The research, published in Frontiers in Space Technologies by Jonah Ekelund and Stefano Markidis of KTH’s Department of Computational Science and Technology, tackles the bounded one-revolution elliptic Lambert problem, the workhorse formulation used when a spacecraft completes less than one full orbit between departure and arrival. Rather than trying to replace high-precision Lambert solvers outright, the authors built neural surrogates: fast approximate models trained to predict the semi-major axis of the transfer ellipse directly from the geometry of the problem and the prescribed time of flight. Their central insight is that the difficulty of learning this mapping lies less in the network architecture than in how the problem itself is presented to the network.

That insight stems from a subtle but troublesome property of Lambert’s problem. The admissible ranges of both the time of flight and the semi-major axis depend heavily on the transfer geometry. For one target position, the valid semi-major axis might span roughly 2.5 to 5.3 canonical distance units, while for another it stretches from 4.1 to 8.2, with time-of-flight ranges shifting just as dramatically. A neural network trained naively on such data must cope with a solution domain that mutates from one example to the next. The researchers’ remedy is a geometry-aware normalization: using the minimum-energy ellipse connecting the two positions and the boundary between the so-called short-way and long-way solutions, each transfer instance is rescaled into a common canonical representation bounded between zero and one.

The effect of this preprocessing proved striking. When identical multilayer perceptrons were trained with four different normalization schemes, the geometry-aware methods improved the final training loss by two orders of magnitude compared with standard z-score normalization and by one order of magnitude compared with ordinary minmax scaling. In the normalized coordinates, solutions for different target positions collapse toward shared patterns, with values converging toward a central value as the target radius approaches its extremes. Structure that was previously hidden in geometry-dependent variation becomes learnable, and the authors argue that this transformation, rather than any particular network design, is the study’s main contribution.

On that normalized foundation, the team compared three architectures with distinctly different inductive biases. The first was a standard multilayer perceptron, the long-established workhorse of deep learning, composed of stacked linear transformations and nonlinear activations. The second was DeepONet, a neural-operator architecture that splits the task between a branch network, which encodes the target position, and a trunk network, which evaluates characteristic functions of the time of flight; the two are combined to form the prediction. The third was a Kolmogorov-Arnold Network, or KAN, an unconventional design in which learnable activation functions, built from B-splines, sit on the edges between nodes rather than fixed activations following learned weights.

Each model was tuned and then evaluated on a validation set of nearly 300,000 samples drawn from a dataset of roughly a million transfer instances generated in canonical units. The multilayer perceptron emerged as the clear accuracy champion, with DeepONet second and KAN third. All three struggled most with small target radii and with target angles near zero or 360 degrees, precisely the regions where the underlying solution varies most steeply. The structured architectures nevertheless offered rewards of their own: the DeepONet’s trunk functions closely mirrored the normalized solution curves, and the KAN’s learned edge functions revealed how the prediction depends on each input, providing an interpretable window into the solution space that the MLP cannot match.

The decisive test came in two realistic mission-design scenarios. In the first, the networks generated porkchop plots, the contour maps of fuel cost that designers use to pick launch and arrival dates, for an Earth-Mars transfer window during 2018. The multilayer perceptron identified exactly the same optimal departure and arrival dates as a classical Lambert solver, May 12 and December 3, 2018, and estimated the required delta-v within a few metres per second of the classical value of 5.7507 kilometres per second. DeepONet and KAN drifted a few days from the optimum and incurred penalties of tens to hundreds of metres per second. Near the minimum-energy boundary of the solution space, where the semi-major axis approaches its lower limit, all the networks showed discontinuities and occasional invalid solutions, a reminder that the surrogates remain approximations rather than solvers.

In the second scenario, the team evaluated the networks on a multi-planetary flyby sequence from Earth to Saturn via Venus, Earth and Jupiter, mirroring the trajectory profile of a classic outer-planet tour. Computing each heliocentric arc between encounters and the gravity-assist geometry at each flyby, the machine-learning models reproduced the total mission delta-v to within roughly 40 metres per second of the classical estimate of 11.0664 kilometres per second. Individual elements carried larger errors; DeepONet’s Jupiter flyby periapsis was off by more than 41,000 kilometres, or 0.8 percent, a figure that looks small in relative terms but is enormous in absolute distance. The authors conclude that the surrogates reliably capture coarse, sequence-level trends, which is exactly what preliminary design demands, while individual trajectory details still require classical refinement.

Speed, of course, is the entire point of a surrogate, and the benchmark results were more modest than a casual observer might hope. On large grids of a million trajectories, the GPU-based MLP achieved a speedup of only 1.15 over a modern multithreaded Lambert solver, with DeepONet and KAN slightly faster. When the classical solver was stripped of its iterative refinement and allowed to return only its initial approximation, the networks’ advantage shrank to 1.10. For small problems, such as a 100-by-100 grid, the traditional solver wins outright. The authors are candid that classical methods remain preferable at small scale, and that the neural surrogates pay for their modest speed gains with measurably higher error.

The broader significance of the work lies in its methodological lesson for machine learning in astrodynamics. Deep-learning approaches are already used to design low-thrust asteroid-belt transfers, asteroid flyby sequences and Jupiter-moon flyby maps, and hybrid schemes have used networks to seed iterative Lambert solvers. What this study demonstrates is that careful, physics-informed normalization can matter more than architectural sophistication: a plain multilayer perceptron, given a geometry-aware representation of the problem, outperformed two more exotic and more interpretable architectures. The authors note that a complete Lambert solver must handle multi-revolution and non-elliptic cases beyond the bounded elliptic domain studied here, and extending the normalization to those broader configurations is future work. For now, the message to mission designers is encouraging: when the mathematics is framed the right way, even simple neural networks can learn the shape of the interplanetary transfer landscape well enough to point the way to promising trajectories before the heavy computation begins.

Subject of Research: Neural network surrogates for solving the elliptic Lambert transfer problem in preliminary spacecraft mission design

Article Title: Neural network surrogates for elliptic Lambert transfers in preliminary mission design

Article References: Ekelund, J., & Markidis, S. (2026). Neural network surrogates for elliptic Lambert transfers in preliminary mission design. Frontiers in Space Technologies, 7, Article 1772629. https://doi.org/10.3389/frspt.2026.1772629

Image Credits: AI Generated

DOI: 10.3389/frspt.2026.1772629

Keywords: Lambert's problem, orbital mechanics, neural networks, mission design, interplanetary trajectories, machine learning, DeepONet, Kolmogorov-Arnold Networks, multilayer perceptron, gravity assists, trajectory optimization, astrodynamics

The core idea is deceptively simple. Instead of solving the full multiple-gravity-assist trajectory optimisation problem for every Kuiper Belt Object, the team trained a Gaussian process regression model to predict how expensive a transfer would be, based only on the orbital characteristics of the target. The training labels for this model came from genuine high-fidelity optimisations: for a set of representative targets, the researchers ran a complete optimisation of an Earth–Jupiter–Kuiper Belt Object transfer, including the deep-space manoeuvres that allow a spacecraft to adjust its course between gravity assists. Each optimisation produced two numbers that matter enormously to mission planners: the total deep-space manoeuvre velocity increment, essentially the fuel cost of the transfer, and the time of flight. Those pairs of numbers became the ground truth from which the surrogate learned.

A crucial design decision was how to represent each target to the machine-learning algorithm. The team compared seven different ways of encoding a Kuiper Belt Object’s orbit, ranging from classical orbital elements such as semimajor axis, eccentricity and inclination, to Cartesian, cylindrical and spherical position-and-velocity states sampled at fixed epochs. This choice matters because a Gaussian process model builds its predictions from similarities between inputs, and a representation that hides the physically relevant structure of the problem will degrade the predictions even if the underlying data are identical. After systematic comparison, a multi-epoch Cartesian representation, in which the target’s heliocentric state is recorded at several fixed times, gave the most balanced performance across both prediction of transfer cost and ranking of targets by accessibility.

Two further refinements sharpened the model’s usefulness. First, the researchers applied a logarithmic scaling to the transfer objective, which concentrated the model’s accuracy in the low-cost region of the distribution, precisely where mission designers care most, because targets with transfer costs below roughly one kilometre per second of deep-space manoeuvring are the ones worth pursuing. Second, they employed uncertainty-guided adaptive sampling, a technique rooted in Bayesian optimisation in which the model itself identifies the regions of input space where its predictions are least confident, and new expensive optimisations are run there to improve the training set. This active-learning strategy measurably improved both the precision and the recall of the screening process at the tested screening fractions of 10 and 20 percent, meaning the model both missed fewer genuinely accessible targets and wasted less attention on inaccessible ones.

The quantitative results are striking. On an independent test set of targets the model had never seen, the Gaussian process regression achieved a global mean absolute error of 0.673 kilometres per second on the transfer objective and 537.5 days on the time of flight. Those global figures, however, mask a far more encouraging picture in the region that matters. For cases whose validated transfer objective was below one kilometre per second, the mean absolute error dropped to just 0.133 kilometres per second, and for cases the surrogate itself selected as having predicted costs below that threshold, the error was 0.245 kilometres per second. The larger global mean reflects the sparse high-cost tail of the objective distribution, where a handful of extremely expensive transfers dominate the average error but are of little practical interest for mission selection.

The demonstration that ties the method together was applied to a catalogue of 2,223 representative Kuiper Belt Object targets. Running the surrogate across this population, the team identified the ten targets with the lowest predicted transfer objectives, candidates that would then be passed on for full multiple-gravity-assist and deep-space-manoeuvre validation. This two-stage workflow, cheap surrogate screening followed by expensive high-fidelity confirmation, is the same philosophy that has transformed other branches of engineering design, and the study shows it can be brought to bear on one of the most computationally demanding problems in astrodynamics. The researchers note that the approach is demonstrated for the prescribed Earth–Jupiter–Kuiper Belt Object transfer scenario, with the gravity-assist sequence fixed in advance.

The technical machinery underneath the method draws on a rich literature. Lambert’s problem, the classical boundary-value problem of orbital mechanics that connects two positions in space with a transfer orbit, underpins the generation of candidate trajectory legs, and modern robust solvers have made such computations fast and reliable. The global optimisation of the full trajectory, with its mixture of continuous variables such as departure epochs, manoeuvre magnitudes and flyby geometry, relies on evolutionary algorithms such as differential evolution, which have proven robust for the multimodal landscapes that trajectory optimisation produces. Gaussian process regression, meanwhile, brings a distinctive advantage over other machine-learning approaches: it provides not just a point prediction but an associated uncertainty estimate for every input, which is exactly what makes uncertainty-guided adaptive sampling possible and what allows a mission designer to know how much to trust a given screening result.

Why does this matter beyond the mathematics? The Kuiper Belt is one of the last great unexplored reservoirs of the solar system, a ring of icy bodies beyond Neptune that preserves a record of the primordial material from which the planets formed. NASA’s New Horizons mission gave humanity its first close look at a Kuiper Belt Object when it flew past Arrokoth in 2019, and concepts for an interstellar probe and follow-on Kuiper Belt missions continue to circulate in the planetary science community. But the sheer number of known and expected objects in this region means that target selection is itself a formidable optimisation problem. Every candidate target implies a different launch window, a different gravity-assist choreography and a different fuel budget, and choosing poorly can cost a mission a decade of flight time or render it infeasible altogether. Rapid, reliable accessibility assessment is therefore not a luxury but a prerequisite for ambitious outer-solar-system exploration.

The study also fits into a broader trend in which machine learning serves as an accelerant for, rather than a replacement of, classical astrodynamics. Earlier work demonstrated that Gaussian process regression could assess the accessibility of main-belt asteroids, and neural networks have been applied to orbit uncertainty propagation and estimation. The present research extends that paradigm to the most distant and most expensive class of targets, and its careful comparison of state representations and sampling strategies offers a practical template for other teams. The authors acknowledge support from the National Natural Science Foundation of China, and the datasets generated during the study are available from the corresponding author on reasonable request.

There remain, of course, limits to what any surrogate can do. The model was trained and validated for a specific transfer scenario with a prescribed gravity-assist sequence, and its accuracy in the low-cost region, while excellent, comes at the price of larger errors in the sparse high-cost tail. The ten targets flagged by the surrogate still require full optimisation before any mission commitment, and the screening fractions of 10 and 20 percent represent a trade-off between computational savings and the risk of overlooking a hidden gem. Yet the headline numbers speak for themselves: a model that predicts transfer costs to within a few hundred metres per second for the targets that matter, ranks thousands of destinations in seconds, and hands mission designers a principled, uncertainty-aware shortlist. As the catalogue of Kuiper Belt Objects continues to grow, tools like this one may well decide which frozen worlds humanity visits next.

Subject of Research: Surrogate-based machine learning for rapid assessment of spacecraft transfer accessibility to Kuiper Belt Objects

Article Title: Surrogate-based rapid accessibility assessment of Kuiper Belt object transfers

Article References: Meng, L., Zhou, X., Qiao, D., & Li, X. (2026). Surrogate-based rapid accessibility assessment of Kuiper Belt object transfers. Astrophysics and Space Science, 371(10), Article 116. https://doi.org/10.1007/s10509-026-04650-9

Image Credits: AI Generated

DOI: 10.1007/s10509-026-04650-9

Keywords: Kuiper Belt objects, trajectory design, Gaussian process regression, surrogate modelling, gravity assists, deep-space manoeuvres, adaptive sampling, astrodynamics, mission design, machine learning, interplanetary transfers, Bayesian optimisation

The appeal of so-called frozen orbits is well established. These are orbits whose size, shape, and orientation remain nearly constant in a frame rotating with the Moon, requiring almost no propellant to maintain. The Lunar Reconnaissance Orbiter demonstrated the payoff: by carefully selecting a polar frozen orbit, the mission used only 33.9 kilograms of hydrazine over twelve years for all of its station-keeping, phasing, and momentum management. Frozen orbits could similarly support extended surface observation campaigns, communications relays, and even the tracking of orbital debris or derelict hardware from past missions, such as the Apollo-era lunar module ascent stages that may still circle the Moon.

Traditionally, frozen orbits have been found analytically. Researchers construct an averaged dynamical model that strips out short-period oscillations, then solve for equilibrium points in the averaged equations. These classical analyses predict that frozen orbits occur under specific conditions: when the mean argument of perilune, the point of closest approach, sits near plus or minus 90 degrees; when the mean eccentricity is zero; or when the argument of perilune equals 0 or 180 degrees. The trouble is that averaging discards real physics, and the resulting predictions must be verified by propagating trajectories in a high-fidelity model that includes a detailed lunar gravity field and the true ephemerides of the Earth and Sun. Scaling that verification across the whole phase space has always been the bottleneck.

Bosanac and Miceli attacked the bottleneck directly. They defined over 1.1 million initial conditions at a fixed epoch, sampling orbital elements coarsely across semi-major axes up to 2,038 kilometers, eccentricities, inclinations from 0 to 180 degrees, and the full range of nodal longitudes and arguments of perilune. Each initial state was propagated for up to 180 days in a demanding dynamical model: a 100 by 100 degree-and-order spherical harmonic lunar gravity field, drawn from the GRGM900C model derived from GRAIL data, combined with the point-mass gravity of the Earth and Sun positioned according to the DE421 ephemerides. Integrations used a high-order Runge-Kutta scheme with tight tolerances of ten to the minus eleventh, terminating early if the spacecraft struck a spherical Moon of radius 1,738 kilometers.

Of the initial conditions tested, 638,181 trajectories survived at least three lunar rotational periods, roughly 81.9 days, and entered the analysis. Rather than storing every point along every path, the researchers represented each trajectory by its sequence of perilunes, the moments of closest approach, which capture the long-period evolution of the orbit’s eccentricity and orientation. Each perilune was described by two compact coordinates, p and q, equal to the eccentricity times the cosine and sine of the argument of perilune. These coordinates encode both the shape of the orbit and the direction in which its low point points, and they trace out distinctive loops and curves as the orbit evolves in the Moon-fixed frame.

The clustering itself relied on DBSCAN, a density-based algorithm that groups points lying in sufficiently crowded neighborhoods while labeling isolated points as noise. Because the dataset was too large to process at once, it was split into 64 partitions of roughly 10,000 trajectories each. Within each partition, trajectories were grouped if their perilune sequences moved through the pq-plane along similar paths at similar times, judged by unit tangent vectors sampled every half day. Local clusters were then merged across partitions and across different initial phasings, using shape-based feature vectors and a graph structure whose connected components defined the final groupings. The outcome was striking: 638,181 trajectories collapsed into just 885 global clusters, a reduction of three orders of magnitude in the information an analyst must examine.

From selected clusters, the team manually extracted sets of long-lifetime orbits, defined as trajectories showing limited drift in the pq-plane over the full 180 days. These orbits appear at discrete combinations of initial semi-major axis and inclination, and in many cases, shifting the initial longitude of the ascending node rotates the entire groundtrack geometry without changing the underlying orbital evolution. At a semi-major axis of 1,808 kilometers and an inclination of 85 degrees, for example, three different nodal longitudes produced long-lifetime orbits whose perilune groundtracks circled the north pole, circled the south pole, or swept across both hemispheres, all while the eccentricity and argument of perilune evolved in nearly identical fashion. That flexibility matters for mission design, because the groundtrack determines which terrain a spacecraft observes.

The comparison with classical theory was largely reassuring, with intriguing exceptions. Most of the recovered long-lifetime orbits cluster around arguments of perilune near plus or minus 90 degrees or encircle zero eccentricity in the pq-plane, exactly as analytical frozen orbit theories predict. The overall shapes of the mean eccentricity curves as a function of inclination also match the families derived by Lara in 2011 and later refined by Yarndley and colleagues. But at initial inclinations of 65 and 115 degrees, the numerical search surfaced low-drift orbits with mean arguments of perilune as far as 30 to 60 degrees away from the predicted minus 90 degrees. When the authors extended two of these trajectories to nearly twenty years, some remained tightly bounded, hinting at frozen orbits that the averaged theories miss, while others drifted substantially under longer-period oscillations governed by multiple frequencies.

The method also recovered orbits that analytical approaches cannot touch by construction. Resonant trajectories, whose perilune sequences form looping paths in the pq-plane, are typically excluded from averaged derivations, yet the clustering search found them naturally at low inclinations. Several results echoed earlier numerical work: orbits at 95 degrees inclination showed oscillation periods of 27.43 to 27.54 days, close to the 27.35-day repeat groundtrack period reported by Russell and Lara in 2007, and near-equatorial long-lifetime orbits resembled those designed for the DARE radio astronomy concept by Plice and colleagues.

The authors are careful to frame the work as a proof of concept. The initial conditions were sampled coarsely, at a single epoch, and only over 180 days, so some families may be missing and some orbits may not persist indefinitely. Extracting representative orbits from clusters is still manual, and automating that step is ongoing work. Even so, the demonstration is compelling: a data-mining pipeline can compress a million brute-force simulations into a compact, interpretable summary that agrees with decades of theory while exposing solutions theory overlooked. As NASA and its partners plan sustained lunar operations under Artemis, tools like this could quietly become as important as the rockets, ensuring that the spacecraft placed around the Moon stay there without burning fuel they cannot spare.

Subject of Research: Numerical identification of long-lifetime low lunar orbits using trajectory clustering in a high-fidelity lunar gravity model

Article Title: Numerical exploration of low lunar, long-lifetime orbits

Article References: Bosanac, N., & Miceli, G. E. (2026). Numerical exploration of low lunar, long-lifetime orbits. Celestial Mechanics and Dynamical Astronomy, 138(5), Article 61. https://doi.org/10.1007/s10569-026-10334-x

Image Credits: AI Generated

DOI: 10.1007/s10569-026-10334-x

Keywords: lunar orbits, frozen orbits, trajectory clustering, celestial mechanics, lunar gravity field, mission design, DBSCAN, perilune, space situational awareness, astrodynamics, Lunar Reconnaissance Orbiter, ephemeris model