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

Neural operators have quietly become one of the most consequential tools in computational science. Instead of learning a mapping between finite vectors, they learn mappings between functions, which makes them natural solvers for partial differential equations. Given a family of PDEs describing, say, fluid flow through porous rock or heat diffusing through a material, a trained neural operator can predict the full solution field for a new set of parameters almost instantly, bypassing the expensive numerical solvers that would otherwise be required. Classical architectures such as DeepONet and the Fourier neural operator have transformed this landscape, but their quantum counterparts have lagged behind, hampered by processing overheads and theoretical gaps about what quantum circuits can actually represent.

The central obstacle has been scaling. Many quantum machine-learning paradigms demand qubit counts or circuit depths that grow so quickly with problem size that they collapse into impracticality on near-term hardware. The Shanghai team, led by Ruocheng Wang, Xiaoqiu Zhong, Zhuo Xia and Junchi Yan, attacked the problem from two directions at once: they built a leaner architecture and, crucially, they proved something rigorous about its power.

The theoretical centerpiece is an extension of the universal approximation theorem to the quantum domain for continuous nonlinear operators. Universal approximation, first established for classical neural networks in the 1990s, guarantees that sufficiently wide networks can represent a broad class of functions; the operator version underpins DeepONet. Proving an analogous guarantee for quantum circuits closes a foundational gap, but the team went further. Departing from the conventional view that quantum advantage must flow from the exponential size of Hilbert space, they showed that the architecture’s density matrix implicitly constructs a quadratic feature frame.

That quadratic frame yields a striking expressivity bound. For operators, the implicit feature space scales as O(p²) in the number of parameters p, circumventing the O(p) linear capacity limits of matched classical models. In plain terms, each additional parameter in the quantum model buys roughly a square’s worth of representational capacity compared with a classical model of the same size. This is a subtle but meaningful advantage: it does not rely on exotic claims about exponentially large state spaces, but on a concrete, provable property of how the quantum circuit encodes features.

The second innovation addresses a practical bottleneck known as spectrum alignment. Quantum models encode input data through frequency modulation, and capturing high-frequency components of a solution, the sharp gradients and fine oscillations that matter in real PDEs, typically requires either deep circuits or many parameters. The researchers introduced a trainable-frequency strategy, dubbed TF-QuanONet, in which the base frequencies of the encoding adaptively space themselves during training. The network effectively learns which frequencies to emphasize, capturing relatively high-frequency structure without inflating the parameter count or the circuit depth.

The benchmarks are where the claims meet reality. Across extensive experiments, TF-QuanONet notably outperformed competing quantum baselines and achieved accuracy competitive with classical frameworks under strictly matched-parameter conditions, a fairness constraint that quantum machine-learning comparisons often fail to honor. More intriguing still is what happened as the problems grew. In high-dimensional scaling regimes, with latent dimensions p approaching 256, the quantum architecture exhibited superior optimization robustness, consistently converging to its intrinsic error floor while classical baselines suffered from high variance. The quantum model was not just accurate; it was reliably trainable where its classical competitors became erratic.

The team also tested the architecture on real IBM quantum processors, including the ibm_fez device, as a qualitative proof of concept. Comparisons between noise-free simulations and physical hardware demonstrated that QuanONet retains functional resilience on near-term machines, a nontrivial achievement given that real qubits decohere, gates misfire, and measurement noise corrupts outputs. The experiments spanned a range of canonical problems, including dynamical systems, advection equations and Darcy flow, with visualizations confirming that predictions track ground-truth solutions across varying input frequencies.

The work arrives amid a broader reckoning in quantum machine learning, where researchers have grown wary of claims that evaporate under fair comparison or realistic hardware assumptions. By grounding its architecture in a proven expressivity theorem, keeping resource requirements modest, and validating on commercial hardware, the study offers a template for what credible quantum advantage in scientific machine learning might look like. The code has been released publicly on GitHub and archived on Zenodo, and all datasets were generated directly from it, inviting the community to scrutinize and extend the results.

Whether the quadratic expressivity edge translates into decisive practical wins as quantum hardware improves remains an open question, but the study reframes the debate. Rather than waiting for fault-tolerant machines or betting everything on exponential Hilbert spaces, it demonstrates that carefully designed quantum architectures, with theory and engineering aligned, can already hold their own against strong classical baselines on the noisy processors available today.

Subject of Research: A quantum neural operator architecture with a proven quadratic expressivity bound for solving partial differential equations on near-term quantum hardware.

Article Title: Quantum neural operators with implicit quadratic frame and expressivity advantages

Article References: Wang, R., Zhong, X., Xia, Z., & Yan, J. (2026). Quantum neural operators with implicit quadratic frame and expressivity advantages. Nature Machine Intelligence. https://doi.org/10.1038/s42256-026-01289-7

Image Credits: AI Generated

DOI: 10.1038/s42256-026-01289-7

Keywords: quantum machine learning, neural operators, partial differential equations, QuanONet, NISQ era, universal approximation, expressivity, trainable frequencies, IBM quantum processors, DeepONet, quantum computing, scientific machine learning