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Continuous-Time Probabilistic Correctors for Uncertainty-Aware Physics-Based Spacecraft Trajectory Forecasting

This paper introduces a continuous-time probabilistic corrector based on Latent Neural Controlled Differential Equations that augments physics-based deterministic spacecraft propagators to significantly improve long-horizon trajectory forecast accuracy and provide calibrated uncertainty estimates.

Original authors: Muhammad Bilal Shahid, Zhanhong Jiang, Soumik Sarkar, Cody Fleming

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Muhammad Bilal Shahid, Zhanhong Jiang, Soumik Sarkar, Cody Fleming

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

The Big Picture: Predicting the Future of Spacecraft

Imagine you are trying to predict exactly where a runaway shopping cart will be in four days. You know the laws of physics (gravity, friction), and you have a super-smart calculator (NASA's GMAT) that simulates the cart's path perfectly right now.

However, once you let go of the cart and stop watching it, tiny things start to go wrong. A gust of wind you didn't measure, a bump in the pavement, or a slight miscalculation in your calculator's math causes the cart to drift off course. Over four days, that tiny drift becomes a huge error.

The problem is that the super-smart calculator (GMAT) is very confident in its answer, even when it's wrong. It says, "I know exactly where the cart is," without admitting, "But I might be off by a few miles." In space, this is dangerous. If we think a satellite is safe when it's actually on a collision course, we could lose a multi-million dollar asset.

This paper introduces a new "Co-Pilot" system. It doesn't replace the super-smart calculator; instead, it wraps around it to say, "Hey, based on how the calculator has messed up in the past, here is a better guess of where the cart is, and here is a realistic map of how uncertain we are."

The Core Idea: The Predictor and the Corrector

The authors built a two-part system called a Predictor–Corrector framework.

1. The Predictor (The Physics Engine)

Think of this as a high-tech GPS. It uses the laws of physics to calculate where a spacecraft should be. It's very good at the short term, but as time goes on without new data, it starts to drift.

The Flaw: It gives a single point answer (e.g., "It will be at this exact spot") and doesn't admit when it's getting unsure.

2. The Corrector (The "Time-Traveling" Learner)

This is the new invention in the paper. Think of it as a smart detective that studies the GPS's past mistakes.

How it learns: It looks at the history of where the GPS was wrong. It notices patterns: "Oh, every time the GPS predicts for 3 days, it tends to drift slightly to the left."
The Magic Tool (Latent NCDE): The paper uses a special mathematical tool called a Latent Neural Controlled Differential Equation (NCDE).
- Analogy: Imagine the GPS's path is a river. A standard computer model tries to predict the river's flow by taking snapshots every second. But rivers flow continuously. The NCDE is like a smooth video of the river. It understands that the water flows between the snapshots, not just at the snapshots. This allows it to handle messy data where observations are missing or arrive at weird times.
The Result: The Corrector doesn't just guess a new spot; it draws an uncertainty bubble (an ellipsoid) around the prediction. It says, "The spacecraft is likely here, but it could be anywhere inside this bubble."

Why This is Better Than Old Methods

The paper compares their new "Co-Pilot" against older methods (like Latent ODEs) and the raw GPS (GMAT).

The Old Way (GMAT): "I am 100% sure the satellite is at Point A." (Often wrong, and dangerously overconfident).
The "Almost Right" Way (Latent ODE): "I think it's at Point A, but I'm not sure." (Better, but the "not sure" part is often too small or too big, making it unreliable).
The New Way (CTPC with NCDE): "It's likely at Point A, and I've drawn a bubble that perfectly captures the range of where it could actually be."
- Sharpness: The bubble isn't a giant, useless cloud; it's tight and precise.
- Calibration: If the bubble says there's a 90% chance the satellite is inside, then 90% of the time, it is inside. It's honest about its uncertainty.

The "Student-t" Secret Sauce

To make these bubbles accurate, the team used a specific statistical tool called the Student-t distribution.

Analogy: Imagine you are betting on a horse race. A normal bell curve assumes the horse will run at a steady speed. But in reality, a horse might trip or sprint unexpectedly (heavy tails). The Student-t distribution is like a safety net that accounts for these rare, wild swings. It prevents the system from being shocked by sudden, unexpected errors.

The Real-World Test

The authors didn't just run this on fake data. They tested it on real NASA data from actual spacecraft.

The Challenge: They tried to predict the path of satellites for 2 to 4 days into the future without any new telescope data to help them.
The Outcome: Their new system consistently beat the standard NASA tools. It reduced the error in the prediction by up to 82% in some cases and, crucially, provided a much more honest and reliable "safety bubble" around the prediction.

Summary in One Sentence

This paper teaches a computer to act like a humble, experienced co-pilot that watches a high-tech physics engine, learns from its past drifts, and draws a perfectly sized "safety bubble" around future predictions so we know exactly how much to trust them.

Technical Summary: Continuous-Time Probabilistic Correctors for Uncertainty-Aware Physics-Based Spacecraft Trajectory Forecasting

Problem Statement

Long-horizon spacecraft trajectory forecasting is critical for Space Domain Awareness (SDA), including collision-risk assessment and conjunction analysis. While high-fidelity physics-based propagators, such as NASA's General Mission Analysis Tool (GMAT), provide accurate deterministic forecasts, they lack reliable, calibrated uncertainty estimates over long, observation-free horizons. In operational regimes where no corrective observations are available, unmodeled dynamics, parametric mismatches, and numerical approximations cause forecast errors to accumulate. Consequently, the error covariance reported by standard propagators often becomes overconfident, understating the true uncertainty and associated collision risks. Existing machine learning approaches often fail to provide full-covariance uncertainty estimates or struggle with irregularly sampled data and missing features common in real-world telemetry.

Methodology

The authors propose a Predictor–Corrector framework designed to wrap around existing deterministic physics-based simulators (specifically GMAT) to enhance both accuracy and uncertainty quantification.

1. Framework Architecture

The framework consists of two components:

Predictor: A continuous-time deterministic forecaster (e.g., GMAT) that generates the baseline trajectory.
Corrector: A learned continuous-time probabilistic module that models the temporal evolution of the predictor's forecast errors.

2. Core Architecture: Latent Neural Controlled Differential Equations (Latent NCDEs)

The primary contribution is the CTPC-CDE++ variant, which utilizes Latent NCDEs. This architecture employs an encoder-decoder structure:

Encoder: Processes the history of GMAT forecast errors (observed up to time $t_{T'}$ ) using an NCDE. It maps this history into a latent distribution (a diagonal Gaussian) that serves as a prior.
Decoder: Models the probabilistic evolution of errors over the forecast horizon ( $t_{T'}$ to $t_T$ ). The decoder is an NCDE driven by a control path constructed from the GMAT's deterministic forecasts. By sampling from the latent distribution to initialize the decoder, the model generates diverse error trajectories, capturing epistemic-like uncertainty.
Prediction Head: Maps the decoder's hidden state to the parameters of a multivariate Student- $t$ distribution (mean $\mu_t$ and full covariance $\Sigma_t$ ) for the forecast error.

3. Training Objective

The model is trained using a composite loss function designed to ensure both sharpness and calibration:

Student- $t$ Negative Log-Likelihood (NLL): Captures heavy-tailed error distributions.
Continuous Ranked Probability Score (CRPS): A strictly proper scoring rule that encourages well-calibrated predictive distributions.
KL Regularization: Ensures the latent posterior remains close to a standard normal prior.

4. Inference

During inference, the framework aggregates $K$ Monte Carlo samples from the latent space. The final uncertainty estimate combines the within-sample (aleatoric) variance and the between-sample (latent-induced) variance, effectively creating an ensemble-like effect without training multiple distinct networks.

Key Contributions

Continuous-Time Probabilistic Correctors (CTPC): A framework that augments deterministic physics simulators with learned continuous-time probabilistic error dynamics.
CTPC-CDE++ Architecture: A specific implementation based on Latent NCDEs that naturally supports irregularly sampled observations and missing features via spline-based control paths, unlike standard Latent ODEs which require explicit masking.
Calibration and Sharpness: The introduction of a loss function combining Student- $t$ likelihood and CRPS, which produces sharp, full-covariance uncertainty estimates that remain calibrated over long horizons (up to 4 days).
Coordinate Frame Robustness: Theoretical and empirical demonstration that NCDE-based decoders can model error dynamics robustly across different coordinate frames (ECI and RTN), whereas NODE-based decoders are restricted to specific frames (e.g., RTN) due to their lack of explicit control path conditioning.

Experimental Results

The framework was evaluated on real-world spacecraft trajectory data from NASA's Crustal Dynamics Data Information System (CDDIS) across six rolling test windows with forecast horizons ranging from 16 hours to 4 days (5760 minutes).

Accuracy: All CTPC variants significantly reduced the Mean Squared Error (MSE) compared to the baseline GMAT. The CTPC-CDE++ variant achieved the best performance, reducing MSE by approximately 64–82% across various horizons compared to GMAT alone.
Calibration: The proposed method produced well-calibrated uncertainty estimates. The normalized squared Mahalanobis distance ( $\bar{d}^2$ ) remained close to the ideal value of 1.0 across all horizons. In contrast, baseline Latent ODE models (even with CRPS loss) exhibited severe overconfidence, with $\bar{d}^2$ values reaching orders of magnitude higher than 1.
Sharpness: The CTPC models maintained sharp uncertainty ellipsoids (low log-determinant of covariance) without sacrificing calibration, demonstrating that the uncertainty was not artificially inflated to compensate for poor accuracy.
Latent Variability: Experiments showed that aggregating total predictive variance (including latent-induced variability) was critical for calibration. Using only the mean predictive variance led to underestimation of uncertainty and overconfident predictions.
Coordinate Robustness: When tested in the Earth-Centered Inertial (ECI) frame, the NCDE-based CTPC-CDE maintained substantial accuracy and calibration improvements, whereas the NODE-based CTPC-ODE failed to generalize, showing negligible MSE reduction and severe miscalibration.

Significance and Claims

The paper claims that the proposed Continuous-Time Probabilistic Corrector effectively bridges the gap between high-fidelity physics-based propagation and the need for reliable, calibrated uncertainty in safety-critical space operations. By wrapping around existing tools like GMAT, the method improves forecast accuracy while providing full-covariance uncertainty estimates that are robust to long observation gaps and irregular data sampling.

The authors emphasize that their approach addresses the specific failure modes of current methods: the overconfidence of deterministic baselines and the poor calibration of standard probabilistic models over long horizons. The theoretical analysis provides guarantees on decoder stability and the bounded growth of covariance, ensuring that uncertainty propagation remains interpretable and does not explode numerically. The work positions CTPC-CDE++ as a flexible, generalizable solution for uncertainty-aware trajectory forecasting that can be adapted to various coordinate frames and operational constraints without architectural modification.