Continuous-Time Probabilistic Correctors for Uncertainty-Aware Physics-Based Spacecraft Trajectory Forecasting
Authors: Muhammad Bilal Shahid, Zhanhong Jiang, Soumik Sarkar, Cody Fleming
Organizations: Iowa State University, Ames, Iowa, 50011, USA
Abstract
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 deterministic forecaster is augmented with a learned continuous-time 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.
Most trajectory forecasting models are trained on clean annotated histories, and are often evaluated under the same idealized assumption, although practical deployments rely on trajectories produced by imperfect multi-object trackers. The real-world observations exhibit localization jitter, missed or unstable detections, and data-association ambiguity, which are usually either ignored or removed through denoising. This paper instead treats tracking-derived reliability cues as an informative signal to be propagated to the predictor. We propose a plug-in uncertainty-aware formulation in which each observed state is encoded as an uncertain state representation, modeled by a Gaussian distribution whose covariance combines detection-level localization uncertainty and association-level ambiguity through the law of total variance. Existing backbones are adapted with minimal architectural changes: input trajectories are represented as Gaussian observations, and predicted trajectories are produced as Gaussian forecasts rather than deterministic coordinates. To train predictors that remain robust under structured observation noise, we combine temporally correlated Ornstein-Uhlenbeck perturbations with response-based knowledge distillation from a teacher trained on clean trajectories. Experiments on Oxford Town Centre and VIRAT using real tracker outputs, together with a complementary ETH/UCY pseudo-detection protocol, show that the proposed formulation improves displacement accuracy and the reliability-sharpness trade-off of probabilistic forecasts.
Stephane Da Silva Martins, Victor Petrovic, Emanuel Aldea +1
Coupled spatiotemporal forecasting is important for predicting the future evolution of multiple interacting dynamical systems, such as in climate models. However, existing methods are severely constrained by the persistent bottleneck of compounding errors. In coupled systems, errors from each subsystem simulator propagate and amplify one another, a phenomenon we term Reciprocal Error Amplification, leading to a rapid collapse of long-range predictions. To address this challenge, we propose a universal framework called PnP-Corrector (Plug-and-Play Corrector). The core idea of our framework is to decouple the physical simulation from the error correction process: it freezes pre-trained physics simulation engines and exclusively trains a correction agent to proactively counteract the systematic biases emerging from the coupled system. Furthermore, we design an efficient predictive model architecture, DSLCast, to serve as the backbone of this framework. Extensive experiments demonstrate that our method significantly enhances the long-term stability and accuracy of coupled forecasting systems. For instance, in the challenging task of a 300-day global ocean-atmosphere coupled forecast, our PnP-Corrector framework reduces the prediction error of the baseline model by 28% and surpasses state-of-the-art models on several key metrics.
Time series forecasting plays a central role in many real-world applications and has been extensively studied. Most existing approaches rely on deterministic models. However, real-world environments exhibit inherently uncertain and complex future behaviors, making single-point predictions insufficient. This highlights the need for probabilistic forecasting methods that can quantify and represent uncertainty. In this work, we propose PaP-NF, a probabilistic forecasting framework that aligns continuous time series representations with a frozen large language model (LLM) using a Prefix-as-Prompt mechanism, and conditions a normalizing flow decoder on the global context extracted by the LLM. The quality of the resulting predictive distributions is evaluated using the Continuous Ranked Probability Score (CRPS), a standard metric in probabilistic forecasting. Across a variety of long-term forecasting benchmarks, PaP-NF robustly captures multi-modal uncertainty while maintaining competitive point forecasting accuracy. The official implementation is available at: https://github.com/democracy04/PaP-NF