Neural Ordinary Differential Equations (Neural ODEs) introduce a new family of deep neural network models that parameterize the derivative of the hidden state using a neural network. Instead of using discrete layers, these models use continuous-time dynamics defined by an ordinary differential equation (ODE). The output is computed using a black-box ODE solver, allowing for constant memory cost and adaptive computation. Neural ODEs can trade numerical precision for speed and are used in continuous-depth residual networks and continuous-time latent variable models. They also enable continuous normalizing flows, which can be trained by maximum likelihood without partitioning data dimensions. The paper demonstrates how to backpropagate through ODE solvers efficiently, enabling end-to-end training of ODEs within larger models. The approach is shown to be effective in supervised learning, density estimation, and generative modeling. The method allows for adaptive computation, memory efficiency, and scalable training. Continuous normalizing flows are derived, which simplify the computation of the change of variables formula and enable more expressive models. The paper also presents a generative latent function time-series model that can handle irregularly sampled data. The results show that Neural ODEs can achieve performance comparable to traditional models while offering advantages in memory and computational efficiency. The method is applicable to a wide range of tasks, including time-series modeling, density estimation, and generative modeling. The paper also discusses the limitations and scope of the approach, including challenges with mini-batching, uniqueness of solutions, and error tolerance settings. Overall, Neural ODEs provide a flexible and efficient framework for modeling complex data with continuous-time dynamics.Neural Ordinary Differential Equations (Neural ODEs) introduce a new family of deep neural network models that parameterize the derivative of the hidden state using a neural network. Instead of using discrete layers, these models use continuous-time dynamics defined by an ordinary differential equation (ODE). The output is computed using a black-box ODE solver, allowing for constant memory cost and adaptive computation. Neural ODEs can trade numerical precision for speed and are used in continuous-depth residual networks and continuous-time latent variable models. They also enable continuous normalizing flows, which can be trained by maximum likelihood without partitioning data dimensions. The paper demonstrates how to backpropagate through ODE solvers efficiently, enabling end-to-end training of ODEs within larger models. The approach is shown to be effective in supervised learning, density estimation, and generative modeling. The method allows for adaptive computation, memory efficiency, and scalable training. Continuous normalizing flows are derived, which simplify the computation of the change of variables formula and enable more expressive models. The paper also presents a generative latent function time-series model that can handle irregularly sampled data. The results show that Neural ODEs can achieve performance comparable to traditional models while offering advantages in memory and computational efficiency. The method is applicable to a wide range of tasks, including time-series modeling, density estimation, and generative modeling. The paper also discusses the limitations and scope of the approach, including challenges with mini-batching, uniqueness of solutions, and error tolerance settings. Overall, Neural ODEs provide a flexible and efficient framework for modeling complex data with continuous-time dynamics.