# Efficient deep data assimilation with sparse observations and time-varying sensors

arXiv: https://arxiv.org/abs/2310.16187 · HTML: https://mlsanity.com/paper/2310.16187/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 56 references that are in the map.
- First posted on arXiv 24 Oct 2023.

## Builds on (most cited first, 20 of 20)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [Neural Encoding and Decoding with Deep Learning for Dynamic Natural Vision](https://mlsanity.com/paper/1608.03425/) (2016)
- [A deep-learning-based surrogate model for data assimilation in dynamic subsurface flow problems](https://mlsanity.com/paper/1908.05823/) (2019)
- [Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning](https://mlsanity.com/paper/2101.00554/) (2021)
- [Fully Convolutional Mesh Autoencoder using Efficient Spatially Varying Kernels](https://mlsanity.com/paper/2006.04325/) (2020)
- [GNN-Surrogate: A Hierarchical and Adaptive Graph Neural Network for Parameter Space Exploration of Unstructured-Mesh Ocean Simulations](https://mlsanity.com/paper/2202.08956/) (2022)
- [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)
- [Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation](https://mlsanity.com/paper/2007.07383/) (2020)
- [Generalised Latent Assimilation in Heterogeneous Reduced Spaces with Machine Learning Surrogate Models](https://mlsanity.com/paper/2204.03497/) (2022)
- [Learning ReLU Networks on Linearly Separable Data: Algorithm, Optimality, and Generalization](https://mlsanity.com/paper/1808.04685/) (2018)
- [A comparison of combined data assimilation and machine learning methods for offline and online model error correction](https://mlsanity.com/paper/2107.11114/) (2021)
- [Multimodal 4DVarNets for the reconstruction of sea surface dynamics from SST-SSH synergies](https://mlsanity.com/paper/2207.01372/) (2022)
- [Deep learning-enhanced ensemble-based data assimilation for high-dimensional nonlinear dynamical systems](https://mlsanity.com/paper/2206.04811/) (2022)
- [Observation Error Covariance Specification in Dynamical Systems for Data assimilation using Recurrent Neural Networks](https://mlsanity.com/paper/2111.06447/) (2021)
- [Low-rank Approximations for Computing Observation Impact in 4D-Var Data Assimilation](https://mlsanity.com/paper/1307.5076/) (2013)

## Built on by (most cited first, 14 of 14)

- [Spatially-Aware Diffusion Models with Cross-Attention for Global Field Reconstruction with Sparse Observations](https://mlsanity.com/paper/2409.00230/) (2024)
- [Machine learning for modelling unstructured grid data in computational physics: a review](https://mlsanity.com/paper/2502.09346/) (2025)
- [Deep Operator Learning for High-Fidelity Fluid Flow Field Reconstruction from Sparse Sensor Measurements](https://mlsanity.com/paper/2412.08009/) (2024)
- [From Models To Experiments: Shallow Recurrent Decoder Networks on the DYNASTY Experimental Facility](https://mlsanity.com/paper/2503.08907/) (2025)
- [Dynamical system prediction from sparse observations using deep neural networks with Voronoi tessellation and physics constraint](https://mlsanity.com/paper/2409.00458/) (2024)
- [Interpretable Diagnostics and Adaptive Data Assimilation for Neural ODEs via Discrete Empirical Interpolation](https://mlsanity.com/paper/2510.21852/) (2025)
- [Rooftop Wind Field Reconstruction Using Sparse Sensors: From Deterministic to Generative Learning Methods](https://mlsanity.com/paper/2603.13077/) (2026)
- [Neural Operator Processes for Probabilistic Operator Learning under Partial Observations](https://mlsanity.com/paper/2606.22946/) (2026)
- [Learning Mappings in Mesh-based Simulations](https://mlsanity.com/paper/2506.12652/) (2025)
- [Learning to Reconstruct Temperature Field from Sparse Observations with Implicit Physics Priors](https://mlsanity.com/paper/2512.01196/) (2025)
- [Latent attention on masked patches for flow reconstruction](https://mlsanity.com/paper/2603.02028/) (2026)
- [Spatiotemporal System Forecasting with Irregular Time Steps via Masked Autoencoder](https://mlsanity.com/paper/2603.25597/) (2026)
- [GLU: Global-Local-Uncertainty Fusion for Scalable Spatiotemporal Reconstruction and Forecasting](https://mlsanity.com/paper/2603.26023/) (2026)
- [Variational Parameter Calibration with Physics-Aware Latent-Space Surrogates](https://mlsanity.com/paper/2608.11435/) (2026)

## Abstract (arXiv)

Variational Data Assimilation (DA) has been broadly used in engineering problems for field reconstruction and prediction by performing a weighted combination of multiple sources of noisy data. In recent years, the integration of deep learning (DL) techniques in DA has shown promise in improving the efficiency and accuracy in high-dimensional dynamical systems. Nevertheless, existing deep DA approaches face difficulties in dealing with unstructured observation data, especially when the placement and number of sensors are dynamic over time. We introduce a novel variational DA scheme, named Voronoi-tessellation Inverse operator for VariatIonal Data assimilation (VIVID), that incorporates a DL inverse operator into the assimilation objective function. By leveraging the capabilities of the Voronoi-tessellation and convolutional neural networks, VIVID is adept at handling sparse, unstructured, and time-varying sensor data. Furthermore, the incorporation of the DL inverse operator establishes a direct link between observation and state space, leading to a reduction in the number of minimization steps required for DA. Additionally, VIVID can be seamlessly integrated with Proper Orthogonal Decomposition (POD) to develop an end-to-end reduced-order DA scheme, which can further expedite field reconstruction. Numerical experiments in a fluid dynamics system demonstrate that VIVID can significantly outperform existing DA and DL algorithms. The robustness of VIVID is also accessed through the application of various levels of prior error, the utilization of varying numbers of sensors, and the misspecification of error covariance in DA.