arXiv subjects

Muhammad Bilal Shahid

Publications and source records attributed to Muhammad Bilal Shahid.

4 recordsLinked to original sources

arXiv2026

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.

cs.LG↗

arXiv2026

Neural CDEs as Correctors for Learned Time Series Models

Learned time-series models, whether continuous or discrete, are widely used for forecasting the states of dynamical systems but suffer from error accumulation in multi-step forecasts. To address this issue, we propose a Predictor-Corrector framework in which the Predictor is a learned time-series model that generates multi-step forecasts and the Corrector is a neural controlled differential equation that corrects the forecast errors. The Corrector works with irregularly sampled time series and is compatible with both continuous- and discrete-time Predictors. We further introduce two regularization strategies that improve the Corrector's extrapolation performance and accelerate its training. We also provide theoretical guarantees on the stability and convergence of the proposed framework. Experiments on synthetic, physics-based, and real-world datasets show that the proposed framework consistently improves forecasting performance across diverse Predictors, including neural ordinary differential equations, ContiFormer, and DLinear, demonstrating its predictor-agnostic nature.

cs.LG↗

arXiv2025

HopCast: Calibration of Autoregressive Dynamics Models

Deep learning models are often trained to approximate dynamical systems that can be modeled using differential equations. Many of these models are optimized to predict one step ahead; such approaches produce calibrated one-step 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 to produce calibrated multi-step predictions. This work introduces an alternative Predictor-Corrector approach named \hop{} that uses Modern Hopfield Networks (MHN) to learn the errors of a deterministic Predictor that approximates the dynamical system. The Corrector predicts a set of errors for the Predictor's output based on a context state at any timestep during autoregression. The set of errors creates sharper and well-calibrated prediction intervals with higher predictive accuracy compared to baselines without uncertainty propagation. The calibration and prediction performances are evaluated across a set of dynamical systems. This work is also the first to benchmark existing uncertainty propagation methods based on calibration errors.

cs.LG↗

arXiv2024

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.

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