# SODEN: A Scalable Continuous-Time Survival Model through Ordinary Differential Equation Networks

arXiv: https://arxiv.org/abs/2008.08637 · HTML: https://mlsanity.com/paper/2008.08637/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Aug 2020.

## Built on by (most cited first, 26 of 26)

- [Survival Regression with Proper Scoring Rules and Monotonic Neural Networks](https://mlsanity.com/paper/2103.14755/) (2021)
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- [Copula-Based Deep Survival Models for Dependent Censoring](https://mlsanity.com/paper/2306.11912/) (2023)
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- [Tabular Foundation Models Can Do Survival Analysis](https://mlsanity.com/paper/2601.22259/) (2026)
- [Deep Copula-Based Survival Analysis for Dependent Censoring with Identifiability Guarantees](https://mlsanity.com/paper/2312.15566/) (2023)
- [Conditional Distribution Function Estimation Using Neural Networks for Censored and Uncensored Data](https://mlsanity.com/paper/2207.02384/) (2022)
- [Dynamical Survival Analysis with Controlled Latent States](https://mlsanity.com/paper/2401.17077/) (2024)
- [Neural Frailty Machine: Beyond proportional hazard assumption in neural survival regressions](https://mlsanity.com/paper/2303.10358/) (2023)
- [Survival In-Context: Amortized Bayesian Survival Analysis via Prior-Fitted Networks](https://mlsanity.com/paper/2603.29475/) (2026)
- [Censored Quantile Regression Neural Networks for Distribution-Free Survival Analysis](https://mlsanity.com/paper/2205.13496/) (2022)
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- [Maximum Likelihood Estimation of Flexible Survival Densities with Importance Sampling](https://mlsanity.com/paper/2311.01660/) (2023)
- [Probabilistic Temporal Prediction of Continuous Disease Trajectories and Treatment Effects Using Neural SDEs](https://mlsanity.com/paper/2406.12807/) (2024)
- [SeqRisk: Transformer-augmented latent variable model for robust survival prediction with longitudinal data](https://mlsanity.com/paper/2409.12709/) (2024)
- [Forecasting Disease Progression with Parallel Hyperplanes in Longitudinal Retinal OCT](https://mlsanity.com/paper/2409.20195/) (2024)
- [Value-Monotonicity Matters: A Concordance Loss for Deep Survival Prediction](https://mlsanity.com/paper/2607.16802/) (2026)
- [KL-divergence Based Deep Learning for Discrete Time Model](https://mlsanity.com/paper/2208.05100/) (2022)
- [Neural Network-Based Piecewise Survival Models](https://mlsanity.com/paper/2403.18664/) (2024)
- [Spectral Survival Analysis](https://mlsanity.com/paper/2505.22641/) (2025)
- [ICODEN: Ordinary Differential Equation Neural Networks for Interval-Censored Data](https://mlsanity.com/paper/2602.10303/) (2026)
- [A Scalable Nonparametric Continuous-Time Survival Model through Numerical Quadrature](https://mlsanity.com/paper/2605.16208/) (2026)
- [KAPLAN: Kolmogorov-Arnold Prognostic Learnable Activation Networks for Survival Analysis](https://mlsanity.com/paper/2605.23082/) (2026)

## Abstract (arXiv)

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.