# Neural Continuous-Discrete State Space Models for Irregularly-Sampled Time Series

arXiv: https://arxiv.org/abs/2301.11308 · HTML: https://mlsanity.com/paper/2301.11308/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 48 references that are in the map.
- First posted on arXiv 26 Jan 2023.

## Builds on (most cited first, 22 of 22)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [HiPPO: Recurrent Memory with Optimal Polynomial Projections](https://mlsanity.com/paper/2008.07669/) (2020)
- [Recurrent Neural Networks for Multivariate Time Series with Missing Values](https://mlsanity.com/paper/1606.01865/) (2016)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [A Recurrent Latent Variable Model for Sequential Data](https://mlsanity.com/paper/1506.02216/) (2015)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [Structured Inference Networks for Nonlinear State Space Models](https://mlsanity.com/paper/1609.09869/) (2016)
- [Deep Kalman Filters](https://mlsanity.com/paper/1511.05121/) (2015)
- [Deep Variational Bayes Filters: Unsupervised Learning of State Space Models from Raw Data](https://mlsanity.com/paper/1605.06432/) (2016)
- [A Disentangled Recognition and Nonlinear Dynamics Model for Unsupervised Learning](https://mlsanity.com/paper/1710.05741/) (2017)
- [Modeling Irregular Time Series with Continuous Recurrent Units](https://mlsanity.com/paper/2111.11344/) (2021)
- [Probabilistic Recurrent State-Space Models](https://mlsanity.com/paper/1801.10395/) (2018)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Neural Jump Ordinary Differential Equations: Consistent Continuous-Time Prediction and Filtering](https://mlsanity.com/paper/2006.04727/) (2020)
- [Collapsed Amortized Variational Inference for Switching Nonlinear Dynamical Systems](https://mlsanity.com/paper/1910.09588/) (2019)
- [Deep Explicit Duration Switching Models for Time Series](https://mlsanity.com/paper/2110.13878/) (2021)
- [Scalable Inference in SDEs by Direct Matching of the Fokker-Planck-Kolmogorov Equation](https://mlsanity.com/paper/2110.15739/) (2021)
- [Intermittent Demand Forecasting with Deep Renewal Processes](https://mlsanity.com/paper/1911.10416/) (2019)
- [Learning Continuous-Time Dynamics by Stochastic Differential Networks](https://mlsanity.com/paper/2006.06145/) (2020)

## Built on by (most cited first, 29 of 29)

- [ContiFormer: Continuous-Time Transformer for Irregular Time Series Modeling](https://mlsanity.com/paper/2402.10635/) (2024)
- [Rethinking Irregular Time Series Forecasting: A Simple yet Effective Baseline](https://mlsanity.com/paper/2505.11250/) (2025)
- [Stable Neural Stochastic Differential Equations in Analyzing Irregular Time Series Data](https://mlsanity.com/paper/2402.14989/) (2024)
- [Scalable Event-by-event Processing of Neuromorphic Sensory Signals With Deep State-Space Models](https://mlsanity.com/paper/2404.18508/) (2024)
- [Universal Time-Series Representation Learning: A Survey](https://mlsanity.com/paper/2401.03717/) (2024)
- [Parsimony or Capability? Decomposition Delivers Both in Long-term Time Series Forecasting](https://mlsanity.com/paper/2401.11929/) (2024)
- [Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series](https://mlsanity.com/paper/2410.05602/) (2024)
- [LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation](https://mlsanity.com/paper/2506.17039/) (2025)
- [ASTGI: Adaptive Spatio-Temporal Graph Interactions for Irregular Multivariate Time Series Forecasting](https://mlsanity.com/paper/2509.23313/) (2025)
- [Temporal Test-Time Adaptation with State-Space Models](https://mlsanity.com/paper/2407.12492/) (2024)
- [Neural Stochastic Flows: Solver-Free Modelling and Inference for SDE Solutions](https://mlsanity.com/paper/2510.25769/) (2025)
- [Latent Space Energy-based Neural ODEs](https://mlsanity.com/paper/2409.03845/) (2024)
- [Neural Predictive Control to Coordinate Discrete- and Continuous-Time Models for Time-Series Analysis with Control-Theoretical Improvements](https://mlsanity.com/paper/2508.01833/) (2025)
- [Technologies on Effectiveness and Efficiency: A Survey of State Spaces Models](https://mlsanity.com/paper/2503.11224/) (2025)
- [Modeling Irregular Astronomical Time Series with Neural Stochastic Delay Differential Equations](https://mlsanity.com/paper/2508.17521/) (2025)
- [Neural Stochastic Processes for Satellite Precipitation Refinement](https://mlsanity.com/paper/2604.10414/) (2026)
- [MILM: Large Language Models for Multimodal Irregular Time Series with Informative Sampling](https://mlsanity.com/paper/2605.13711/) (2026)
- [Preparing for Black Swans: The Antifragility Imperative for Machine Learning](https://mlsanity.com/paper/2405.11397/) (2024)
- [Stochastic Optimal Control for Continuous-Time fMRI Representation Learning](https://mlsanity.com/paper/2502.04892/) (2025)
- [Auto-Configured Networks for Multi-Scale Multi-Output Time-Series Forecasting](https://mlsanity.com/paper/2604.07610/) (2026)
- [Analyzing the Impact of Climate Change With Major Emphasis on Pollution: A Comparative Study of ML and Statistical Models in Time Series Data](https://mlsanity.com/paper/2405.15835/) (2024)
- [Marginalization Consistent Probabilistic Forecasting of Irregular Time Series via Mixture of Separable flows](https://mlsanity.com/paper/2406.07246/) (2024)
- [Neural Stochastic Differential Equations on Compact State Spaces: Theory, Methods, and Application to Suicide Risk Modeling](https://mlsanity.com/paper/2508.17090/) (2025)
- [Deep ZakaiJ: Structured Filtering for Jump-Diffusion Time Series Forecasting](https://mlsanity.com/paper/2605.24548/) (2026)
- [Physically-Constrained Mamba-SDE for Remaining Useful Life Prediction under Irregular Observations](https://mlsanity.com/paper/2606.01894/) (2026)
- [Deep Spectral Learning of Embedded Latent Transfer Operators for Stochastic Dynamical Systems](https://mlsanity.com/paper/2606.14079/) (2026)
- [Dynestyx: A Probabilistic Programming Library for Dynamical Systems](https://mlsanity.com/paper/2606.16985/) (2026)
- [Continuous-Time Machine Learning: A Unified Mathematical Perspective](https://mlsanity.com/paper/2609.16710/) (2026)
- [Scan-Parallel Variational Inference for event-driven Latent Jump Models](https://mlsanity.com/paper/2610.12986/) (2026)

## Abstract (arXiv)

Learning accurate predictive models of real-world dynamic phenomena (e.g., climate, biological) remains a challenging task. One key issue is that the data generated by both natural and artificial processes often comprise time series that are irregularly sampled and/or contain missing observations. In this work, we propose the Neural Continuous-Discrete State Space Model (NCDSSM) for continuous-time modeling of time series through discrete-time observations. NCDSSM employs auxiliary variables to disentangle recognition from dynamics, thus requiring amortized inference only for the auxiliary variables. Leveraging techniques from continuous-discrete filtering theory, we demonstrate how to perform accurate Bayesian inference for the dynamic states. We propose three flexible parameterizations of the latent dynamics and an efficient training objective that marginalizes the dynamic states during inference. Empirical results on multiple benchmark datasets across various domains show improved imputation and forecasting performance of NCDSSM over existing models.