# EM-like Learning Chaotic Dynamics from Noisy and Partial Observations

arXiv: https://arxiv.org/abs/1903.10335 · HTML: https://mlsanity.com/paper/1903.10335/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Mar 2019.

## Built on by (most cited first, 11 of 11)

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)
- [Using Machine Learning to Anticipate Tipping Points and Extrapolate to Post-Tipping Dynamics of Non-Stationary Dynamical Systems](https://mlsanity.com/paper/2207.00521/) (2022)
- [Out-of-Domain Generalization in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2402.18377/) (2024)
- [Auto-differentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2107.07687/) (2021)
- [Reduced-Order Autodifferentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2301.11961/) (2023)
- [Incorporating Transformer and LSTM to Kalman Filter with EM algorithm for state estimation](https://mlsanity.com/paper/2105.00250/) (2021)
- [Leveraging Neural Koopman Operators to Learn Continuous Representations of Dynamical Systems from Scarce Data](https://mlsanity.com/paper/2303.06972/) (2023)
- [Long-time accuracy of ensemble Kalman filters for chaotic and machine-learned dynamical systems](https://mlsanity.com/paper/2412.14318/) (2024)
- [Deep Gaussian Markov Random Fields for Graph-Structured Dynamical Systems](https://mlsanity.com/paper/2306.08445/) (2023)

## Abstract (arXiv)

The identification of the governing equations of chaotic dynamical systems from data has recently emerged as a hot topic. While the seminal work by Brunton et al. reported proof-of-concepts for idealized observation setting for fully-observed systems, em i.e. large signal-to-noise ratios and high-frequency sampling of all system variables, we here address the learning of data-driven representations of chaotic dynamics for partially-observed systems, including significant noise patterns and possibly lower and irregular sampling setting. Instead of considering training losses based on short-term prediction error like state-of-the-art learning-based schemes, we adopt a Bayesian formulation and state this issue as a data assimilation problem with unknown model parameters. To solve for the joint inference of the hidden dynamics and of model parameters, we combine neural-network representations and state-of-the-art assimilation schemes. Using iterative Expectation-Maximization (EM)-like procedures, the key feature of the proposed inference schemes is the derivation of the posterior of the hidden dynamics. Using a neural-network-based Ordinary Differential Equation (ODE) representation of these dynamics, we investigate two strategies: their combination to Ensemble Kalman Smoothers and Long Short-Term Memory (LSTM)-based variational approximations of the posterior. Through numerical experiments on the Lorenz-63 system with different noise and time sampling settings, we demonstrate the ability of the proposed schemes to recover and reproduce the hidden chaotic dynamics, including their Lyapunov characteristic exponents, when classic machine learning approaches fail.