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

Taylor-Lagrange Neural Ordinary Differential Equations: Toward Fast Training and Evaluation of Neural ODEs