# Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations

arXiv: https://arxiv.org/abs/2108.03561 · HTML: https://mlsanity.com/paper/2108.03561/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 67 references that are in the map.
- First posted on arXiv 8 Aug 2021.

## Builds on (most cited first, 25 of 25)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Next Generation Reservoir Computing](https://mlsanity.com/paper/2106.07688/) (2021)
- [Deep learning to represent sub-grid processes in climate models](https://mlsanity.com/paper/1806.04731/) (2018)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020)
- [Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model](https://mlsanity.com/paper/2001.01520/) (2020)
- [Hybrid Forecasting of Chaotic Processes: Using Machine Learning in Conjunction with a Knowledge-Based Model](https://mlsanity.com/paper/1803.04779/) (2018)
- [Earth System Modeling 2.0: A Blueprint for Models That Learn From Observations and Targeted High-Resolution Simulations](https://mlsanity.com/paper/1709.00037/) (2017)
- [Machine Learning for Stochastic Parameterization: Generative Adversarial Networks in the Lorenz '96 Model](https://mlsanity.com/paper/1909.04711/) (2019)
- [Combining data assimilation and machine learning to infer unresolved scale parametrisation](https://mlsanity.com/paper/2009.04318/) (2020)
- [Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization](https://mlsanity.com/paper/2001.06270/) (2020)
- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [Machine Learning for Prediction with Missing Dynamics](https://mlsanity.com/paper/1910.05861/) (2019)
- [Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation](https://mlsanity.com/paper/2007.07383/) (2020)
- [Combining Machine Learning with Knowledge-Based Modeling for Scalable Forecasting and Subgrid-Scale Closure of Large, Complex, Spatiotemporal Systems](https://mlsanity.com/paper/2002.05514/) (2020)
- [Using Data Assimilation to Train a Hybrid Forecast System that Combines Machine-Learning and Knowledge-Based Components](https://mlsanity.com/paper/2102.07819/) (2021)
- [Online learning of both state and dynamics using ensemble Kalman filters](https://mlsanity.com/paper/2006.03859/) (2020)
- [Machine Learning, Deepest Learning: Statistical Data Assimilation Problems](https://mlsanity.com/paper/1707.01415/) (2017)
- [Reservoir Computing as a Tool for Climate Predictability Studies](https://mlsanity.com/paper/2103.06206/) (2021)
- [Kernel-based Prediction of Non-Markovian Time Series](https://mlsanity.com/paper/2007.04286/) (2020)
- [Leveraging Bayesian Analysis To Improve Accuracy of Approximate Models](https://mlsanity.com/paper/1905.08227/) (2019)
- [Forecasting Using Reservoir Computing: The Role of Generalized Synchronization](https://mlsanity.com/paper/2102.08930/) (2021)
- [Combining Ensemble Kalman Filter and Reservoir Computing to predict spatio-temporal chaotic systems from imperfect observations and models](https://mlsanity.com/paper/2006.14276/) (2020)

## Built on by (most cited first, 21 of 21)

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [Catch-22s of reservoir computing](https://mlsanity.com/paper/2210.10211/) (2022)
- [LD-EnSF: Synergizing Latent Dynamics with Ensemble Score Filters for Fast Data Assimilation with Sparse Observations](https://mlsanity.com/paper/2411.19305/) (2024)
- [Deep learning delay coordinate dynamics for chaotic attractors from partial observable data](https://mlsanity.com/paper/2211.11061/) (2022)
- [Learning dynamical systems with hit-and-run random feature maps](https://mlsanity.com/paper/2501.06661/) (2025)
- [CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilation](https://mlsanity.com/paper/2410.20072/) (2024)
- [Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case](https://mlsanity.com/paper/2110.00546/) (2021)
- [CGNSDE: Conditional Gaussian Neural Stochastic Differential Equation for Modeling Complex Systems and Data Assimilation](https://mlsanity.com/paper/2404.06749/) (2024)
- [How more data can hurt: Instability and regularization in next-generation reservoir computing](https://mlsanity.com/paper/2407.08641/) (2024)
- [Stable generative modeling using Schr"odinger bridges](https://mlsanity.com/paper/2401.04372/) (2024)
- [iPINNER: An Iterative Physics-Informed Neural Network with Ensemble Kalman Filter](https://mlsanity.com/paper/2506.00731/) (2025)
- [On the choice of the non-trainable internal weights in random feature maps](https://mlsanity.com/paper/2408.03626/) (2024)
- [RL-DAUNCE: Reinforcement Learning-Driven Data Assimilation with Uncertainty-Aware Constrained Ensembles](https://mlsanity.com/paper/2505.05452/) (2025)
- [Topological Approach for Data Assimilation](https://mlsanity.com/paper/2411.18627/) (2024)
- [Long-time accuracy of ensemble Kalman filters for chaotic and machine-learned dynamical systems](https://mlsanity.com/paper/2412.14318/) (2024)
- [Auto-differentiable data assimilation: Co-learning of states, dynamics, and filtering algorithms](https://mlsanity.com/paper/2603.20891/) (2026)
- [Learning dynamical systems from noisy data with Weak-form Kernel Ridge Regression](https://mlsanity.com/paper/2607.00257/) (2026)
- [Autoregressive with Slack Time Series Model for Forecasting a Partially-Observed Dynamical Time Series](https://mlsanity.com/paper/2306.16593/) (2023)
- [Forecasting the Forced van der Pol Equation with Frequent Phase Shifts Using Reservoir Computing](https://mlsanity.com/paper/2404.14651/) (2024)
- [Inference of Unknown Dynamical Components Using Next Generation Reservoir Computing: From Chaotic Systems to Climate Data](https://mlsanity.com/paper/2609.24754/) (2026)
- [LD-EnFF: Latent-Dynamics Ensemble Flow Filtering for Data Assimilation with Sparse Observations](https://mlsanity.com/paper/2610.04034/) (2026)

## Abstract (arXiv)

We present a supervised learning method to learn the propagator map of a dynamical system from partial and noisy observations. In our computationally cheap and easy-to-implement framework a neural network consisting of random feature maps is trained sequentially by incoming observations within a data assimilation procedure. By employing Takens' embedding theorem, the network is trained on delay coordinates. We show that the combination of random feature maps and data assimilation, called RAFDA, outperforms standard random feature maps for which the dynamics is learned using batch data.