Tensor-Var: Efficient Four-Dimensional Variational Data Assimilation

Yiming Yang, Xiaoyuan Cheng, Daniel Giles, Sibo Cheng, Yi He, Xiao Xue, Boli Chen, Yukun Hu
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:70649-70679, 2025.

Abstract

Variational data assimilation estimates the dynamical system states by minimizing a cost function that fits the numerical models with the observational data. Although four-dimensional variational assimilation (4D-Var) is widely used, it faces high computational costs in complex nonlinear systems and depends on imperfect state-observation mappings. Deep learning (DL) offers more expressive approximators, while integrating DL models into 4D-Var is challenging due to their nonlinearities and lack of theoretical guarantees in assimilation results. In this paper, we propose Tensor-Var, a novel framework that integrates kernel conditional mean embedding (CME) with 4D-Var to linearize nonlinear dynamics, achieving convex optimization in a learned feature space. Moreover, our method provides a new perspective for solving 4D-Var in a linear way, offering theoretical guarantees of consistent assimilation results between the original and feature spaces. To handle large-scale problems, we propose a method to learn deep features (DFs) using neural networks within the Tensor-Var framework. Experiments on chaotic systems and global weather prediction with real-time observations show that Tensor-Var outperforms conventional and DL hybrid 4D-Var baselines in accuracy while achieving a 10- to 20-fold speed improvement.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-yang25h, title = {Tensor-Var: Efficient Four-Dimensional Variational Data Assimilation}, author = {Yang, Yiming and Cheng, Xiaoyuan and Giles, Daniel and Cheng, Sibo and He, Yi and Xue, Xiao and Chen, Boli and Hu, Yukun}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {70649--70679}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/yang25h/yang25h.pdf}, url = {https://proceedings.mlr.press/v267/yang25h.html}, abstract = {Variational data assimilation estimates the dynamical system states by minimizing a cost function that fits the numerical models with the observational data. Although four-dimensional variational assimilation (4D-Var) is widely used, it faces high computational costs in complex nonlinear systems and depends on imperfect state-observation mappings. Deep learning (DL) offers more expressive approximators, while integrating DL models into 4D-Var is challenging due to their nonlinearities and lack of theoretical guarantees in assimilation results. In this paper, we propose Tensor-Var, a novel framework that integrates kernel conditional mean embedding (CME) with 4D-Var to linearize nonlinear dynamics, achieving convex optimization in a learned feature space. Moreover, our method provides a new perspective for solving 4D-Var in a linear way, offering theoretical guarantees of consistent assimilation results between the original and feature spaces. To handle large-scale problems, we propose a method to learn deep features (DFs) using neural networks within the Tensor-Var framework. Experiments on chaotic systems and global weather prediction with real-time observations show that Tensor-Var outperforms conventional and DL hybrid 4D-Var baselines in accuracy while achieving a 10- to 20-fold speed improvement.} }
Endnote
%0 Conference Paper %T Tensor-Var: Efficient Four-Dimensional Variational Data Assimilation %A Yiming Yang %A Xiaoyuan Cheng %A Daniel Giles %A Sibo Cheng %A Yi He %A Xiao Xue %A Boli Chen %A Yukun Hu %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-yang25h %I PMLR %P 70649--70679 %U https://proceedings.mlr.press/v267/yang25h.html %V 267 %X Variational data assimilation estimates the dynamical system states by minimizing a cost function that fits the numerical models with the observational data. Although four-dimensional variational assimilation (4D-Var) is widely used, it faces high computational costs in complex nonlinear systems and depends on imperfect state-observation mappings. Deep learning (DL) offers more expressive approximators, while integrating DL models into 4D-Var is challenging due to their nonlinearities and lack of theoretical guarantees in assimilation results. In this paper, we propose Tensor-Var, a novel framework that integrates kernel conditional mean embedding (CME) with 4D-Var to linearize nonlinear dynamics, achieving convex optimization in a learned feature space. Moreover, our method provides a new perspective for solving 4D-Var in a linear way, offering theoretical guarantees of consistent assimilation results between the original and feature spaces. To handle large-scale problems, we propose a method to learn deep features (DFs) using neural networks within the Tensor-Var framework. Experiments on chaotic systems and global weather prediction with real-time observations show that Tensor-Var outperforms conventional and DL hybrid 4D-Var baselines in accuracy while achieving a 10- to 20-fold speed improvement.
APA
Yang, Y., Cheng, X., Giles, D., Cheng, S., He, Y., Xue, X., Chen, B. & Hu, Y.. (2025). Tensor-Var: Efficient Four-Dimensional Variational Data Assimilation. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:70649-70679 Available from https://proceedings.mlr.press/v267/yang25h.html.

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