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Differential equations in general and neural ODEs in particular are anessential technique in continuous-time system identification. While manydeterministic learning algorithms have been designed based on numericalintegration via the adjoint method, many downstream tasks such as activelearning, exploration in reinforcement learning, robust control, or filteringrequire accurate estimates of predictive uncertainties. In this work, wepropose a novel approach towards estimating epistemically uncertain neuralODEs, avoiding the numerical integration bottleneck. Instead of modelinguncertainty in the ODE parameters, we directly model uncertainties in the statespace. Our algorithm - distributional gradient matching (DGM) - jointly trainsa smoother and a dynamics model and matches their gradients via minimizing aWasserstein loss. Our experiments show that, compared to traditionalapproximate inference methods based on numerical integration, our approach isfaster to train, faster at predicting previously unseen trajectories, and inthe context of neural ODEs, significantly more accurate.

Distributional Gradient Matching for Learning Uncertain Neural Dynamics Models