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

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.

Neural CDEs as Correctors for Learned Time Series Models

Learned time-series models, whether continuous- or discrete-time, are widely used to forecast the states of a dynamical system. Such models generate multi-step forecasts either directly, by predicting the full horizon at once, or iteratively, by feeding back their own predictions at each step. In both cases, the multi-step forecasts are prone to errors. To address this, we propose a Predictor-Corrector mechanism where the Predictor is any learned time-series model and the Corrector is a neural controlled differential equation. The Predictor forecasts, and the Corrector predicts the errors of the forecasts. Adding these errors to the forecasts improves forecast performance. The proposed Corrector works with irregularly sampled time series and continuous- and discrete-time Predictors. Additionally, we introduce two regularization strategies to improve the extrapolation performance of the Corrector with accelerated training. We evaluate our Corrector with diverse Predictors, e.g., neural ordinary differential equations, Contiformer, and DLinear, on synthetic, physics simulation, and real-world forecasting datasets. The experiments demonstrate that the Predictor-Corrector mechanism consistently improves the performance compared to Predictor alone.

HopCast: Calibration of Autoregressive Dynamics Models

Deep learning models are often trained to approximate dynamical systems that can be modeled using differential equations. These models are optimized to predict one step ahead and produce calibrated predictions if the predictive model can quantify uncertainty, such as deep ensembles. At inference time, multi-step predictions are generated via autoregression, which needs a sound uncertainty propagation method (e.g., Trajectory Sampling) to produce calibrated multi-step predictions. This paper introduces an approach named HopCast that uses the Modern Hopfield Network (MHN) to learn the residuals of a deterministic model that approximates the dynamical system. The MHN predicts the density of residuals based on a context vector at any timestep during autoregression. This approach produces calibrated multi-step predictions without uncertainty propagation and turns a deterministic model into a calibrated probabilistic model. This work is also the first to benchmark existing uncertainty propagation methods based on calibration errors with deep ensembles for multi-step predictions.

Towards Robust Car Following Dynamics Modeling via Blackbox Models: Methodology, Analysis, and Recommendations

The selection of the target variable is important while learning parameters of the classical car following models like GIPPS, IDM, etc. There is a vast body of literature on which target variable is optimal for classical car following models, but there is no study that empirically evaluates the selection of optimal target variables for black-box models, such as LSTM, etc. The black-box models, like LSTM and Gaussian Process (GP) are increasingly being used to model car following behavior without wise selection of target variables. The current work tests different target variables, like acceleration, velocity, and headway, for three black-box models, i.e., GP, LSTM, and Kernel Ridge Regression. These models have different objective functions and work in different vector spaces, e.g., GP works in function space, and LSTM works in parameter space. The experiments show that the optimal target variable recommendations for black-box models differ from classical car following models depending on the objective function and the vector space. It is worth mentioning that models and datasets used during evaluation are diverse in nature: the datasets contained both automated and human-driven vehicle trajectories; the black-box models belong to both parametric and non-parametric classes of models. This diversity is important during the analysis of variance, wherein we try to find the interaction between datasets, models, and target variables. It is shown that the models and target variables interact and recommended target variables don't depend on the dataset under consideration.