🔢 mathematics

Comparative analysis of Neural Networks approaches for Initial Orbit Determination of Near-Earth Objects

This paper presents a comparative analysis demonstrating that a physics-informed neural network outperforms both a purely data-driven neural network and classical methods (Gauss's and Laplace's) in Initial Orbit Determination for Near-Earth Objects using Very Short Arcs, particularly by achieving higher rates of dynamically admissible predictions and superior performance on fast, nearby objects.

Original authors: Francesco Geroni, Roberto Paoli, Riccardo Massidda, Giacomo Tommei

Published 2026-09-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Francesco Geroni, Roberto Paoli, Riccardo Massidda, Giacomo Tommei

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Every night, powerful telescopes sweep the sky, scanning for the faint, moving specks of rock that circle our planet. These are near-Earth objects, and while most are harmless, some pose a genuine threat if they were to collide with us. The challenge for astronomers is not just finding them, but quickly understanding where they are going. When a new object is spotted, it is often seen for only a few hours on a single night before it disappears behind the sun or into the glare of dawn. This fleeting glimpse, known as a very short arc, provides a handful of angles but almost no information about how far away the object actually is. Without knowing the distance, scientists cannot calculate the object's speed or its future path, leaving it a mystery that might be lost forever before a follow-up observation can be scheduled.

To solve this, researchers must turn a few points of light into a complete picture of motion. They need to determine the distance to the object and how fast that distance is changing. This is a notoriously difficult mathematical puzzle because the limited data allows for many different possible answers, most of which are physically impossible. For decades, the standard tools used to solve this have struggled when the data is this sparse, often failing to find a solution or settling on an answer that places the object impossibly close to Earth. A team of researchers at the University of Pisa has now tested a new approach using artificial intelligence to tackle this specific problem, aiming to provide a reliable first guess for these dangerous, newly discovered rocks.

The team developed and compared two different types of neural networks, which are computer systems designed to learn patterns from data. Both systems were fed the same limited information: three snapshots of an object's position taken over a single night. The first system was a standard data-driven model, trained purely to memorize the relationship between the angles seen in the sky and the true distances recorded in a massive database of known asteroids. It learned by looking at hundreds of thousands of real observations from the NEODyS-2 catalogue, a comprehensive library of asteroid tracking data. The second system was a more sophisticated version that did not just look at the data but also obeyed the laws of physics. This model was built with a specific rule baked into its learning process: the object's motion must follow the gravitational rules that govern how things move around the sun and the Earth. This physical constraint was designed to stop the computer from guessing wildly or settling on a mathematically possible but physically nonsensical answer.

When the researchers tested these models against a set of new, unseen data, the results revealed a clear trade-off. The standard data-driven model was slightly better at predicting the exact distance for the average object, but it did so by shrinking its answers toward a safe, middle-ground average. It tended to ignore the extremes, which meant it often failed to predict the true distance for objects that were either very close or moving very fast. The physics-informed model, however, behaved differently. While its average error was slightly higher, it was far more successful at producing answers that were physically possible. On the test set, the physics-based model found a valid, bound orbit for 85.1% of the arcs, compared to only 76.5% for the standard model. More importantly, the physics-based model excelled in the most critical scenarios: when the object was close and moving quickly. These are the exact situations that matter most for planetary defense, as a fast, nearby object is the one most likely to hit Earth soon.

The study also showed that the old, classical methods of calculating orbits, which have been used for centuries, are largely ineffective in this specific regime. When applied to the same short arcs, the traditional methods failed to find a solution for more than half of the cases. When they did succeed, they frequently collapsed onto a degenerate solution, placing the object impossibly close to the observer, effectively guessing that the rock was right on top of the telescope. The new neural network approaches, by contrast, were able to provide a solution for every single arc tested. While neither artificial intelligence model solved the problem perfectly—the inherent lack of data still leaves some uncertainty—the physics-informed approach proved to be a significant step forward. It offers a way to generate a reliable, physically consistent starting point for tracking these objects, turning a single night of observations into a usable prediction that can guide future telescopes to the right spot in the sky.

The researchers emphasize that this is not a final solution to the problem of tracking asteroids, but a powerful new tool for the first step. The models are fast, capable of making predictions in milliseconds, which makes them suitable for real-time use in automated survey systems. By combining the pattern recognition of machine learning with the hard constraints of orbital mechanics, the team has created a system that is more robust than previous methods. The work highlights that while the data from a single night is incomplete, it is not useless. With the right mathematical framework, it is possible to extract enough information to keep a newly discovered object from vanishing into the void, giving humanity a better chance to monitor and, if necessary, defend against the rocks that share our neighborhood.

Technical Summary: Comparative Analysis of Neural Network Approaches for Initial Orbit Determination of Near-Earth Objects

Problem Statement
Initial Orbit Determination (IOD) from Very Short Arcs (VSAs)—observations spanning only a few hours within a single night—remains a critical challenge in asteroid surveillance. Classical analytical methods, such as those by Gauss and Laplace, rely on sufficient orbital curvature to solve the ranging problem (recovering geocentric range $\rho$ and range-rate $\dot{\rho}$ from angular data). However, in the VSA regime, curvature information is weak, rendering these methods ill-conditioned or causing them to fail entirely. Furthermore, the problem is inherently degenerate; a single night of observations often admits multiple distinct orbital solutions, including a "degenerate near-observer" root that is physically implausible but mathematically valid. Consequently, many newly discovered Near-Earth Objects (NEOs) are lost before follow-up observations can be scheduled.

Methodology
The authors propose and compare two Neural Network (NN) architectures designed to directly predict the geocentric range and range-rate $(\rho, \dot{\rho})$ from a triplet of time-tagged angular measurements $(t_i, \alpha_i, \delta_i)$ and the observer's heliocentric position. The input vector excludes absolute calendar time to prevent the network from learning secular trends rather than orbital geometry.

Multi-Layer Perceptron (MLP): A standard data-driven model trained to minimize the Mean Squared Error (MSE) between predicted and ground-truth $(\rho, \dot{\rho})$ values derived from the NEODyS-2 catalogue.
Physics-Informed Neural Network (PINN): An extension of the MLP that incorporates physical constraints into the loss function. The total loss $L(\theta)$ $L (θ)$ is a weighted sum of:
- Data-driven terms: A residual loss using log-cosh to balance quadratic sensitivity and outlier robustness.
- Physics-informed terms:
  - Equation of Motion (EOM): Enforces the Gauss scalar range ODE, derived by projecting the two-body equations of motion onto the line of sight.
  - Kinematics: Enforces consistency between the predicted $\dot{\rho}^*$ and the time derivative of $\rho^*$ .
- Distributional term: A Wasserstein-1 ( $W_1$ ) distance loss designed to counteract "mode collapse," where the network converges to a single degenerate solution (e.g., the near-observer root) rather than capturing the multimodal distribution of possible orbits. This term includes a conditioned component penalizing violations of the physical prior that faster-moving objects are generally closer.

The models were trained on a dataset of 568,127 VSAs from 39,031 real NEOs, using an object-disjoint partition to prevent data leakage.

Key Results
The study evaluates performance on a held-out test set of 56,773 arcs using metrics including Mean Absolute Error (MAE), Median Absolute Error (medAE), correlation, and the fraction of arcs yielding a dynamically admissible solution (PAR).

Admissibility: The PINN significantly outperforms the MLP in generating physically valid orbits. The PINN places 85.1% of its predictions inside the admissible region, compared to 76.5% for the MLP. In contrast, classical Gauss and Laplace methods return solutions for only 45.0% and 47.3% of arcs, respectively, and when they do succeed, they collapse onto the degenerate near-observer root in approximately 90% of cases.
Error Characteristics: The MLP achieves a lower MAE (0.286 AU vs. 0.304 AU for PINN) but a higher medAE. This indicates the MLP shrinks predictions toward the population mean, reducing the error tail but losing the ability to reproduce the true distribution of orbits. The PINN, while having a longer error tail, better preserves the spread of the data and provides more accurate median estimates for specific regimes.
Stratified Analysis: Performance varies by proper motion ( $\eta$ $η$ ) and true range ( $\rho$ $ρ$ ).
- For slow and distant arcs, the MLP performs comparably or better.
- For fast and nearby arcs (the most operationally relevant for planetary defense), the PINN demonstrates a marked advantage in both admissibility (PAR) and median error, outperforming the MLP across all range bins in the fast stratum.
Uncertainty Quantification: The authors employ Mondrian conformal prediction to generate distribution-free prediction intervals calibrated per proper-motion bin. This allows for adaptive uncertainty estimates that narrow for informative geometries (fast arcs) and widen for degenerate ones.

Significance and Claims
The paper positions this work as a first step toward fast, robust tools for real-time IOD of newly discovered NEOs. The authors claim that while the intrinsic degeneracy of the VSA problem is only partially addressed (as the mapping from attributable to orbital elements remains non-unique), the PINN approach offers a superior trade-off for operational scenarios. Specifically, the physics-informed objective sacrifices some raw accuracy (longer error tail) to gain feasibility and regime-dependent accuracy for the fast, nearby objects that are critical for planetary defense.

The study concludes that classical methods are largely ineffective for single-night VSAs, often failing or returning degenerate roots. While the developed NN models do not "solve" the ranging problem perfectly, they provide a more reliable initial guess than classical methods, capable of producing predictions in milliseconds suitable for operational pipelines. The authors modestly note that the current uncertainty estimates are marginal and that future work will focus on refining physical terms (e.g., including Jupiter's perturbation), addressing multimodal solution structures more naturally, and validating the initial guesses via differential corrections convergence.

More like this