# Time and State Dependent Neural Delay Differential Equations

arXiv: https://arxiv.org/abs/2306.14545 · HTML: https://mlsanity.com/paper/2306.14545/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 37 references that are in the map.
- First posted on arXiv 26 Jun 2023.

## Builds on (most cited first, 13 of 13)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Learning Differential Equations that are Easy to Solve](https://mlsanity.com/paper/2007.04504/) (2020)
- [Neural Delay Differential Equations](https://mlsanity.com/paper/2102.10801/) (2021)
- [Adversarial Noise Layer: Regularize Neural Network By Adding Noise](https://mlsanity.com/paper/1805.08000/) (2018)
- [Neural Laplace: Learning diverse classes of differential equations in the Laplace domain](https://mlsanity.com/paper/2206.04843/) (2022)
- [Neural Ordinary Differential Equations for Semantic Segmentation of Individual Colon Glands](https://mlsanity.com/paper/1910.10470/) (2019)
- [Mining Causality from Continuous-time Dynamics Models: An Application to Tsunami Forecasting](https://mlsanity.com/paper/2210.04958/) (2022)

## Built on by (most cited first, 3 of 3)

- [Neural delay differential equations: learning non-Markovian closures for partially known dynamical systems](https://mlsanity.com/paper/2410.02843/) (2024)
- [Breaking Free: Decoupling Forced Systems with Laplace Neural Networks](https://mlsanity.com/paper/2503.13158/) (2025)
- [Learning non-Markovian Dynamical Systems with Signature-based Encoders](https://mlsanity.com/paper/2509.12022/) (2025)

## Abstract (arXiv)

Discontinuities and delayed terms are encountered in the governing equations of a large class of problems ranging from physics and engineering to medicine and economics. These systems cannot be properly modelled and simulated with standard Ordinary Differential Equations (ODE), or data-driven approximations such as Neural Ordinary Differential Equations (NODE). To circumvent this issue, latent variables are typically introduced to solve the dynamics of the system in a higher dimensional space and obtain the solution as a projection to the original space. However, this solution lacks physical interpretability. In contrast, Delay Differential Equations (DDEs), and their data-driven approximated counterparts, naturally appear as good candidates to characterize such systems. In this work we revisit the recently proposed Neural DDE by introducing Neural State-Dependent DDE (SDDDE), a general and flexible framework that can model multiple and state- and time-dependent delays. We show that our method is competitive and outperforms other continuous-class models on a wide variety of delayed dynamical systems. Code is available at the repository hrefhttps://github.com/thibmonsel/Time-and-State-Dependent-Neural-Delay-Differential-Equationshere.