# Accurate deep learning-based filtering for chaotic dynamics by identifying instabilities without an ensemble

arXiv: https://arxiv.org/abs/2408.04739 · HTML: https://mlsanity.com/paper/2408.04739/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 66 references that are in the map.
- First posted on arXiv 8 Aug 2024.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Mish: A Self Regularized Non-Monotonic Activation Function](https://mlsanity.com/paper/1908.08681/) (2019)
- [On Feature Collapse and Deep Kernel Learning for Single Forward Pass Uncertainty](https://mlsanity.com/paper/2102.11409/) (2021)
- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [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)
- [Backprop KF: Learning Discriminative Deterministic State Estimators](https://mlsanity.com/paper/1605.07148/) (2016)
- [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)
- [Learning Variational Data Assimilation Models and Solvers](https://mlsanity.com/paper/2007.12941/) (2020)
- [Latent Space Data Assimilation by using Deep Learning](https://mlsanity.com/paper/2104.00430/) (2021)
- [Variational Data Assimilation with a Learned Inverse Observation Operator](https://mlsanity.com/paper/2102.11192/) (2021)
- [Auto-differentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2107.07687/) (2021)
- [Adaptively Truncating Backpropagation Through Time to Control Gradient Bias](https://mlsanity.com/paper/1905.07473/) (2019)
- [Data driven weather forecasts trained and initialised directly from observations](https://mlsanity.com/paper/2407.15586/) (2024)
- [Data Assimilation Networks](https://mlsanity.com/paper/2010.09694/) (2020)
- [Learning to Assimilate in Chaotic Dynamical Systems](https://mlsanity.com/paper/2111.01058/) (2021)
- [Learning Optimal Filters Using Variational Inference](https://mlsanity.com/paper/2406.18066/) (2024)
- [On Training Recurrent Networks with Truncated Backpropagation Through Time in Speech Recognition](https://mlsanity.com/paper/1807.03396/) (2018)
- [Degenerate Kalman filter error covariances and their convergence onto the unstable subspace](https://mlsanity.com/paper/1604.02578/) (2016)
- [Learning 4DVAR inversion directly from observations](https://mlsanity.com/paper/2211.09741/) (2022)
- [Rank deficiency of Kalman error covariance matrices in linear time-varying system with deterministic evolution](https://mlsanity.com/paper/1503.05029/) (2015)

## Built on by (most cited first, 7 of 7)

- [Learning Enhanced Ensemble Filters](https://mlsanity.com/paper/2504.17836/) (2025)
- [Ensemble Kalman filter in latent space using a variational autoencoder pair](https://mlsanity.com/paper/2502.12987/) (2025)
- [Closed-form conditional diffusion models for data assimilation](https://mlsanity.com/paper/2603.21291/) (2026)
- [Neural Operator-Based Nonlinear Nudging for Chaotic Dynamical Systems](https://mlsanity.com/paper/2508.05778/) (2025)
- [The Dynamic-Probabilistic Consistency Gap in Chaotic Surrogate Modeling](https://mlsanity.com/paper/2605.31547/) (2026)
- [Hybrid machine learning data assimilation for marine biogeochemistry](https://mlsanity.com/paper/2504.05218/) (2025)
- [On the Effect of Neural Field Reparameterization for 4DVAR](https://mlsanity.com/paper/2509.21751/) (2025)

## Abstract (arXiv)

We investigate the ability to discover data assimilation (DA) schemes meant for chaotic dynamics with deep learning. The focus is on learning the analysis step of sequential DA, from state trajectories and their observations, using a simple residual convolutional neural network, while assuming the dynamics to be known. Experiments are performed with the Lorenz 96 dynamics, which display spatiotemporal chaos and for which solid benchmarks for DA performance exist. The accuracy of the states obtained from the learned analysis approaches that of the best possibly tuned ensemble Kalman filter, and is far better than that of variational DA alternatives. Critically, this can be achieved while propagating even just a single state in the forecast step. We investigate the reason for achieving ensemble filtering accuracy without an ensemble. We diagnose that the analysis scheme actually identifies key dynamical perturbations, mildly aligned with the unstable subspace, from the forecast state alone, without any ensemble-based covariances representation. This reveals that the analysis scheme has learned some multiplicative ergodic theorem associated to the DA process seen as a non-autonomous random dynamical system.