# Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation

arXiv: https://arxiv.org/abs/2007.07383 · HTML: https://mlsanity.com/paper/2007.07383/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jul 2020.

## Built on by (most cited first, 27 of 27)

- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [Auto-differentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2107.07687/) (2021)
- [Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations](https://mlsanity.com/paper/2108.03561/) (2021)
- [Learning to Forecast Dynamical Systems from Streaming Data](https://mlsanity.com/paper/2109.09703/) (2021)
- [Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effects](https://mlsanity.com/paper/2203.05164/) (2022)
- [Online model error correction with neural networks in the incremental 4D-Var framework](https://mlsanity.com/paper/2210.13817/) (2022)
- [Efficient deep data assimilation with sparse observations and time-varying sensors](https://mlsanity.com/paper/2310.16187/) (2023)
- [Reservoir Computing as a Tool for Climate Predictability Studies](https://mlsanity.com/paper/2103.06206/) (2021)
- [A comparison of combined data assimilation and machine learning methods for offline and online model error correction](https://mlsanity.com/paper/2107.11114/) (2021)
- [A Causality-Based Learning Approach for Discovering the Underlying Dynamics of Complex Systems from Partial Observations with Stochastic Parameterization](https://mlsanity.com/paper/2208.09104/) (2022)
- [Learning dynamical systems with hit-and-run random feature maps](https://mlsanity.com/paper/2501.06661/) (2025)
- [State, global and local parameter estimation using local ensemble Kalman filters: applications to online machine learning of chaotic dynamics](https://mlsanity.com/paper/2107.11253/) (2021)
- [CEBoosting: Online Sparse Identification of Dynamical Systems with Regime Switching by Causation Entropy Boosting](https://mlsanity.com/paper/2304.07863/) (2023)
- [Stable generative modeling using Schr"odinger bridges](https://mlsanity.com/paper/2401.04372/) (2024)
- [Localized Schr"odinger Bridge Sampler](https://mlsanity.com/paper/2409.07968/) (2024)
- [Reservoir Computing with Error Correction: Long-term Behaviors of Stochastic Dynamical Systems](https://mlsanity.com/paper/2305.00669/) (2023)
- [Development of an offline and online hybrid model for the Integrated Forecasting System](https://mlsanity.com/paper/2403.03702/) (2024)
- [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)
- [Ensemble forecasts in reproducing kernel Hilbert space family](https://mlsanity.com/paper/2207.14653/) (2022)
- [Likelihood-based generalization of Markov parameter estimation and multiple shooting objectives in system identification](https://mlsanity.com/paper/2212.13902/) (2022)
- [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)
- [A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Series](https://mlsanity.com/paper/2511.06609/) (2025)
- [The Dynamic-Probabilistic Consistency Gap in Chaotic Surrogate Modeling](https://mlsanity.com/paper/2605.31547/) (2026)
- [Learning dynamical systems from noisy data with Weak-form Kernel Ridge Regression](https://mlsanity.com/paper/2607.00257/) (2026)
- [Variability of echo state network prediction horizon for partially observed dynamical systems](https://mlsanity.com/paper/2306.10797/) (2023)

## Abstract (arXiv)

Data-driven prediction and physics-agnostic machine-learning methods have attracted increased interest in recent years achieving forecast horizons going well beyond those to be expected for chaotic dynamical systems. In a separate strand of research data-assimilation has been successfully used to optimally combine forecast models and their inherent uncertainty with incoming noisy observations. The key idea in our work here is to achieve increased forecast capabilities by judiciously combining machine-learning algorithms and data assimilation. We combine the physics-agnostic data-driven approach of random feature maps as a forecast model within an ensemble Kalman filter data assimilation procedure. The machine-learning model is learned sequentially by incorporating incoming noisy observations. We show that the obtained forecast model has remarkably good forecast skill while being computationally cheap once trained. Going beyond the task of forecasting, we show that our method can be used to generate reliable ensembles for probabilistic forecasting as well as to learn effective model closure in multi-scale systems.