Neural ordinary differential equations (NODEs) -- parametrizations of differential equations using neural networks -- have shown tremendous promise in learning models of unknown continuous-time dynamical systems from data. However, every forward evaluation of a NODE requires numerical integration of the neural network used to capture the system dynamics, making their training prohibitively expensive. Existing works rely on off-the-shelf adaptive step-size numerical integration schemes, which often require an excessive number of evaluations of the underlying dynamics network to obtain sufficient accuracy for training. By contrast, we accelerate the evaluation and the training of NODEs by proposing a data-driven approach to their numerical integration. The proposed Taylor-Lagrange NODEs (TL-NODEs) use a fixed-order Taylor expansion for numerical integration, while also learning to estimate the expansion's approximation error. As a result, the proposed approach achieves the same accuracy as adaptive step-size schemes while employing only low-order Taylor expansions, thus greatly reducing the computational cost necessary to integrate the NODE. A suite of numerical experiments, including modeling dynamical systems, image classification, and density estimation, demonstrate that TL-NODEs can be trained more than an order of magnitude faster than state-of-the-art approaches, without any loss in performance.

提出了一种数据驱动的积分方法，称为Taylor-Lagrange NODEs (TL-NODEs)，它使用定阶Taylor扩展进行数值积分，同时学习估计扩展的近似误差，从而在保持准确性的前提下，仅使用低阶Taylor扩展，大大降低了计算成本。一系列数值实验表明，TL-NODEs比现有方法快一个数量级以上，性能也不会降低。

Taylor-Lagrange神经常微分方程: 迈向快速训练和评估神经ODE