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

Neural Ordinary Differential Equations replace the right-hand side of a\nconventional ODE with a neural net, which by virtue of the universal\napproximation theorem, can be trained to the representation of any function.\nWhen we do not know the function itself, but have state trajectories (time\nevolution) of the ODE system we can still train the neural net to learn the\nrepresentation of the underlying but unknown ODE. However if the state of the\nsystem is incompletely known then the right-hand side of the ODE cannot be\ncalculated. The derivatives to propagate the system are unavailable. We show\nthat a specially augmented Neural ODE can learn the system when given\nincomplete state information. As a worked example we apply neural ODEs to the\nLotka-Voltera problem of 3 species, rabbits, wolves, and bears. We show that\neven when the data for the bear time series is removed the remaining time\nseries of the rabbits and wolves is sufficient to learn the dynamical system\ndespite the missing the incomplete state information. This is surprising since\na conventional ODE system cannot output the correct derivatives without the\nfull state as the input. We implement augmented neural ODEs and differential\nequation solvers in the julia programming language.\n