# Survival Analysis via Ordinary Differential Equations

arXiv: https://arxiv.org/abs/2009.03449 · HTML: https://mlsanity.com/paper/2009.03449/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Sep 2020.

## Built on by (most cited first, 6 of 6)

- [Fairness in Survival Analysis with Distributionally Robust Optimization](https://mlsanity.com/paper/2409.10538/) (2024)
- [Survival Mixture Density Networks](https://mlsanity.com/paper/2208.10759/) (2022)
- [Conditional Distribution Function Estimation Using Neural Networks for Censored and Uncensored Data](https://mlsanity.com/paper/2207.02384/) (2022)
- [Hazard Gradient Penalty for Survival Analysis](https://mlsanity.com/paper/2205.13717/) (2022)
- [A Scalable Nonparametric Continuous-Time Survival Model through Numerical Quadrature](https://mlsanity.com/paper/2605.16208/) (2026)
- [Distributionally Robust Survival Models under Subpopulation Shift and Outlier Contamination](https://mlsanity.com/paper/2610.02868/) (2026)

## Abstract (arXiv)

This paper introduces an Ordinary Differential Equation (ODE) notion for survival analysis. The ODE notion not only provides a unified modeling framework, but more importantly, also enables the development of a widely applicable, scalable, and easy-to-implement procedure for estimation and inference. Specifically, the ODE modeling framework unifies many existing survival models, such as the proportional hazards model, the linear transformation model, the accelerated failure time model, and the time-varying coefficient model as special cases. The generality of the proposed framework serves as the foundation of a widely applicable estimation procedure. As an illustrative example, we develop a sieve maximum likelihood estimator for a general semi-parametric class of ODE models. In comparison to existing estimation methods, the proposed procedure has advantages in terms of computational scalability and numerical stability. Moreover, to address unique theoretical challenges induced by the ODE notion, we establish a new general sieve M-theorem for bundled parameters and show that the proposed sieve estimator is consistent and asymptotically normal, and achieves the semi-parametric efficiency bound. The finite sample performance of the proposed estimator is examined in simulation studies and a real-world data example.