On Numerical Integration in Neural Ordinary Differential Equations
This addresses a fundamental issue in Neural ODEs for researchers and practitioners, providing theoretical insights into numerical integration effects, though it is incremental as it builds on existing work.
The paper tackles the challenge of understanding how numerical integration affects Neural ODE performance by proposing inverse modified differential equations (IMDE), showing that training approximates IMDE rather than the true ODE, with theoretical bounds on discrepancies and failure to learn conservation laws using non-symplectic methods.
The combination of ordinary differential equations and neural networks, i.e., neural ordinary differential equations (Neural ODE), has been widely studied from various angles. However, deciphering the numerical integration in Neural ODE is still an open challenge, as many researches demonstrated that numerical integration significantly affects the performance of the model. In this paper, we propose the inverse modified differential equations (IMDE) to clarify the influence of numerical integration on training Neural ODE models. IMDE is determined by the learning task and the employed ODE solver. It is shown that training a Neural ODE model actually returns a close approximation of the IMDE, rather than the true ODE. With the help of IMDE, we deduce that (i) the discrepancy between the learned model and the true ODE is bounded by the sum of discretization error and learning loss; (ii) Neural ODE using non-symplectic numerical integration fail to learn conservation laws theoretically. Several experiments are performed to numerically verify our theoretical analysis.