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Neural Ordinary Differential Equations replace the right-hand side of aconventional ODE with a neural net, which by virtue of the universalapproximation theorem, can be trained to the representation of any function.When we do not know the function itself, but have state trajectories (timeevolution) of the ODE system we can still train the neural net to learn therepresentation of the underlying but unknown ODE. However if the state of thesystem is incompletely known then the right-hand side of the ODE cannot becalculated. The derivatives to propagate the system are unavailable. We showthat a specially augmented Neural ODE can learn the system when givenincomplete state information. As a worked example we apply neural ODEs to theLotka-Voltera problem of 3 species, rabbits, wolves, and bears. We show thateven when the data for the bear time series is removed the remaining timeseries of the rabbits and wolves is sufficient to learn the dynamical systemdespite the missing the incomplete state information. This is surprising sincea conventional ODE system cannot output the correct derivatives without thefull state as the input. We implement augmented neural ODEs and differentialequation solvers in the julia programming language.

Augmenting Neural Differential Equations to Model Unknown Dynamical Systems with Incomplete State Information