g., optimizing for one set of mission parameters) to solve a related but different problem (e.g., new mission parameters). Here, it transfers learned solution structures across varying mission parameter values by using samples from one parameter value to initialize the next.
Diffusion Models
Deep generative machine learning models trained to learn complex data distributions. In this context, they are used to learn a conditional distribution over high-quality initial costates (like starting points for trajectories) by gradually adding and removing Gaussian noise from samples.

Terminology

Summary

The gist: This work introduces a transfer-learning framework that combines homotopy in a mission parameter with Markov chain Monte Carlo (MCMC) to generate training data more efficiently for diffusion models, enabling them to learn a global representation of the underlying solution distribution and efficiently generate new solutions for indirect trajectory optimization problems with varying parameters.

How it works

The proposed framework reformulates a multiobjective optimization problem as sampling from an unnormalized target distribution in costate space The approach combines parameter homotopy with MCMC to leverage existing training data when constructing solutions for new parameter values. Performing homotopy over full solution distributions, rather than over individual solutions, allows data to be generated across a continuous range of mission-parameter values while keeping successive problems sufficiently similar. At each homotopy step, samples obtained for one parameter value are used to initialize the Markov chains for the next, thereby transferring the learned solution structure across the mission-parameter space.

Diffusion Models for Costate Generation

Diffusion models are employed to learn a conditional distribution over high-quality initial costates λ for LT transfers. The model is defined through a forward diffusion process that gradually corrupts costate samples with Gaussian noise and a learned reverse process that reconstructs the costate distribution from noise. A neural network is trained to predict the Gaussian noise added at each diffusion step, and classifier-free guidance (CFG) is used for conditional sample generation to combine conditional and unconditional denoising predictions.

MCMC Methods for Costate Sampling

MCMC methods are used to generate samples from a target distribution with unnormalized density π(λ) by constructing a Markov chain whose stationary distribution is π(λ). Three MCMC variants are compared: Random-Walk Metropolis (RWM), Metropolis-Adjusted Langevin Algorithm (MALA), and Hamiltonian Monte Carlo (HMC). Gradient-based samplers like MALA and HMC are preferred because they incorporate the gradient information into the proposal mechanism, allowing the chain to better exploit local information about the geometry of the target distribution. For example, MALA uses a modified proposal that augments a random walk with a gradient drift step before drawing the stochastic proposal.

Reward-Weighted Fine-Tuning

After the homotopy-MCMC stage has generated costate samples for new mission-parameter values, the baseline diffusion model is adapted to this data through supervised fine-tuning. The loss function is defined as L(θ) = EP new h R(λ0; α) ϵ − ϵθ(λn, n, α) 2. The reward function R(λ; α) is chosen as a rescaled version of the target distribution, where costates with lower objective values contribute more strongly during fine-tuning.

Objective Function Evaluation

Each MCMC iteration requires evaluating the target density π(λ α), which reduces to evaluating the objective function J∗(λ; α). This is computed through a preliminary screening procedure where shooting and coast times are selected to minimize J(λ, τs, τf; α). The gradient approximation involves fixing the shooting and coast times at the current sample’s optimal values to define a continuously differentiable frozen-time objective J k(λ).

Comparison of MCMC Methods

The comparison shows that incorporating gradient information leads to overall improved performance, even when accounting for the additional computational overhead. Among the three algorithms tested, MALA achieved the strongest overall performance, providing the best balance of feasibility, solution quality, Pareto-front coverage, and computational cost. MALA achieves by far the highest feasibility rate and produces a denser Pareto front than state-of-the-art approaches like Russell ACT.

Diffusion-Model Fine-Tuning

The final step involves using the costate samples generated by MALA, together with their reward values, to fine-tune the baseline diffusion model according to Eq. (28). The fine-tuned model is able to generate additional high-quality samples that fill gaps in the solution space and lead to a denser Pareto front after warm-starting the numerical solver.

The results demonstrate that MCMC is well suited for transfer learning in indirect multiobjective trajectory optimization and that combining it with diffusion-model fine-tuning provides an effective way to generate solution data across a continuous range of mission-parameter values.

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Preliminary low-thrust spacecraft mission design is a global search problem characterized by a complex solution landscape, multiple objectives, and numerous local minima >

It is therefore of interest to mission designers to understand how to efficiently and thoroughly identify Pareto-optimal solutions with respect to these objectives >

Direct methods transcribe the optimal control problem as stated into a higher-dimensional parameter optimization problem >

By defining a probability density function supported on solution clusters, the global search problem is recast as sampling from a distribution with unnormalized density >

This makes diffusion models, which are deep generative machine learning models designed to learn complex data distributions, a natural choice for accelerating the generation of new solutions >

Improvements for AI systems

  1. Bold header: Transfer-Learning for Training Data Generation

This improves AI systems by enabling efficient and automated generation of new training data for diffusion models through combining parameter homotopy with MCMC, allowing extrapolation to new mission parameters beyond the originally represented dataset.

  1. Bold header: High-Quality Costate Initialization via MCMC

The system can generate high-quality initial costates for a numerical solver by using gradient-based variants [like MALA and HMC] which are shown to achieve the best trade-off between sample quality and computational cost, specifically achieving a significantly higher feasibility rate than state-of-the-art adjoint control transformations.

  1. Bold header: Global Solution Distribution Learning

The diffusion model can be fine-tuned to learn a global representation of the underlying solution distribution across varying mission parameters, enabling it to generate new solutions for previously unseen or sparsely sampled parameter values.

  1. Bold header: Objective Function Evaluation via Surrogate Gradients

The MCMC sampling is accelerated by computing a differentiable surrogate by fixing shooting and coast times at the current sample, defining the frozen-time objective Jk(λ) = J(λ, τks, τkf), which ensures that the gradient approximation provides a descent direction for J∗ at λk whenever d != 0.

  1. Bold header: Enhanced Pareto Front Coverage

The resulting AI system can produce a denser Pareto front and achieve a higher quality Pareto front than a state-of-the-art indirect approach, as MALA achieves the greatest diversity in the objective and fills regions of the solution space not reached by standard methods.

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