Technical Summary: Physics-Informed Conditional Normalizing Flows for Angles-Only Cislunar Orbit Determination

Problem Statement

This work addresses the challenge of Initial Orbit Determination (IOD) within the cislunar environment, specifically for spacecraft operating in Near Rectilinear Halo Orbits (NRHOs) around the Earth-Moon $L_2$ point. The problem is formulated as an angles-only estimation task, where the objective is to infer the probability distribution of a target's initial geocentric state ($X_1 \in \mathbb{R}^6$) based solely on a short arc of azimuth and elevation measurements ($\theta, \phi$) collected by a ground station (Cebreros, Spain).

The problem is characterized by significant difficulties:

• Data Scarcity: Observations are limited to short arcs (50–150 minutes) with sparse cadence.
• Dynamical Complexity: The cislunar environment is governed by the Circular Restricted Three-Body Problem (CR3BP), exhibiting intrinsic instability and chaotic behavior.
• Lack of Priors: Unlike routine Orbit Determination (OD), IOD lacks prior state information, making the inverse problem highly non-linear and prone to local minima in traditional optimization.
• Absence of GNSS: There is no dedicated Global Navigation Satellite System for the cislunar region, necessitating reliance on optical tracking.

Methodology

The authors propose a Generative Astrodynamics framework that treats orbit determination as a conditional density estimation problem. Instead of directly solving for a single state vector, the model learns the conditional probability distribution $p(\bar{X}_1 | Y)$, where $Y$ represents the sequence of angle-only observations.

1. Generative Architecture: Conditional Normalizing Flows

The core of the approach is a Conditional Normalizing Flow (CNF) model, specifically a Real Non-Volume Preserving (Real-NVP) architecture.

• Transformation: The model defines an invertible mapping $f: \mathbb{R}^6 \to \mathbb{R}^6$ that transforms the target initial state $\bar{X}_1$ into a latent variable $w$ drawn from a known base distribution (standard multivariate Gaussian).
• Conditioning: The flow is conditioned on the observation arc $Y_k$. To handle variable-length observation sequences and diverse observer locations, a Transformer-based encoder processes the observation sequence $Y_k$ into a fixed-size representation $\tilde{Y}_k$, which is concatenated with the masked state components at each coupling layer.
• Sampling: During inference, latent samples are generated by perturbing latent codes derived from the training data ($w_{inf} = w_{train} + 0.1\varepsilon$). These are mapped back through the inverse flow $g(w|Y_{new})$ to generate a set of statistically consistent initial state hypotheses.

2. Physics-Informed Regularization

A critical component of the methodology is the integration of physical laws into the training loss function to ensure dynamical consistency.

• Standard Loss: The model minimizes the negative log-likelihood of the conditional distribution ($L_{flow}$).
• Physics Loss ($L_{phys}$): An additional term is introduced to penalize states that, while statistically probable, violate the equations of motion. This is calculated by:
  1. Propagating the generated initial state forward in time using the CR3BP Equations of Motion (integrated via RK4).
  2. Computing the predicted angles-only observables.
  3. Calculating the residual vector $\bar{\sigma}$ between predicted and actual measurements.
  4. Minimizing the squared norm of these residuals.
• Total Objective: The combined loss is $L = L_{flow} + \beta L_{phys}$, where $\beta$ is a weighting coefficient that increases linearly during the first 100 epochs of training.

3. Hybrid Estimation Pipeline

The generative model serves as a warm start for classical estimation algorithms. The workflow is:

1. Generation: The CNF generates multiple plausible initial state hypotheses conditioned on new measurements.
2. Refinement: These hypotheses are fed into a Nonlinear Least-Squares Estimation (NLSE) algorithm (Gauss-Newton iterative scheme) to refine the estimates and minimize the measurement residuals.

Key Contributions

1. Generative Formulation of IOD: The paper extends Generative Astrodynamics to the cislunar domain, framing IOD as a conditional density estimation task rather than a deterministic inverse problem. This allows for the representation of multimodal posteriors and inherent uncertainty quantification.
2. Physics-Informed Normalizing Flows: The authors demonstrate that augmenting the likelihood-based training of normalizing flows with a physics-informed loss term ($L_{phys}$) is essential. While the flow can learn to reconstruct training data without this term, the physics loss acts as a regularizer that ensures the learned latent manifold preserves dynamical consistency, preventing the generation of physically inadmissible states during inference.
3. Robust Initialization Strategy: The study validates the CNF as a superior initialization strategy for classical NLSE compared to naive approximations (e.g., using the Moon's state as a proxy).

Results

The model was trained on a synthetic dataset of 4,119 valid observation arcs derived from 10 distinct NRHO trajectories.

• Training Dynamics: The inclusion of $L_{phys}$ resulted in a stable training process where the latent distribution remained close to the base Gaussian while capturing necessary deviations for physical fidelity.
• Performance Comparison:
  • Vs. Moon-Based Approximation: The flow-based initialization significantly outperformed initialization based on the Moon's state, yielding lower position/velocity errors and much smaller post-fit residuals. The Moon-based approach frequently converged to local minima.
  • Vs. Exact Initialization: The flow-based approach achieved accuracy comparable to initializing the NLSE with the exact true state, demonstrating its ability to provide high-quality starting points.
• Ablation Study: Removing the physics-informed loss term led to a degradation in robustness. While average residuals remained low, the model produced large maximum errors (up to ~9,600 km in position), indicating a failure to generalize to physically consistent solutions outside the immediate training manifold.

Significance and Claims

The paper claims that this work represents a significant step forward in Generative Astrodynamics by successfully applying conditional normalizing flows to a complex, real-world navigation problem (cislunar IOD).

The primary significance lies in the synergy between generative AI and physical laws. The authors argue that:

• Purely data-driven generative models may produce statistically plausible but dynamically impossible states.
• The physics-informed loss is not merely a performance booster but a necessary constraint to ensure the generative model learns the underlying dynamical structure of the CR3BP.
• The proposed method provides a probabilistic alternative to deterministic OD, offering a natural mechanism for uncertainty quantification and the ability to handle multi-modal solution spaces (e.g., multiple valid orbits fitting the same short arc).
• As a warm start for classical algorithms, the method mitigates the risk of convergence to local minima, a critical advantage in the highly non-linear cislunar environment where observation arcs are short and noisy.

The authors conclude that this approach has the potential to be extended to broader space navigation scenarios, including sparse observations and autonomous onboard navigation, provided future work validates performance on real observational data.