# Machine Learning, Deepest Learning: Statistical Data Assimilation Problems

arXiv: https://arxiv.org/abs/1707.01415 · HTML: https://mlsanity.com/paper/1707.01415/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jul 2017.

## Built on by (most cited first, 13 of 13)

- [On the difficulty of learning chaotic dynamics with RNNs](https://mlsanity.com/paper/2110.07238/) (2021)
- [Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations](https://mlsanity.com/paper/2108.03561/) (2021)
- [Integrating Recurrent Neural Networks with Data Assimilation for Scalable Data-Driven State Estimation](https://mlsanity.com/paper/2109.12269/) (2021)
- [Learning Spatiotemporal Chaos Using Next-Generation Reservoir Computing](https://mlsanity.com/paper/2203.13294/) (2022)
- [Towards physically consistent data-driven weather forecasting: Integrating data assimilation with equivariance-preserving deep spatial transformers](https://mlsanity.com/paper/2103.09360/) (2021)
- [Bridging observation, theory and numerical simulation of the ocean using Machine Learning](https://mlsanity.com/paper/2104.12506/) (2021)
- [Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution](https://mlsanity.com/paper/2103.14629/) (2021)
- [Online model error correction with neural networks in the incremental 4D-Var framework](https://mlsanity.com/paper/2210.13817/) (2022)
- [A comparison of combined data assimilation and machine learning methods for offline and online model error correction](https://mlsanity.com/paper/2107.11114/) (2021)
- [Learning 4DVAR inversion directly from observations](https://mlsanity.com/paper/2211.09741/) (2022)
- [Machine learning-based conditional mean filter: a generalization of the ensemble Kalman filter for nonlinear data assimilation](https://mlsanity.com/paper/2106.07908/) (2021)
- [Message Propagation Through Time: An Algorithm for Sequence Dependency Retention in Time Series Modeling](https://mlsanity.com/paper/2309.16882/) (2023)
- [Probabilistic Emulation of the Community Radiative Transfer Model Using Machine Learning](https://mlsanity.com/paper/2504.16192/) (2025)

## Abstract (arXiv)

We formulate a strong equivalence between machine learning, artificial intelligence methods and the formulation of statistical data assimilation as used widely in physical and biological sciences. The correspondence is that layer number in the artificial network setting is the analog of time in the data assimilation setting. Within the discussion of this equivalence we show that adding more layers (making the network deeper) is analogous to adding temporal resolution in a data assimilation framework. How one can find a candidate for the global minimum of the cost functions in the machine learning context using a method from data assimilation is discussed. Calculations on simple models from each side of the equivalence are reported. Also discussed is a framework in which the time or layer label is taken to be continuous, providing a differential equation, the Euler-Lagrange equation, which shows that the problem being solved is a two point boundary value problem familiar in the discussion of variational methods. The use of continuous layers is denoted "deepest learning". These problems respect a symplectic symmetry in continuous time/layer phase space. Both Lagrangian versions and Hamiltonian versions of these problems are presented. Their well-studied implementation in a discrete time/layer, while respected the symplectic structure, is addressed. The Hamiltonian version provides a direct rationale for back propagation as a solution method for the canonical momentum.