Can Generative AI Discover Better Cislunar Orbits? Machine Learning, Stable Earth-Moon Cyclers, and Lunar Communications Infrastructure

Generative Astrodynamics for Cislunar Infrastructure: AI-Assisted Discovery, Classification, and Optimization of Periodic Orbits, Invariant-Manifold Transfers, and Low-Energy Communications Architectures in the Earth-Moon CR3BP

Summary

A generative model can plausibly help find orbit families that are unremarkable in Δv terms but valuable as persistent communications infrastructure, but no published work has yet shown one doing so.

The best current example of such an orbit class was found by classical dynamical-systems geometry, not by machine learning. Here is what each piece of the answer rests on. It is demonstrated, in the CR3BP only, that a variational autoencoder trained on a catalog of Earth-Moon periodic orbits can produce samples that converge under differential correction into periodic orbits absent from its training set (46 of 100 samples converged) [5]. It is also demonstrated in the CR3BP that deep generative models can warm-start global searches for low-thrust transfers [6][7][8]. What has been modeled, but not flown, is the infrastructure value of classically discovered orbits such as the stable prograde Earth-Moon cyclers [1][2]. What is funded and proposed is operational relay and navigation constellations in elliptical lunar frozen orbits and related geometries [25][27][28]. What remains conceptual is the combined thing: a generative model scoring or conditioning candidates on coverage, eclipse, power, thermal, station-keeping and network geometry at the same time.

Four things: First, the hard part is not generating candidates but verifying and scoring them. Every generated orbit must pass through differential correction and monodromy analysis, and every infrastructure metric apart from stability and period needs a Sun model, an ephemeris, or a link budget that the CR3BP does not provide. Second, the orbits actually selected for cislunar infrastructure (the 9:2 near-rectilinear halo orbit, elliptical lunar frozen orbits, distant retrograde orbits, and the L2 halo used by Queqiao-1) were already chosen on infrastructure criteria such as eclipse avoidance, polar coverage and maintenance cost [14][15][23]. That weakens the premise that Δv-centric design has left a large blind spot. Third, value lives at the constellation level. Published multi-objective constellation studies already trade coverage, dilution of precision and cost using genetic algorithms and grid searches [31][33][34], so a generative method must beat these baselines, not an idealized Δv-only strawman. Fourth, the programmatic pull is weak and shifting. NASA paused Gateway in March 2026 [20], and China's Chang'e-7 slipped to 2027 [24]. No funding line was identified that asks for machine-generated orbit families.

2. Background

In the circular restricted three-body problem (CR3BP), a massless spacecraft moves under the gravity of Earth and Moon, which are fixed in a rotating frame. The only parameter is the mass ratio (about 0.01215 for Earth-Moon), and the Jacobi constant is the only integral of motion [1]. Periodic orbits come in one-parameter families: Lyapunov, halo, vertical, butterfly, distant retrograde and prograde, resonant and cycler families, among others. Families are traced by continuation from one converged member, and each member's linear stability is set by the eigenvalues of its monodromy matrix (the state transition matrix over one period) [1]. Unstable periodic orbits carry stable and unstable invariant manifolds. In the planar problem these are tubes that separate transit from non-transit trajectories through the L1 and L2 necks, and they underpin low-energy transfer design [1][4]. JPL publishes a catalog of such families [38]. One generative study reports training on 44,112 periodic initial conditions in 40 Earth-Moon families taken from that catalog [5].

The communications problem in cislunar space is about geometry. The lunar far side never sees Earth. The south pole sees Earth only intermittently and at low elevation. Relays therefore must see both the user and Earth, must spend enough time over priority regions, and must survive Earth and lunar eclipses. NASA's Gateway analysis treated eclipses longer than 90 minutes as undesirable for power and thermal reasons [14]. That single constraint shows how quickly an infrastructure objective departs from Δv.

3. Technical Core

3.1 Four machine learning roles that should not be conflated

Generative discovery means sampling new candidate states or trajectories from a learned distribution. The only published example that targets periodic orbits directly is Gil and colleagues' variational autoencoder, which ESA's OrbitGPT project (grant 40001435) partly funded [5]. It learned a two-dimensional latent space from the JPL catalog. Its raw samples were "fuzzy," with an average dynamical error of 34 (nondimensional, per node) against the integrator tolerance of 10^-13. After multiple-shooting correction capped at 20 iterations, 46 of the 100 generated samples converged, averaging 10.1 iterations. The authors report that all refined orbits were new, meaning absent from the training set [5]. That is not the same as a new family. The paper does not claim any converged orbit lies outside the known families, and the plausible reading is that the model interpolates within them.

Classification means labeling and clustering computed trajectories. Bosanac and Bonasera applied clustering and dimension reduction to higher-dimensional Poincaré maps [9][10]. Spear and Bosanac clustered Earth-Moon CR3BP trajectories generated from local sets of initial states by their geometric and temporal features [11]. The same variational autoencoder's latent space grouped orbits by family without labels, reaching a normalized mutual information of 0.78 and a clustering accuracy of 0.56 across 40 classes [5]. This is mature, useful work, and it is not generative.

Surrogate modeling means fast approximation of expensive evaluations such as stability indices, manifold arcs or coverage statistics. No published neural surrogate was identified that predicts coverage, eclipse exposure or link availability for CR3BP periodic orbits. That gap matters for the key question, because infrastructure scoring is where most of the computation would sit.

Optimization means search. The Beeson group's amortized global search uses conditional generative models (a conditional variational autoencoder, later diffusion models) to predict where good local solutions lie for new problem parameters. In a CR3BP minimum-fuel low-thrust transfer, it found 27 locally optimal good solutions in the first 20 seconds, compared with 1 for uniform initialization [6][7][8]. Reinforcement learning has produced closed-loop low-thrust guidance in the planar CR3BP [12] and multi-objective low-thrust transfer design near Earth-Moon L2 [13]. Both are optimization roles. The Beeson work is the most rigorous generative result in multi-body dynamics, but it is a generative model in service of optimization, and it amortizes Δv-type objectives. It does not discover orbit families.

3.2 From generated candidate to verified periodic orbit

A generative model's output is at best an initial guess. It becomes a periodic orbit only after differential correction (single or multiple shooting) closes it to tolerance, pseudo-arclength continuation shows it belongs to a family, and the monodromy matrix gives its stability. Ross and Roberts-Tsoukkas show the classical version of this pipeline in detail. Symmetric initial conditions from manifold-tube intersections are corrected by Newton iterations to about 10^-8 (about 10^-5 m/s), usually within five iterations, and stability comes from the half-period state transition matrix through the orbit's symmetry [1]. The verification burden is therefore concrete. Each candidate needs a converged correction, a family membership check against known catalogs (so that "new" means new), stability and bifurcation analysis, and then transition into an ephemeris model where the orbit becomes quasi-periodic or disappears. A 46 percent convergence rate [5] is acceptable for a candidate generator. It isn't measuring how many converged orbits are dynamically novel or operationally useful.

3.3 The objective space: what the CR3BP can and cannot compute

Some infrastructure objectives are native to the CR3BP: periodicity, Jacobi constant, period, stability indices, bifurcation proximity, and the geometric parts of visibility. With the Moon fixed in the rotating frame, line of sight from a relay to a lunar-surface point or to Earth's center is a CR3BP geometry computation. Commensurability with the synodic month can also be checked from the period alone. Ross and Roberts-Tsoukkas do this for the (1,1) cycler, whose stable midpoint period of 44.75 days lies within 1 percent of a 2:3 ratio with the 29.53-day synodic month [1].

Eclipse exposure and power need a Sun model. At minimum that is a rotating Sun direction or a bicircular model, and properly it is an ephemeris. The NRHO case shows why: eclipse avoidance depends on phasing within the orbit, not only on the orbit's shape, and Gateway analysts had to add phase control to orbit maintenance to keep the reference eclipse-free [14]. Station-keeping cost in the CR3BP is only proxied by the stability index. Real cost depends on ephemeris perturbations, navigation error and maneuver execution error, as CAPSTONE's flight record shows (Section 4). Thermal loading needs attitude, Sun, Earth infrared and albedo models. Link availability needs a link budget: frequency, antenna gain, transmit power, range and pointing. Dilution of precision is a geometric quantity, but it is defined only for a constellation and depends on clock and ephemeris error models [32][33]. Lunar libration and the Moon's real orientation also enter surface visibility at the precision service providers need.

A credible multi-objective fitness function would therefore be layered. The CR3BP would supply candidate generation and stability screening, a bicircular or ephemeris propagation would supply eclipse and maintenance cost, and a link-and-navigation layer would supply availability and dilution of precision. The published literature has pieces of this. Iiyama and Gao trade coverage, position dilution of precision, orbit-determination accuracy, receiver noise and insertion cost across frozen-orbit Walker constellations [33]. Chen and colleagues optimize a hybrid DRO, ELFO and NRHO navigation constellation with an improved genetic algorithm under dilution of precision, signal attenuation and cost constraints [34]. Bhamidipati, Mina and Gao trade navigation, communication and SmallSat cost factors for an ELFO constellation [32]. All of these select among known orbit families. No published source was identified that combines eclipse exposure, relay availability, station-keeping cost and thermal load in a single optimization objective, or that uses any such objective to condition a generative orbit-discovery model. That function would have to be built.

3.4 The classical benchmark: stable prograde cyclers

Ross and Roberts-Tsoukkas reported fully ballistic, prograde Earth-Moon (k1, k2)-cyclers, built from intersections of the stable and unstable manifold tubes of the L1 Lyapunov orbit, with linearly stable subfamilies [1]. Their stable windows differ widely in size. The (1,1) stable subfamily spans only about 0.13 km in perilune distance. The (3,1) window spans roughly 750 to 1,000 km in perilune altitude. The largest (3,3) window spans about 2,041 km, at perilune altitudes of about 4,200 to 6,200 km [1]. The authors state that the (3,2) stable orbits closely match a 2:5 synodic resonance, and that the exact resonance is met by two nearby unstable members with stability indices of order 10 to 100 [1]. By this report's own conversion of the tabulated period (17.90 nondimensional time units at 4.348 days per unit, or about 77.8 days, against 73.8 days for an exact 2:5 ratio), the representative stable (3,2) orbit is only loosely commensurate. That should temper claims of repeating Sun-Earth-Moon geometry for that family. The 2026 arXiv preprint extends the construction to spatial stability. It reports that planar and out-of-plane stability separate, that out-of-plane instability appears only through isolated parametric resonances, that every family examined has a subfamily stable in both directions, and that stable examples exist from the Sun-Jupiter mass ratio to the equal-mass case. It also conjectures that saddle-center birth of stable cyclers is universal [2].

All of these results are in the planar or spatial CR3BP. The AAS paper cites evidence that a perturbed version of one earlier (3,2)-type orbit persists in a Sun-Earth-Moon quasi-bicircular model [1], but persistence of the stable families under solar gravity and lunar eccentricity has not been shown in an ephemeris model. Second, the (3,1) row of the AAS paper's Jacobi-constant tables is internally inconsistent. The upper-bound value in Table 3 does not match the value implied by Table 4's tabulated offset from C1, which the (1,1) and (2,1) rows do satisfy. Both versions also conflict with the text's statement that every (k1,1) family lies below the L2 Jacobi value [1]. This report does not reproduce those figures. Third, the (1,1) stable window is operationally meaningless as a design margin. At 0.13 km, it is an order of magnitude smaller than CAPSTONE's flown mean 3-sigma post-fit position uncertainty of 1.484 km in the NRHO, which ranged from 0.0925 km to 56.993 km while still meeting the 10 km 3-sigma requirement for more than 98 percent of the time in the NRHO [17].

Could a generative model have found these orbits? Probably not from the JPL catalog alone. A model trained on 40 known families [5] learns their distribution and has no mechanism for proposing a topologically distinct class, such as orbits alternating temporary capture about both primaries, that is absent from its training data. The cyclers came from a structural insight, namely that stable orbits can sit inside the transit-tube intersection region beyond the reach of the Global Orbit Structure Theorem's unstable constructions [1]. That is exactly what a density model does not encode. The more plausible generative role is extension. The cycler work restricted itself to symmetric orbits and a handful of (k1, k2) pairs [1]. Asymmetric, spatial and non-resonant cyclers, and high-k combinations, form a combinatorially large, poorly charted space in which a model trained on the classical families and conditioned on stability and resonance could propose correction seeds more cheaply than grid searches over Poincaré sections. That is a reasoned forward-looking hypothesis, not a demonstrated result. For the key question, the implication is that infrastructure-relevant orbit classes that are not Δv-optimal do exist and were missed for decades, which supports the premise. But the discovery engine was geometric, which undercuts the claim that generative models are needed to find them.

3.5 What real infrastructure selections reveal

Designers of flown and planned systems already optimized beyond Δv. NASA selected the 9:2 southern L2 NRHO for Gateway partly because its resonance with the synodic month sets up a repeating geometry that avoids long Earth eclipses, and partly for low orbit maintenance and favorable communications with Earth and the south pole [14][15]. The orbit is slightly unstable, and Davis and colleagues report that an unmaintained spacecraft departs within several months "as a result of inherent instability as well as perturbations and errors" [16]. CAPSTONE, launched June 28, 2022, entered this NRHO on November 13, 2022 [17]. NASA's activities on it concluded in June 2026, after nearly four years, and Advanced Space continues to operate it as a testbed [18]. Its flown record is the best public evidence on NRHO operations. Orbit maintenance maneuvers ranged from 6.02 to 78.8 cm/s, with a mean of 28.2 cm/s. Maneuvers that optimized below a 5 cm/s minimum were skipped, as were some in favor of payload activities; the stuck thruster instead degraded execution, with a mean magnitude error of about 13 percent and a mean pointing error of 6.08 degrees, and nearly half its fuel (about 100 m/s) remained in mid-2025 [17]. From those figures, this report estimates maintenance at about 5 m/s per year, which is an inference and not a published figure. Gateway itself was paused "in its current form" on March 24, 2026 in favor of a surface-first architecture [20][21]. So the one crewed infrastructure node that justified the NRHO no longer has a schedule.

China's relays show the other design lineage. Queqiao-1 (2018) supported Chang'e-4 from an Earth-Moon L2 halo orbit [23]. Queqiao-2, launched March 20, 2024, entered a 24-hour elliptical frozen lunar orbit on April 2, 2024 and relayed for Chang'e-6 [23]. It is planned to shift to a 12-hour orbit for south-pole missions [22], but Chang'e-7 was postponed to 2027 in August 2026 [24]. China's Chinese Academy of Sciences also reports running a three-satellite constellation using a lunar distant retrograde orbit since 2024 [30]. ESA's Moonlight programme puts priority on south-pole coverage, plans initial services by the end of 2028 and full operations by 2030, and commits to LunaNet interoperability [25]. Its precursor, SSTL's Lunar Pathfinder, was reported in July 2026 to be riding Firefly's Blue Ghost Mission 2, "planned for late this year or next" [26]. NASA's LCRNS project is validating commercial relay services procured under Near Space Network Services, with Intuitive Machines (NASDAQ:LUNR) selected in 2024 [27][28]. The company planned to deploy the first of five relay satellites on its third lunar mission [29].

Would a generative, multi-objective approach have changed these selections?
Evidence suggests it probably would not have changed the family choices. Frozen orbits are chosen because lunar gravity-field and third-body effects cancel to leave eccentricity, inclination and argument of periapsis nearly constant, and because a south-pole-favoring apolune suits polar service [34]; Iiyama and Gao, citing Ceresoli and colleagues, put frozen-orbit station-keeping "below 1 m/s per year," so that lunar orbit insertion Δv dominates the budget [33]. The NRHO's advantages are well understood analytically [14][15]. The more plausible gains lie in phasing, in the number and plane of satellites, and in mixed-family combinations, which is a constellation-level combinatorial problem.

3.6 Network-level architecture

Infrastructure value is a property of the network, and existing studies already treat it that way. Xu and colleagues combined halo and DRO orbits and selected a six-orbit, 16-satellite lunar navigation design, finding that 8-day halo orbits give the best polar coverage and DROs give stable equatorial coverage [31]. Iiyama and Gao found that a south-ELFO Walker constellation needs at least 12 satellites for 99 percent or better four-fold south-pole coverage with semi-major axes up to 16,000 km. A configuration with a 15,000 km semi-major axis and 67.5-degree inclination can still give about 90 percent with six satellites and about 95 percent with eight, and larger semi-major axes degrade orbit determination and received power [33]. Chen and colleagues report a nine-satellite mixed DRO, ELFO and NRHO configuration [34]. Braik and Ross provide a complementary structural tool: a reachable-set network over 13 planar CR3BP families at a common Jacobi constant, which identifies hub, gateway and relay roles using finite-Δv and time-of-flight accessibility [3]. That network is built on transport cost, not communications, but it is the natural substrate for mixed-family constellations that include cyclers. No published work was identified that applies generative models to multi-satellite cislunar architectures. The integration problem (phasing, inter-satellite visibility, ground-segment scheduling, handover and interoperability standards) is combinatorial and constraint-heavy. Evolutionary and mixed-integer methods already handle it, so a generative model would need to show it speeds up search over that space rather than over single orbits.

4. Development Status and Maturity

Demonstrated in flight: the NRHO (CAPSTONE, nearly four years under NASA [17][18]), the L2 halo relay (Queqiao-1 [23]), the elliptical frozen relay orbit (Queqiao-2 [23]), and a DRO-based constellation per Chinese Academy of Sciences statements [30]. CAPSTONE's crosslink navigation with the Lunar Reconnaissance Orbiter achieved an average Doppler residual standard deviation of 0.29 mm/s across five successful crosslink passes in ground processing, against Deep Space Network Doppler noise of about 0.1 mm/s. The authors concluded it "does not present the navigation capabilities necessary to fully supplant Earth ground tracking" [17]. Demonstrated in high-fidelity simulation: NRHO orbit maintenance and phase control [14][16] and frozen-orbit constellation coverage and navigation trades [32][33]. Demonstrated only in the CR3BP: every machine learning result reviewed here, including generative periodic-orbit sampling [5], amortized generative warm-starts [6][7][8], clustering [9][10][11] and reinforcement-learning guidance [12][13], as well as the stable cycler families [1][2]. Funded and proposed: Moonlight, Lunar Pathfinder, LCRNS and commercial relays [25][26][27][29]. Conceptual: infrastructure-conditioned generative discovery and generative constellation design. Where NASA technology readiness levels apply at all, the machine learning tools sit at roughly TRL 2 to 3 (analytical proof of concept in a simplified model). The flown orbit families are operational heritage. A TRL for the integrated concept would be meaningless, because no system exists to rate.

5. Key Programs, Institutions, and Actors

Virginia Tech's Ross group supplies the classical benchmark: cyclers [1][2], orbital networks [3] and a cislunar primer [4]. The cycler work was funded through the U.S. Space Force and Air Force Research Laboratory University Consortium Research Opportunity under the xRADAR space domain awareness cooperative agreement [1]. This is the clearest case of defense cislunar domain awareness money pulling orbit-discovery research forward. The University of Colorado Boulder's Bosanac group leads data-driven classification of multi-body trajectories [9][10][11], and its reinforcement learning work (with JPL co-authorship) addresses transfer design [13]. Purdue's Howell group produced much of the NRHO eclipse and maintenance analysis [14][15][16] and reinforcement learning guidance with MIT [12]. Princeton's Beeson group leads generative amortized global search [6][7][8]. Universidad Politécnica de Madrid and the University of Strathclyde, under ESA's OrbitGPT activity, produced the only direct generative periodic-orbit study [5]. JPL's catalog is the training substrate for that work [5][38]. Advanced Space operates CAPSTONE [17][18]. On the service side, NASA's Space Communications and Navigation program (through LCRNS) [27][28], ESA with Telespazio and SSTL (Moonlight and Lunar Pathfinder) [25][26], CNSA (the Queqiao relays) [22][23] and the Chinese Academy of Sciences' Technology and Engineering Center for Space Utilization (the DRO constellation) [30] define the requirements any new orbit class would have to meet.

6. Path to Realization

Five things stand between the research and a working capability. First, a validated multi-objective fitness function: the layered CR3BP, ephemeris and link-budget evaluator described in Section 3.3, benchmarked against flown NRHO and ELFO performance. Second, training data. The JPL catalog covers known families [38] and would bias any generative model toward them. Adding asymmetric, spatial and cycler families, plus ephemeris-corrected quasi-periodic counterparts, is a prerequisite for anything beyond interpolation. Third, verification and certification. Machine-generated trajectories are harmless if they only seed classical correctors, because the verified product is then a conventionally computed orbit. Certification risk arises only if learned models enter onboard guidance, as in reinforcement learning work [12]. Fourth, the fidelity gap. Any candidate family must survive transition to an ephemeris model with a maintenance cost comparable to CAPSTONE's flown maneuvers [17]. Fifth, demand pull. Relay procurements are specified by service levels [27][28], and providers default to heritage orbits. The most credible near-term driver is defense cislunar domain awareness, which already funds cycler research [1], because cyclers' broad reach suits observance more than relay.

7. Risks and Failure Modes

Classical capture of value is the most important risk: the possibility that dynamical-systems methods plus conventional optimization already capture most of the value. Its likelihood is high. The cycler case [1][2], the analytic selection of NRHO and frozen orbits [14][15], and existing genetic-algorithm constellation optimizers [31][34] all show that the high-value discoveries and trades are within reach without generative models. The impact would be moderate: it confines generative methods to acceleration roles rather than eliminating them. The mitigation is to target problems where classical methods scale poorly, such as asymmetric and high-order spatial families and mixed-family constellation phasing, and to report head-to-head compute and yield against Poincaré-section and continuation baselines.

The fidelity gap is also likely and high-impact. Stable CR3BP families may lose stability or coherence under solar gravity and lunar eccentricity, and the evidence for cycler persistence is limited to one orbit in a quasi-bicircular model [1]. The mitigation is to require ephemeris continuation before making infrastructure claims.

Scoring error in the absence of surrogates is moderately likely and moderate in impact. Without validated surrogates for eclipse and link metrics, conditioning a generative model on infrastructure scores risks optimizing a proxy. The mitigation is to validate surrogates against ephemeris-propagated truth on the flown orbit families.

Novelty inflation is likely and moderate in impact. Reporting converged orbits as "new" when they belong to known families [5] invites overstatement. The mitigation is mandatory family-membership checks against catalogs.

Programmatic volatility is likely, and its impact is high for any single-architecture bet. Gateway's pause [20] and Chang'e-7's slip [24] show how fast the anchoring missions move. The mitigation is to tie research to service-level infrastructure (LunaNet-compliant relays) rather than to a single crewed node.

8. Regulatory Landscape

Spectrum rules shape which relay geometries are usable. WRC-27 agenda item 1.15 will consider new or modified space research service (space-to-space) allocations for communications on the lunar surface and between lunar orbit and the surface [35]. The United States supports those studies while protecting the Shielded Zone of the Moon for radio astronomy [36]. NTIA's Institute for Telecommunication Sciences is developing lunar propagation models to inform them [37]. Far-side radio-quiet protection could limit when a relay may transmit over the far side, which is a geometry-dependent constraint that an infrastructure objective function would have to encode. Interoperability standards (LunaNet, which Moonlight commits to [25]) similarly fix frequency plans and link parameters that any orbit selection inherits.

9. Strategic Dimensions

Parallel U.S. and Chinese programs matter for feasibility in one concrete way: they decide which orbit classes gain flight heritage first. China has flown relays in L2 halo and elliptical frozen orbits and operates a DRO constellation [23][30]. The United States has flown only CAPSTONE in the NRHO, has paused Gateway, and relies on commercial relays not yet launched [18][20][29]. ISR objectives such as reach and revisit across cislunar volume may therefore shape the next orbit classes more than communications requirements do.

10. Implications and Outlook

For astrodynamicists and machine learning researchers, the productive framing is not "generative models discover orbits" but "generative models amortize expensive classical pipelines." The Beeson group's results [6][7][8] show the pattern: learn where good basins lie, then let verified solvers finish. Applied to infrastructure, that means conditioning on stability, resonance and coverage surrogates and seeding correctors in poorly charted regions, especially asymmetric and spatial cycler extensions [2]. For mission designers, the evidence does not suggest that current NRHO or frozen-orbit selections were mistakes. Gains are more likely in constellation phasing and mixed-family networks, building on reachable-set tools [3] and existing multi-objective optimizers [33][34].

Four open questions will decide the outcome. Do stable cyclers persist in ephemeris models with maintenance cost competitive with the NRHO? Can a generative model produce a verified orbit outside every known family? Can eclipse and link-availability surrogates be validated to operational accuracy? Will any relay, navigation or domain-awareness program write a requirement that heritage orbits cannot meet?

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