  In this paper, we propose a flexible model for survival analysis using neural
networks along with scalable optimization algorithms. One key technical
challenge for directly applying maximum likelihood estimation (MLE) to censored
data is that evaluating the objective function and its gradients with respect
to model parameters requires the calculation of integrals. To address this
challenge, we recognize that the MLE for censored data can be viewed as a
differential-equation constrained optimization problem, a novel perspective.
Following this connection, we model the distribution of event time through an
ordinary differential equation and utilize efficient ODE solvers and adjoint
sensitivity analysis to numerically evaluate the likelihood and the gradients.
Using this approach, we are able to 1) provide a broad family of
continuous-time survival distributions without strong structural assumptions,
2) obtain powerful feature representations using neural networks, and 3) allow
efficient estimation of the model in large-scale applications using stochastic
gradient descent. Through both simulation studies and real-world data examples,
we demonstrate the effectiveness of the proposed method in comparison to
existing state-of-the-art deep learning survival analysis models. The
implementation of the proposed SODEN approach has been made publicly available
at https://github.com/jiaqima/SODEN.
