Neural Networks Learn to Chart Spacecraft Trajectories Faster Than Classic Math

Every robotic mission to another planet begins its life as a blizzard of calculations. Long before a spacecraft is built, mission designers must sweep through thousands, sometimes millions, of candidate trajectories across launch windows, arrival dates and possible gravity-assist sequences, searching for the routes that cost the least fuel. The mathematical backbone of that search is Lambert’s problem, a boundary-value puzzle first posed by Johann Lambert in 1761: given a starting position, a destination and a travel time, what orbit connects them? The equations that answer this question cannot be inverted analytically, so classical solvers grind through iterations of numerical refinement for every single candidate transfer. A new study from KTH Royal Institute of Technology in Stockholm suggests that neural networks, paired with a cleverly designed way of normalizing the problem, could take over much of that burden during the earliest phase of mission design.

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