# A flexible state space model for learning nonlinear dynamical systems

arXiv: https://arxiv.org/abs/1603.05486 · HTML: https://mlsanity.com/paper/1603.05486/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Mar 2016.

## Built on by (most cited first, 18 of 18)

- [Sparse Bayesian Deep Learning for Dynamic System Identification](https://mlsanity.com/paper/2107.12910/) (2021)
- [Transport with Support: Data-Conditional Diffusion Bridges](https://mlsanity.com/paper/2301.13636/) (2023)
- [Dynamical System Identification, Model Selection and Model Uncertainty Quantification by Bayesian Inference](https://mlsanity.com/paper/2401.16943/) (2024)
- [Continuous-time identification of dynamic state-space models by deep subspace encoding](https://mlsanity.com/paper/2204.09405/) (2022)
- [Sequential Estimation of Gaussian Process-based Deep State-Space Models](https://mlsanity.com/paper/2301.12528/) (2023)
- [Ensemble Kalman Filtering Meets Gaussian Process SSM for Non-Mean-Field and Online Inference](https://mlsanity.com/paper/2312.05910/) (2023)
- [Learning Nonparametric Volterra Kernels with Gaussian Processes](https://mlsanity.com/paper/2106.05582/) (2021)
- [On Foundation Models for Dynamical Systems from Purely Synthetic Data](https://mlsanity.com/paper/2412.00395/) (2024)
- [Learning-Based Optimal Control with Performance Guarantees for Unknown Systems with Latent States](https://mlsanity.com/paper/2303.17963/) (2023)
- [Out of Distribution Detection via Domain-Informed Gaussian Process State Space Models](https://mlsanity.com/paper/2309.06655/) (2023)
- [Efficient Transformed Gaussian Process State-Space Models for Non-Stationary High-Dimensional Dynamical Systems](https://mlsanity.com/paper/2503.18309/) (2025)
- [Bayesian Inference and Learning in Nonlinear Dynamical Systems: A Framework for Incorporating Explicit and Implicit Prior Knowledge](https://mlsanity.com/paper/2508.15345/) (2025)
- [Learning Dynamics from Input-Output Data with Hamiltonian Gaussian Processes](https://mlsanity.com/paper/2511.05330/) (2025)
- [Distributed Identification of Contracting and/or Monotone Network Dynamics](https://mlsanity.com/paper/2107.14309/) (2021)
- [Unsupervised learning of observation functions in state-space models by nonparametric moment methods](https://mlsanity.com/paper/2207.05242/) (2022)
- [Friction-Adaptive Stochastic Nonlinear Model Predictive Control for Autonomous Vehicles](https://mlsanity.com/paper/2305.03798/) (2023)
- [State Derivative Normalization for Continuous-Time Deep Neural Networks](https://mlsanity.com/paper/2401.02902/) (2024)
- [Recursive Gaussian Process State Space Model](https://mlsanity.com/paper/2411.14679/) (2024)

## Abstract (arXiv)

We consider a nonlinear state-space model with the state transition and observation functions expressed as basis function expansions. The coefficients in the basis function expansions are learned from data. Using a connection to Gaussian processes we also develop priors on the coefficients, for tuning the model flexibility and to prevent overfitting to data, akin to a Gaussian process state-space model. The priors can alternatively be seen as a regularization, and helps the model in generalizing the data without sacrificing the richness offered by the basis function expansion. To learn the coefficients and other unknown parameters efficiently, we tailor an algorithm using state-of-the-art sequential Monte Carlo methods, which comes with theoretical guarantees on the learning. Our approach indicates promising results when evaluated on a classical benchmark as well as real data.