# Double projection for reconstructing dynamical systems: between stochastic and deterministic regimes

arXiv: https://arxiv.org/abs/2510.01089 · HTML: https://mlsanity.com/paper/2510.01089/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 57 references that are in the map.
- First posted on arXiv 1 Oct 2025.

## Builds on (most cited first, 28 of 28)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [Learning Latent Dynamics for Planning from Pixels](https://mlsanity.com/paper/1811.04551/) (2018)
- [Scheduled Sampling for Sequence Prediction with Recurrent Neural Networks](https://mlsanity.com/paper/1506.03099/) (2015)
- [Generating Sequences With Recurrent Neural Networks](https://mlsanity.com/paper/1308.0850/) (2013)
- [A Recurrent Latent Variable Model for Sequential Data](https://mlsanity.com/paper/1506.02216/) (2015)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [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)
- [Dynamical Variational Autoencoders: A Comprehensive Review](https://mlsanity.com/paper/2008.12595/) (2020)
- [Variational Sequential Monte Carlo](https://mlsanity.com/paper/1705.11140/) (2017)
- [Chaos as an interpretable benchmark for forecasting and data-driven modelling](https://mlsanity.com/paper/2110.05266/) (2021)
- [On the difficulty of learning chaotic dynamics with RNNs](https://mlsanity.com/paper/2110.07238/) (2021)
- [Probabilistic Recurrent State-Space Models](https://mlsanity.com/paper/1801.10395/) (2018)
- [Learning Stochastic Recurrent Networks](https://mlsanity.com/paper/1411.7610/) (2014)
- [Generalized Teacher Forcing for Learning Chaotic Dynamics](https://mlsanity.com/paper/2306.04406/) (2023)
- [Identifying nonlinear dynamical systems via generative recurrent neural networks with applications to fMRI](https://mlsanity.com/paper/1902.07186/) (2019)
- [Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers](https://mlsanity.com/paper/2008.09768/) (2020)
- [Constructing Neural Network-Based Models for Simulating Dynamical Systems](https://mlsanity.com/paper/2111.01495/) (2021)
- [Reconstructing Nonlinear Dynamical Systems from Multi-Modal Time Series](https://mlsanity.com/paper/2111.02922/) (2021)
- [Inferring stochastic low-rank recurrent neural networks from neural data](https://mlsanity.com/paper/2406.16749/) (2024)
- [Integrating Multimodal Data for Joint Generative Modeling of Complex Dynamics](https://mlsanity.com/paper/2212.07892/) (2022)
- [Learning Awareness Models](https://mlsanity.com/paper/1804.06318/) (2018)
- [Reservoir Computing with Diverse Timescales for Prediction of Multiscale Dynamics](https://mlsanity.com/paper/2108.09446/) (2021)
- [Sparse Identification of Slow Timescale Dynamics](https://mlsanity.com/paper/2006.00940/) (2020)
- [Stochastic embeddings of dynamical phenomena through variational autoencoders](https://mlsanity.com/paper/2010.06265/) (2020)
- [RefreshNet: Learning Multiscale Dynamics through Hierarchical Refreshing](https://mlsanity.com/paper/2401.13282/) (2024)

## Built on by (most cited first, 1 of 1)

- [The Dynamic-Probabilistic Consistency Gap in Chaotic Surrogate Modeling](https://mlsanity.com/paper/2605.31547/) (2026)

## Abstract (arXiv)

Learning stochastic models of dynamical systems from observed data is of interest in many scientific fields. Here, we propose a new method for this task within the family of dynamical variational autoencoders. The proposed double projection method estimates both the system state trajectories and the noise time series from data. This approach naturally allows us to perform multi-step system evolution and to learn models with a comparatively low-dimensional state space. We evaluate the performance of the method on six benchmark problems, including both simulated and experimental data. We further illustrate the effects of the teacher forcing interval of the multi-step scheme on the nature of the internal dynamics and compare the resulting behavior to that of deterministic models of equivalent architecture.