  We propose Characteristic-Neural Ordinary Differential Equations (C-NODEs), a
framework for extending Neural Ordinary Differential Equations (NODEs) beyond
ODEs. While NODEs model the evolution of a latent variables as the solution to
an ODE, C-NODE models the evolution of the latent variables as the solution of
a family of first-order quasi-linear partial differential equations (PDEs)
along curves on which the PDEs reduce to ODEs, referred to as characteristic
curves. This in turn allows the application of the standard frameworks for
solving ODEs, namely the adjoint method. Learning optimal characteristic curves
for given tasks improves the performance and computational efficiency, compared
to state of the art NODE models. We prove that the C-NODE framework extends the
classical NODE on classification tasks by demonstrating explicit C-NODE
representable functions not expressible by NODEs. Additionally, we present
C-NODE-based continuous normalizing flows, which describe the density evolution
of latent variables along multiple dimensions. Empirical results demonstrate
the improvements provided by the proposed method for classification and density
estimation on CIFAR-10, SVHN, and MNIST datasets under a similar computational
budget as the existing NODE methods. The results also provide empirical
evidence that the learned curves improve the efficiency of the system through a
lower number of parameters and function evaluations compared with baselines.
