Title: Manifold-Constrained Gaussian Process Inference for One-shot Learning of Unknown Ordinary Differential Equations
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: Gaussian Process, ODE learning

Abstract: Learning unknown ordinary differential equations (ODEs) from a single trajectory of scarce, noisy data is challenging, especially with partial observability. We introduce MAGI-X, an integration-free framework that couples a neural vector field with a Gaussian process prior over trajectories and enforces ODE consistency via a GP manifold constraint, thereby circumventing traditional numerical integration. Across canonical examples (FitzHugh--Nagumo, Lotka--Volterra, and Hes1), MAGI-X achieves better accuracy in both fitting and forecasting while requiring comparable or less computation time than benchmark methods NPODE and Neural ODE, with runtime scaling linearly in state dimension. MAGI-X offers a practical solution for \emph{partially observed} systems without bespoke priors or imputation heuristics, where existing methods struggle. The GP posterior further yields calibrated uncertainty, and experiments demonstrate robustness across initial conditions. We show practicality on seasonal flu data with rolling multi-week forecasts from noisy signals.
 These properties establish MAGI-X as a fast, accurate, and robust tool for data-driven discovery of nonlinear dynamics from a single noisy trajectory.