Abstract

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Neural Ordinary Differential Equations (NODEs) are a new class of models that transform data continuously through infinite-depth architectures. The continuous nature of NOD Es has made them particularly suitable for learning the dynamics of complex physical systems. While previous work has mostly been focused on first order ODEs, the dynamics of many systems, especially in classical physics, are governed by second order laws. In this work, we consider Second Order Neural ODEs (SONODEs). We show how the adjoint sensitivity method can be extended to SONOD Es and prove that the optimisation of a first order coupled ODE is equivalent and computationally more efficient. Furthermore, we extend the theoretical understanding of the broader class of Augmented NOD Es (ANODEs) by showing they can also learn higher order dynamics with a minimal number of augmented dimensions, but at the cost of interpretability. This indicates that the advantages of ANOD Es go beyond the extra space offered by the augmented dimensions, as originally thought. Finally, we compare SONOD Es and ANOD Es on synthetic and real dynamical systems and demonstrate that the inductive biases of the former generally result in faster training and better performance.

arXiv

NeurIPS pre-proceedings