Near-Earth Objects

Frozen vision-language models (VLMs) remain unreliable on fine-grained left-right claims, and raw claim likelihoods need not reliably rank verification errors. After a horizontal-reflection intervention is fixed, how should its induced likelihood measurements be combined into a selective verification signal? We introduce Relation-Orbit, a closed-form contrast with no learned fusion parameters that assigns eight normalized likelihoods to query-supporting and counterfactual roles determined by reflection, inverse relation, and entity exchange. A claim is asserted only when the signed contrast exceeds a threshold selected on held-out data using pointwise Clopper-Pearson upper confidence bounds. On VSR and GQA across four frozen VLMs, Relation-Orbit yields higher mean test coverage at a 10% selective-risk calibration target than an all-eight Orbit-Max baseline in all eight dataset-backbone settings; gains over a nearly abstain-all one-sided intervention score are reported separately. A separate LLaVA-1.5/COCO evaluation, reduced-orbit controls, and a two-sided partition diagnostic further characterize the structural advantage.

We propose to learn the unknown dynamics in the proximity of a small body directly from tracking data, representing them as a feed-forward neural network embedded in the system Hamiltonian. The equations of motion form a Neural Hamiltonian Ordinary Differential Equation, whose variational equations provide exact training gradients: estimation uses position and velocity arcs at realistic noise levels, without acceleration or potential labels, and a continual learning approach warm-starts the network as new data are acquired. The known part of the Hamiltonian carries whatever is available, from the central term and spin state to the constant-density model of the imaged shape. We assess the method against a normalized spherical harmonics expansion estimated from identical arcs through the same machinery, on scenarios built on the shapes of Itokawa, 67P, Bennu and Eros. The network remains usable inside the Brillouin sphere: it plans ballistic descents at Itokawa to \SI{4.6}{m} median touchdown error from tracking alone, against 5.1--\SI{48.9}{m} for harmonics of degree 4--12, and to \SI{0.9}{m} with the imaged shape as prior, a configuration that also recovers localised density anomalies invisible to any harmonics degree. The two representations are complementary, and we discuss their combined use across the phases of a small-body mission.

Long-duration gravitational-wave modelling must resolve fast orbital motion together with slow dissipative evolution while preventing small numerical errors from accumulating into secular phase drift. Here we ask whether the finite-time evolution map itself can be learned as an explicit, differentiable, structure-preserving object and then repeatedly composed through a complete inspiral. We construct three neural-flow architectures: a symplectic and slimplectic flow on Galley's doubled phase space, [SINFONIA-J0]; a Taylor-anchored flow, [SINFONIA-J1]; and a Magnusian flow that learns the finite-time dissipative correction in the interaction picture, [SINFONIA-J2]. Applied to a 2.5PN neutron-star inspiral, all three expose the same controlling mechanism: long-time accuracy is governed not by pointwise map error alone, but by its signed projection onto a single secular channel fixed by energy--angular-momentum balance. Encoding this structure allows the learned maps to remain accurate through $10^{2}$--$10^{5}$ window compositions to coalescence at timesteps of a full orbital period and beyond, reaching chained phase errors orders of magnitude below a benchmark slimplectic integrator at lower cost. The same secular structure can also be exploited for physics inference: when the channel is left unconstrained, the accumulated phase retains enough information to recover an un-modelled dynamical-friction-like force, both parametrically and as a learned function of separation. Network-off controls isolate the contribution of learning from the analytic structure already built into each map. These results establish a proof of concept for structure-preserving learned evolution maps as tools for fast long-duration integration and physics inference in gravitational-wave source modelling.

Estimating positive largest Lyapunov exponents from data is comparatively natural because neighboring trajectories separate, whereas stable dynamics require resolving contraction before measurement noise or finite precision erases the signal. We introduce a period-aware forecast-error contraction procedure for estimating a dominant negative Lyapunov exponent from ensembles of short scalar trajectories without using governing equations or an analytical Jacobian. A k-nearest-neighbor predictor is trained on trajectory histories, the geometric-mean absolute forecast error is evaluated at phase-consistent horizons, and the exponent is obtained from the slope of the logarithmic error profile. Unlike data-driven approaches that reconstruct local evolution matrices or differentiate a learned surrogate, the proposed method extracts the contraction rate directly from out-of-sample forecast errors. Two adaptations are essential: the forecast step is synchronized with the detected orbit period, and candidate slopes are accepted only when they form a stable consensus across several transient lengths. On the logistic map, the method recovers 92 of 112 negative-exponent parameter values with a mean absolute error of 0.0253 and $R^2=0.886$. On a two-dimensional map without fixed points, independent scalar pipelines based on the three observables $x_n$, $y_n$, and $z_n$ give mean absolute errors of 0.00879--0.01145 and $R^2=0.983$--$0.986$. Because the estimation stage uses only observed trajectories, the framework provides a basis for repeated-relaxation experiments in which short sensor responses are available but the governing equations and analytical Jacobian are unknown. Experimental validation remains a subject of future work.

Satellite constellations require orbital transfers that are both fuel efficient and collision avoidant. Yet, the computational cost of optimization methods traditionally used to plan their trajectories scales poorly with both the number of satellites as well as the number of obstacles to avoid, due to the pairwise safety constraints. In this work, we introduce a permutation-equivariant neural operator for trajectory planning of spacecraft swarms. This neural operator maps distributions of spacecraft initial states, target states, and obstacle initial states to trajectories which avoid collision and conserve fuel. This neural operator output is then paired with a batched Gauss-Newton finish to enforce exact orbital dynamics, and further reduce fuel use. The operator is self-supervised, trained without optimal trajectory labels. When trained on ten spacecraft, the proposed method generalized zero-shot to swarms of 1,000 spacecraft and 11,000 obstacles. The generated trajectories matched a per-agent optimal control solver's accuracy while retaining collision avoidance. Operator learning grounded in physics may offer a fast, scalable alternative to trajectory optimization in the increasingly crowded orbits of the future.

Symbolic Regression (SR) aims to discover analytical equations from observational data and plays a central role in scientific modeling. While recent Large Language Model (LLM) based approaches show promise, they face two limitations. First, they lack data analysis mechanisms for uncovering variable dependencies, which reduces the efficiency of equation discovery. Second, most methods rely on single-objective evaluation focused solely on fitting error. This neglect of structural complexity and generalization often causes models to converge prematurely to local optima, limiting their ability to explore the broader equation space. We propose Multi-Objective Tool-augmented Symbolic Regression (MOT-SR), a unified framework that integrates external analytical tools to extract structural priors and guide equation generation, while jointly optimizing for accuracy, complexity, and generalization via a multi-objective evaluation module that maintains a dynamic Pareto front. MOT-SR employs two collaborative LLM modules: a Meta Strategy Generator, which selects tools and synthesizes structural optimization strategies based on Pareto-optimal equations, and an Equation Generator, which produces new candidate equations accordingly. The system operates in a closed-loop manner, continuously refining both strategies and equation structures. Across 40 standard tasks, MOT-SR outperforms existing SR methods in accuracy, generalization, and efficiency. We further validate MOT-SR on extreme mass-ratio inspiral (EMRI) orbital modeling, an important problem in space-based gravitational-wave astronomy where small local errors can accumulate substantially over long-term evolution. The discovered interpretable correction achieves the lowest trajectory-level integration error on held-out configurations. These results demonstrate the potential of MOT-SR to enable reliable modeling of long-horizon scientific dynamics.

Autonomous free-flying robots in orbital environments require controllers that are both versatile and resource-efficient, yet maintaining a separate, task-specific policy for each mission profile is architecturally brittle and limits operational flexibility as requirements evolve. We introduce HYPER-GNC, a multi-task reinforcement learning framework in which a hypernetwork maps physics-informed task embeddings to the weights of a shared actor-critic policy, enabling a single compact controller to master four distinct GNC tasks: velocity tracking, docking, inspection, and navigation with obstacle avoidance. The continuous embedding space allows the controller to generalize to novel mission configurations at deployment time without any retraining. Extensive experiments demonstrate that HYPER-GNC achieves sample efficiency comparable to single-task specialists while maintaining stability under significant inertial perturbations and external body wrenches. We further validate the framework on a physical satellite emulator, successfully bridging the simulation-to-reality gap across all mission profiles. Code, trained models, and deployment scripts are made publicly available to support reproducibility.

Locating periodic solutions of chaotic dynamical systems normally requires an initial guess close enough to the target orbit for numerical continuation or gradient-based search to converge. We show that Physics-Informed Neural Networks (PINNs) trained on sparse, noisy observations \emph{without} initial conditions recover periodic orbits of the gravitational three-body problem, including orbit families absent from the training data. The method rests on a second-order ODE formulation, fixed-frequency Fourier features, percentile-based adaptive refinement, and a trainable scaling parameter, each validated on forward problems. Across two 100-seed ensembles, $23$--$25\%$ of runs converge to families not present in the training data. We then ask what determines which family emerges. Two $χ^2$ tests give a consistent answer: changing the training data source significantly shifts the distribution of recovered families ($p < 0.001$, Cramér's $V = 0.339$), whereas switching between the two initialization distributions tested does not ($p = 0.620$, $V = 0.094$). The random seed selects which family a given run recovers; the \emph{distribution} the weights are drawn from does not shift the aggregate frequencies, but the training data does. The evidence is empirical: we do not characterize the loss landscape analytically, and PINNs remain slower than conventional integrators on well-posed initial-value problems. What the experiments establish is that the recovered orbits are verifiable rather than merely plausible: the identified ones refine to genuine periodic solutions, a network trained on Lagrange data recovers the figure-eight choreography (Li--Liao class I.A.1, matched to seven significant digits in $T^*$), and one trained on figure-eight data recovers a Broucke--Hadjidemetriou--Hénon orbit closing to $δ_T < 10^{-9}$.

We show that, up to isotropic scaling, the Gaussian RBF reproducing kernel Hilbert space (RKHS) is asymptotically isometric to Euclidean space in the large bandwidth limit. This strongly suggests that kernel-based constructions reliant on metric properties of the RKHS will yield results for Gaussian RBF kernels that similarly approach those of linear kernels for large bandwidths. The asymptotic behavior of Gaussian CKA can be understood in this light. We further consider kernel PCA, showing that Gaussian RBF eigenvalues, eigenprojections, and principal components all converge to those of classical (linear) PCA as bandwidth $σ\rightarrow \infty$. For a given data representation, both the RKHS feature embeddings and the orthogonal PCA eigenframes of the two kernel types differ asymptotically by a geometric similarity transformation, up to a residual of size $O \left (\fracρσ \right )^2$, where $ρ$ is a measure of geometric eccentricity of the representation, equal to the ratio of maximum to median pairwise distance between data examples. Experiments over a diverse collection of data sets demonstrate that $ρ$ provides a simple and reliable predictor of dataset-specific convergence behavior in the top principal directions.

Classical regularization removes the binary-collision singularity from the Kepler problem, but its value as a representation for learned Hamiltonian dynamics has not been systematically isolated. We compare Cartesian and planar Levi--Civita formulations of a perturbed Kepler system with a smooth quadrupole potential. With the perturbation supplied analytically, a Levi--Civita Hamiltonian splitting holds the maximum relative energy error near $2.1\times10^{-5}$ through eccentricity $e=0.99$, while the Cartesian splitting becomes unstable. This advantage persists at matched physical horizon and force-evaluation budget, where the regularized baseline is $3\times10^{-5}$, about $4.7$--$8.3$ orders of magnitude below the Cartesian arm depending on eccentricity. In held-out high-eccentricity tests with matched sampling, regularized models produce finite rollouts in $40/40$ runs versus $0/40$ for Cartesian. However, the fixed-shell construction supplies the regularized model with the exact initial orbit energy, and survival still carries $\mathcal{O}(1)$ energy error. Four neural residual objectives fail to approach the analytic result. Exact-feature controls show that the regularized residual is a four-monomial degree-6 polynomial that a direct least-squares solve fits to the baseline. The remaining exact-feature gap is due to severe raw-basis ill-conditioning: orthogonalization restores baseline fitting for L-BFGS in two iterations. Small MLPs remain at $\mathcal{O}(1)$ rollout error even after gauge symmetrization. Levi--Civita coordinates therefore improve dynamical conditioning while worsening raw-basis optimization conditioning; accurate neural residual learning remains unresolved. This is a controlled falsification-plus-trade-off study, not a solution to learned close-encounter dynamics.

Generative Astrodynamics is advanced in this work by extending generative modelling to an orbit determination problem in the cislunar environment. The task is formulated as conditional density estimation, aiming to infer the probability distribution of the initial state from angles-only measurements over short observation arcs. A normalising flow is trained on perturbed topocentric observations from Near Rectilinear Halo Orbits, enabling a flexible and potentially multimodal posterior representation. Given new measurements, the learned density is sampled to generate statistically consistent and physics-informed state hypotheses. These estimates are refined via nonlinear least-squares minimisation, providing a competitive warm start for classical algorithms.

Long-horizon spacecraft trajectory forecasting suffers from error accumulation due to the absence of corrective observations in the forecast regime, making reliable uncertainty estimation crucial for safety-critical decision-making such as space domain awareness and conjunction assessment. While high-fidelity physics-based orbit propagators provide accurate deterministic forecasts, they typically lack calibrated uncertainty estimates over long horizons. We introduce a Predictor--Corrector framework in which a physics-based continuous-time $\textit{deterministic}$ forecaster is augmented with a learned continuous-time $\textit{probabilistic}$ Corrector that models forecast errors. The proposed Corrector can be wrapped around an existing deterministic propagator to improve forecast accuracy while producing sharp and calibrated full-covariance uncertainty estimates. The Corrector is based on Latent Neural Controlled Differential Equations (Latent NCDEs) and models the probabilistic temporal evolution of forecast errors in continuous time, naturally supporting irregular sampling and missing features. We further introduce a loss function that promotes calibration and sharpness in long-horizon uncertainty propagation. We evaluate the proposed framework on long-horizon spacecraft trajectory forecasting using real-world data from NASA's Crustal Dynamics Data Information System (CDDIS), wrapping the Corrector around NASA's General Mission Analysis Tool (GMAT). Across forecast horizons of 2--4 days without observations and six rolling test windows, the proposed approach consistently improves accuracy and uncertainty calibration compared to deterministic baselines and Latent ODE-based correctors, demonstrating the effectiveness of the continuous-time probabilistic Corrector for trajectory forecasting.

Scale buys interpolation; structure buys certifiable transfer. A world model's average error does not say whether a particular rollout can be trusted, or for how long. For equivariant latent world models we give a predictability certificate: a computable region spanning configuration, horizon, and resolution. Under exact equivariance, rollout error is invariant over the monoid generated by k primitive symmetries and is certified from the k generators (Theorem A); universal orbit-flatness over equivariant targets characterizes equivariance at the function level (Lemma 2), so an unconstrained architecture cannot certify the property by construction. Approximate orbit-transfer defects propagate by the finite-time Lyapunov spectrum (Theorem B): expanding channels give a logarithmic horizon $T_j(ε)\sim\log(1/ε)/λ_j$, neutral channels accumulate recurrent defect linearly, and contracting channels accumulate a bounded nonzero floor. Exact conserved charge values are certified to all horizons only at zero defect; with one-step defect $η$, charge-value error grows at most as $Tη$. Empirically, on a 40-dimensional learned model a $\mathbb{Z}_N$-equivariant network recovers the full Lyapunov spectrum ($R^2=0.98$-$0.99$) where dense and recurrent baselines fail. A cone/adapted-metric certificate reads an a-priori horizon off the model's own Jacobian, tight on uniformly hyperbolic dynamics and self-abstaining elsewhere; the resulting horizon improves a budgeted re-observation decision. For public non-equivariant world models the tangent spectrum gives a training-free candidate horizon, paired with a held-out divergence cross-check that abstains or corrects when the learned loop over-promises.

This paper addresses orbital stabilization of a circular motion primitive for a dynamic extension of the Dubins car model within a transverse-linearization framework. We show that the corresponding transverse linearization is unstable and not stabilizable by linear state feedback. Therefore, the standard linearization-based approach to orbital stabilization cannot be applied directly. The main contribution is a set of explicit and verifiable conditions that characterize when a controller design based on transverse linearization remains applicable. These conditions rely on the specific structure of the dynamics in a neighborhood of the motion and on the use of non-standard transverse coordinates for controller design and analysis. Numerical simulations illustrate the proposed design procedure.

This paper presents a fast, recursive neural solver for the J2-perturbed Lambert problem based on Tiny Recursive Models (TRM), termed the TRM-Perturbed Lambert (TRM-PL) model. TRM is a weight-shared architecture whose effective capacity emerges from iteration depth rather than parameter count: a compact reasoning module is applied repeatedly within a two-level latent hierarchy, refining a candidate departure velocity by simulating the J2 trajectory and correcting it from the resulting tracking error. This unifies initial-guess generation and iterative correction in a single, end-to-end differentiable architecture. The recursive refinement loop is a learned alternative to the homotopy and continuation schemes of classical perturbed-Lambert solvers: rather than following a hand-designed path from the Keplerian to the perturbed solution, the network learns its own sequence of corrections. We evaluate TRM-PL on three test cases of increasing difficulty: single-revolution low-Earth-orbit (LEO) transfers, multi-revolution LEO transfers, and multi-revolution Jovian transfers. Three training paradigms are compared: jointly learning the Lambert solution and the J2 correction; refining the Lambert initial velocity with target-position and J2-corrected velocity supervision; and refining it with target-position supervision alone. Across all cases, the refinement-only approaches are the most reliable. The position-supervised variant reduces the median terminal-position error from 21.7 km to 0.027 km on single-revolution LEO, from 340.9 km to 0.31 km on multi-revolution LEO, all with the same 2.3M-parameter architecture. A single Newton corrector iteration on the TRM-PL output tightens the Jovian median to 0.063 km, yielding compact models accurate enough for embedded deployment.

Aerial navigation on Mars requires vision-based pipelines that are robust to the diverse illumination conditions and terrain morphology of the Martian surface. A key bottleneck for training and evaluating such methods is the scarcity of large-scale, annotated aerial datasets. We present MARTIAN, an open-source Blender-based rendering framework that leverages real HiRISE orbital map products to synthesize realistic aerial views of the Martian terrain under controllable lighting conditions and at varying altitudes. MARTIAN generates observations with accurate pose annotations, directly addressing the scarcity of training data for vision-based navigation on Mars. The framework has been validated through its deployment in concurrent work on map-based localization systems for Ingenuity and future Mars rotorcraft, where synthetically trained deep image matchers were successfully evaluated on real Mars imagery. MARTIAN is publicly available at: https://github.com/nasa-jpl/martian.

Machine learning interatomic potentials (MLIPs) require generating computationally expensive, large-scale training datasets to accurately simulate materials and molecules. Incorporating electronic structure information using multitask learning improves sample efficiency, however, training on full Hamiltonian matrices, which scale quadratically with the number of atoms, is intractable for large datasets. In this work, we show that multitask learning utilizing orbitally resolved semiempirical charges significantly improves sample efficiency and accuracy in MLIPs. To efficiently predict orbital charges, we implement a specialized equivariant model, reducing charge prediction error compared to an invariant baseline. By augmenting training with computationally inexpensive GFN1-xTB orbital charges, which scale linearly with the number of atoms, our model achieves a 46% reduction in energy mean absolute error and requires five times less data to match the performance of energy-only models. Furthermore, our approach outperforms models trained on expensive density functional theory (DFT) atomic charges, capturing orbitally resolved electronic complexity and forcing the network to learn a physically accurate latent space that spontaneously clusters metals by shared chemical properties. Because orbital charges are only required during training, this approach preserves inference efficiency, providing a scalable recipe for developing accurate, data-efficient foundation models for complex chemical systems.

Detecting orbital anomalies, such as maneuvers, atmospheric decay, and attitude upsets, across the rapidly growing population of low-Earth-orbit (LEO) satellites is a prerequisite for collision avoidance, decay forecasting, and conjunction screening. The bottleneck is not modeling capacity but labels: there is no public ground-truth corpus of orbital anomalies, manual review does not scale to approximately 10^4 active satellites, and pure rule-based detectors trade recall for precision so aggressively that they are blind to most behavioral anomalies. We present a multi-tier labeling cascade that composes three weak supervision sources of increasing fidelity: a fast physics rule set (rule_v1), an Interacting Multiple Model Unscented Kalman Filter (IMM-UKF) bank, and a supplemental-element calibration step (supGP), to produce labels at a scale unavailable from any single source. Applied to 232M Two-Line Element (TLE) records spanning 60 years, the cascade yields 8.6M labeled sequences of length 50 (430M timesteps) over 11 features that include explicit time encoding and full mean-element state. On overlapping satellites, IMM-UKF surfaces 42.6x more anomalies than rule_v1 alone. We train a 6.5M-parameter Transformer in two stages, achieving a maneuver recall of 55.4% and decay recall of 62.8% on a held-out test set. An ablation on the time-delta feature alone yields a 107% relative improvement in decay recall. We frame the resulting model as a high-recall triage classifier whose role is to surface candidate events for downstream filtering, not to issue final attributions, and discuss the path toward a Neural-ODE-based orbital world model.

We present You Only Stack Once (YOSO), an automated pipeline designed to detect faint, slow-moving Solar System objects in wide-field astronomical surveys. The pipeline integrates a novel Gaussian Motion Filter (GMoF) that operates at the pixel level to enhance signal-to-noise for objects exhibiting a range of apparent rates of motion. Unlike conventional shift-and-stack methods, which rely on discrete velocity trials, GMoF amplifies trails while suppressing random noise and static background features. Applied to a subset of DEEP observations from the Dark Energy Camera, YOSO recovered 45 out of 73 previously detected objects, as well as 11 new TNOs. It also discovered 216 objects in the near Solar System. Although alternative shift-and-stack methods are sensitive to objects about 0.88 magnitudes fainter, YOSO's false positive rate is extremely low, since it detects only sources that exhibit a trail and are consistent with a point source when shifted at the right rate. We show how this method can be deployed on large surveys like LSST, and adapted for other domains that require motion-based signal enhancement, including exoplanet imaging through Angular Differential Imaging (ADI), and near-Earth object (NEO) detection for missions like NEO Surveyor. YOSO thus provides a versatile, scalable approach for extracting faint, motion-dependent signals in the era of data-intensive astronomy.

Artificial satellites and space debris increasingly contaminate astronomical images, affecting scientific surveys and producing large volumes of streaked exposures. Manual inspection is no longer feasible at scale, and reliable detection and characterisation of streaks has become essential for both data-quality control and the monitoring of objects in Earth orbit. We present StreakMind, an automated pipeline designed to detect Near-Earth Objects and satellite streaks in astronomical images, characterise their geometry, and cross-identify them with known orbital objects. The system integrates all inference results into a structured database suitable for large surveys. A YOLO OBB model was trained on a hybrid dataset of 2335 images and applied to processed FITS frames. Geometric refinement, inter-frame association, satellite cross-identification, and Gaussian-based confidence scoring were then used to produce final identifications stored in a relational database. Observations from La Sagra Observatory were used to develop and test the method. On the test set, the model achieved a precision of 94 percent and a recall of 97 percent. It reliably detected faint streaks, delivered consistent geometric reconstructions, and performed robust satellite cross-identification. StreakMind demonstrates strong potential for large-scale automated analysis of linear streaks produced by both Near-Earth Objects and artificial satellites, contributing to space situational awareness.

Real-world knowledge is often organized as hierarchies such as product taxonomies, medical ontologies, and label trees, yet learning hierarchical representations is challenging due to asymmetric structure and noisy semantics. We introduce Polaris, a polar hyperspherical embedding framework that separates semanticity from hierarchy using angular geometry and radius, enabling the learning of meaning and structure without interference. To map latent representation onto the sphere, we project it to the tangent space at the north pole, apply the exponential map, and learn unit-norm representations using spherical linear layers. Polaris then combines robust local constraints, global regularization that prevents geometric collapse, and uncertainty-aware asymmetric objectives that encourage directional containment. At inference time, Polaris uses structure-guided retrieval to efficiently narrow down candidate parents before final ranking. We evaluate Polaris on different settings of taxonomy expansion - spanning trees, multi-parent DAGs, and multimodal hierarchies, showing consistent improvements of up to ~19 points in top-K retrieval and up to ~60% reduction in mean rank over fourteen strong baselines.

We introduce surrogate functionals: machine-learned energy functionals for orbital-free density functional theory (OF-DFT) which are defined not by universal fidelity to a physical reference, but merely by the requirement that density optimization with a fixed procedure yields the true ground-state density. Helpfully, training surrogate functionals requires only ground-state densities, no energies or gradients away from the ground state. We here propose a gradient-descent-improvement loss that guarantees exponential convergence of the density to the ground state, and combine it with an adaptive sampling scheme that concentrates learning around the optimization trajectories actually visited during inference. On the QM9 and QMugs benchmarks, surrogate functionals achieve density errors competitive with or improving upon the state of the art for fully supervised machine-learned OF-DFT, while eliminating the need for the $O(N^3)$ orthononormalization step required by prior work, yielding improved runtime scaling for larger systems.

Many learning problems are organized by group symmetries. While invariance is often imposed through architectures or group averaging, we ask when it can emerge from training on a finite random subset of an orbit. We study this question in classical Hopfield networks, where strict memorization can be expressed as a linear margin problem. Reparameterizing minimization of energy flow (MEF) as an exponential loss connects gradient descent to the corresponding minimum-norm hard-margin memorizer. Our main result shows that, for independent uniform samples from any finite permutation orbit, the exact sample hard-margin support vector machine (HSVM) concentrates exponentially around the invariant full-orbit HSVM. Consequently, an orbit-size-independent polynomial number of samples suffices both for approximate parameter invariance and for simultaneous memorization of every orbit element; directional convergence transfers this conclusion asymptotically to MEF gradient descent. For graph-isomorphism orbits, we characterize the invariant parameters as a three-dimensional subspace and show that every such orbit is memorizable. For cliques of fixed linear density, additional symmetry sharpens the uniform memorization bound to $O(v^4\log(1/δ))$, exponentially smaller than the orbit size. Together with experiments across several learning rules, these results give a finite-sample account of how optimization bias can recover symmetry from partial group-structured data.

In this work, we propose a flow-matching Markov chain Monte Carlo (FM-MCMC) algorithm for estimating the orbital parameters of exoplanetary systems, especially for those only one exoplanet is involved. Compared to traditional methods that rely on random sampling within the Bayesian framework, our approach first leverages flow matching posterior estimation (FMPE) to efficiently constrain the prior range of physical parameters, and then employs MCMC to accurately infer the posterior distribution. For example, in the orbital parameter inference of beta Pictoris b, our model achieved a substantial speed-up while maintaining comparable accuracy-running 77.8 times faster than Parallel Tempered MCMC (PTMCMC) and 365.4 times faster than nested sampling. Moreover, our FM-MCMC method also attained the highest average log-likelihood among all approaches, demonstrating its superior sampling efficiency and accuracy. This highlights the scalability and efficiency of our approach, making it well-suited for processing the massive datasets expected from future exoplanet surveys. Beyond astrophysics, our methodology establishes a versatile paradigm for synergizing deep generative models with traditional sampling, which can be adopted to tackle complex inference problems in other fields, such as cosmology, biomedical imaging, and particle physics.