Differential equations in general and neural ODEs in particular are an essential technique in continuous-time system identification. While many deterministic learning algorithms have been designed based on numerical integration via the adjoint method, many downstream tasks such as active learning, exploration in reinforcement learning, robust control, or filtering require accurate estimates of predictive uncertainties. In this work, we propose a novel approach towards estimating epistemically uncertain neural ODEs, avoiding the numerical integration bottleneck. Instead of modeling uncertainty in the ODE parameters, we directly model uncertainties in the state space. Our algorithm – distributional gradient matching (DGM) – jointly trains a smoother and a dynamics model and matches their gradients via minimizing a Wasserstein loss. Our experiments show that, compared to traditional approximate inference methods based on numerical integration, our approach is faster to train, faster at predicting previously unseen trajectories, and in the context of neural ODEs, significantly more accurate.

@inproceedings{treven2021distributional,
	author = {Treven, Lenart and Wenk, Philippe and D{\"o}rfler, Florian and Krause, Andreas},
	booktitle = {Proc. Neural Information Processing Systems (NeurIPS)},
	month = {December},
	title = {Distributional Gradient Matching for Learning Uncertain Neural Dynamics Models},
	year = {2021}}