  Neural ordinary differential equations (NODE) have been proposed as a
continuous depth generalization to popular deep learning models such as
Residual networks (ResNets). They provide parameter efficiency and automate the
model selection process in deep learning models to some extent. However, they
lack the much-required uncertainty modelling and robustness capabilities which
are crucial for their use in several real-world applications such as autonomous
driving and healthcare. We propose a novel and unique approach to model
uncertainty in NODE by considering a distribution over the end-time $T$ of the
ODE solver. The proposed approach, latent time NODE (LT-NODE), treats $T$ as a
latent variable and apply Bayesian learning to obtain a posterior distribution
over $T$ from the data. In particular, we use variational inference to learn an
approximate posterior and the model parameters. Prediction is done by
considering the NODE representations from different samples of the posterior
and can be done efficiently using a single forward pass. As $T$ implicitly
defines the depth of a NODE, posterior distribution over $T$ would also help in
model selection in NODE. We also propose, adaptive latent time NODE (ALT-NODE),
which allow each data point to have a distinct posterior distribution over
end-times. ALT-NODE uses amortized variational inference to learn an
approximate posterior using inference networks. We demonstrate the
effectiveness of the proposed approaches in modelling uncertainty and
robustness through experiments on synthetic and several real-world image
classification data.
