# Learning Variational Data Assimilation Models and Solvers

arXiv: https://arxiv.org/abs/2007.12941 · HTML: https://mlsanity.com/paper/2007.12941/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jul 2020.

## Built on by (most cited first, 30 of 36)

- [Score-based Data Assimilation](https://mlsanity.com/paper/2306.10574/) (2023)
- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [A posteriori learning for quasi-geostrophic turbulence parametrization](https://mlsanity.com/paper/2204.03911/) (2022)
- [FengWu-4DVar: Coupling the Data-driven Weather Forecasting Model with 4D Variational Assimilation](https://mlsanity.com/paper/2312.12455/) (2023)
- [Towards an end-to-end artificial intelligence driven global weather forecasting system](https://mlsanity.com/paper/2312.12462/) (2023)
- [FuXi Weather: A data-to-forecast machine learning system for global weather](https://mlsanity.com/paper/2408.05472/) (2024)
- [Fuxi-DA: A Generalized Deep Learning Data Assimilation Framework for Assimilating Satellite Observations](https://mlsanity.com/paper/2404.08522/) (2024)
- [Bridging observation, theory and numerical simulation of the ocean using Machine Learning](https://mlsanity.com/paper/2104.12506/) (2021)
- [Accurate deep learning-based filtering for chaotic dynamics by identifying instabilities without an ensemble](https://mlsanity.com/paper/2408.04739/) (2024)
- [A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problems](https://mlsanity.com/paper/2412.09009/) (2024)
- [Bounded nonlinear forecasts of partially observed geophysical systems with physics-constrained deep learning](https://mlsanity.com/paper/2202.05750/) (2022)
- [Inversion of sea surface currents from satellite-derived SST-SSH synergies with 4DVarNets](https://mlsanity.com/paper/2211.13059/) (2022)
- [Training neural mapping schemes for satellite altimetry with simulation data](https://mlsanity.com/paper/2309.14350/) (2023)
- [Improving the Adaptive Moment Estimation (ADAM) stochastic optimizer through an Implicit-Explicit (IMEX) time-stepping approach](https://mlsanity.com/paper/2403.13704/) (2024)
- [Tensor-Var: Efficient Four-Dimensional Variational Data Assimilation](https://mlsanity.com/paper/2501.13312/) (2025)
- [Augmented Invertible Koopman Autoencoder for long-term time series forecasting](https://mlsanity.com/paper/2503.12930/) (2025)
- [Joint calibration and mapping of satellite altimetry data using trainable variational models](https://mlsanity.com/paper/2110.03405/) (2021)
- [Neural Incremental Data Assimilation](https://mlsanity.com/paper/2406.15076/) (2024)
- [Benchmarking AI-based data assimilation to advance data-driven global weather forecasting](https://mlsanity.com/paper/2408.11438/) (2024)
- [Combined Optimization of Dynamics and Assimilation with End-to-End Learning on Sparse Observations](https://mlsanity.com/paper/2409.07137/) (2024)
- [Generating Unseen Nonlinear Evolution in Sea Surface Temperature Using a Deep Learning-Based Latent Space Data Assimilation Framework](https://mlsanity.com/paper/2412.13477/) (2024)
- [VAE-Var: Variational-Autoencoder-Enhanced Variational Assimilation](https://mlsanity.com/paper/2405.13711/) (2024)
- [Multimodal learning-based inversion models for the space-time reconstruction of satellite-derived geophysical fields](https://mlsanity.com/paper/2203.10640/) (2022)
- [Ensemble forecasts in reproducing kernel Hilbert space family](https://mlsanity.com/paper/2207.14653/) (2022)
- [Uncertainty-aware data assimilation through variational inference](https://mlsanity.com/paper/2510.17268/) (2025)
- [ADDA: a Modular Framework for Representing, Simulating and Assimilating Dynamics with End-to-end Differentiability](https://mlsanity.com/paper/2608.23297/) (2026)
- [Learning-based estimation of in-situ wind speed from underwater acoustics](https://mlsanity.com/paper/2208.08912/) (2022)
- [Learning Regionalization using Accurate Spatial Cost Gradients within a Differentiable High-Resolution Hydrological Model: Application to the French Mediterranean Region](https://mlsanity.com/paper/2308.02040/) (2023)
- [Multi-Modal Learning-based Reconstruction of High-Resolution Spatial Wind Speed Fields](https://mlsanity.com/paper/2312.08933/) (2023)
- [U-Net Kalman Filter (UNetKF): An Example of Machine Learning-assisted Ensemble Data Assimilation](https://mlsanity.com/paper/2403.12366/) (2024)

## Abstract (arXiv)

This paper addresses variational data assimilation from a learning point of view. Data assimilation aims to reconstruct the time evolution of some state given a series of observations, possibly noisy and irregularly-sampled. Using automatic differentiation tools embedded in deep learning frameworks, we introduce end-to-end neural network architectures for data assimilation. It comprises two key components: a variational model and a gradient-based solver both implemented as neural networks. A key feature of the proposed end-to-end learning architecture is that we may train the NN models using both supervised and unsupervised strategies. Our numerical experiments on Lorenz-63 and Lorenz-96 systems report significant gain w.r.t. a classic gradient-based minimization of the variational cost both in terms of reconstruction performance and optimization complexity. Intriguingly, we also show that the variational models issued from the true Lorenz-63 and Lorenz-96 ODE representations may not lead to the best reconstruction performance. We believe these results may open new research avenues for the specification of assimilation models in geoscience.