# Efficient Transformed Gaussian Process State-Space Models for Non-Stationary High-Dimensional Dynamical Systems

arXiv: https://arxiv.org/abs/2503.18309 · HTML: https://mlsanity.com/paper/2503.18309/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 65 references that are in the map.
- First posted on arXiv 24 Mar 2025.

## Builds on (most cited first, 30 of 35)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [A Brief Survey of Deep Reinforcement Learning](https://mlsanity.com/paper/1708.05866/) (2017)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Structured Inference Networks for Nonlinear State Space Models](https://mlsanity.com/paper/1609.09869/) (2016)
- [KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics](https://mlsanity.com/paper/2107.10043/) (2021)
- [Dynamical Variational Autoencoders: A Comprehensive Review](https://mlsanity.com/paper/2008.12595/) (2020)
- [Gaussian Processes for Data-Efficient Learning in Robotics and Control](https://mlsanity.com/paper/1502.02860/) (2015)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Probabilistic Recurrent State-Space Models](https://mlsanity.com/paper/1801.10395/) (2018)
- [Convergence of Sparse Variational Inference in Gaussian Processes Regression](https://mlsanity.com/paper/2008.00323/) (2020)
- [Coupling techniques for nonlinear ensemble filtering](https://mlsanity.com/paper/1907.00389/) (2019)
- [Variational Gaussian Process State-Space Models](https://mlsanity.com/paper/1406.4905/) (2014)
- [Variational Implicit Processes](https://mlsanity.com/paper/1806.02390/) (2018)
- [Identification of Gaussian Process State Space Models](https://mlsanity.com/paper/1705.10888/) (2017)
- [Transforming Gaussian Processes With Normalizing Flows](https://mlsanity.com/paper/2011.01596/) (2020)
- [Auto-differentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2107.07687/) (2021)
- [Bayesian Inference and Learning in Gaussian Process State-Space Models with Particle MCMC](https://mlsanity.com/paper/1306.2861/) (2013)
- [Overcoming Mean-Field Approximations in Recurrent Gaussian Process Models](https://mlsanity.com/paper/1906.05828/) (2019)
- [DANSE: Data-driven Non-linear State Estimation of Model-free Process in Unsupervised Learning Setup](https://mlsanity.com/paper/2306.03897/) (2023)
- [Computationally Efficient Bayesian Learning of Gaussian Process State Space Models](https://mlsanity.com/paper/1506.02267/) (2015)
- [A flexible state space model for learning nonlinear dynamical systems](https://mlsanity.com/paper/1603.05486/) (2016)
- [Compositionally-Warped Gaussian Processes](https://mlsanity.com/paper/1906.09665/) (2019)
- [Online Joint State Inference and Learning of Partially Unknown State-Space Models](https://mlsanity.com/paper/2102.07418/) (2021)
- [Efficient Transformed Gaussian Processes for Non-Stationary Dependent Multi-class Classification](https://mlsanity.com/paper/2205.15008/) (2022)
- [Free-Form Variational Inference for Gaussian Process State-Space Models](https://mlsanity.com/paper/2302.09921/) (2023)
- [Output-Dependent Gaussian Process State-Space Model](https://mlsanity.com/paper/2212.07608/) (2022)
- [Towards Flexibility and Interpretability of Gaussian Process State-Space Model](https://mlsanity.com/paper/2301.08843/) (2023)

## Built on by (most cited first, 2 of 2)

- [pDANSE: Particle-based Data-driven Nonlinear State Estimation from Nonlinear Measurements](https://mlsanity.com/paper/2510.27503/) (2025)
- [Simulation-Free Learning of GP-SDEs from Irregular Observations](https://mlsanity.com/paper/2609.33112/) (2026)

## Abstract (arXiv)

Gaussian process state-space models (GPSSMs) offer a principled framework for learning and inference in nonlinear dynamical systems with uncertainty quantification. However, existing GPSSMs are limited by the use of multiple independent stationary Gaussian processes (GPs), leading to prohibitive computational and parametric complexity in high-dimensional settings and restricted modeling capacity for non-stationary dynamics. To address these challenges, we propose an efficient transformed Gaussian process state-space model (ETGPSSM) for scalable and flexible modeling of high-dimensional, non-stationary dynamical systems. Specifically, our ETGPSSM integrates a single shared GP with input-dependent normalizing flows, yielding an expressive non-stationary implicit process prior that can capture complex transition dynamics while significantly reducing model complexity. For the inference of the implicit process, we develop a variational inference algorithm that jointly approximates the posterior over the underlying GP and the neural network parameters defining the normalizing flows. To avoid explicit variational parameterization of the latent states, we further incorporate the ensemble Kalman filter (EnKF) into the variational framework, enabling accurate and efficient state estimation. Extensive empirical evaluations on synthetic and real-world datasets demonstrate the superior performance of our ETGPSSM in system dynamics learning, high-dimensional state estimation, and time-series forecasting, outperforming existing GPSSMs and neural network-based SSMs in terms of computational efficiency and accuracy.