ICML 2025

Tensor-Var: Efficient Four-Dimensional Variational Data Assimilation
Yiming Yang 1 Xiaoyuan Cheng 1 Daniel Giles 1 Sibo Cheng 2 Yi He 1 Xiao Xue 1 Boli Chen 1 Yukun Hu 1
Abstract
Variational data assimilation estimates the dynam-
ical system states by minimizing a cost function
that fits the numerical models with the obser-
vational data. Although four-dimensional vari-
ational assimilation (4D-Var) is widely used, it
faces high computational costs in complex non-
linear 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 ker-
nel 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 perspec-
tive 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. Ex-
periments on chaotic systems and global weather
prediction with real-time observations show that
Tensor-Var outperforms conventional and DL hy-
brid 4D-Var baselines in accuracy while achieving
a 10- to 20-fold speed improvement.
1. Introduction
Forecasting of dynamical systems is an initial value problem
of practical significance. Many real-world systems, such as
the ocean and atmosphere, are chaotic, which means that
minor errors in current estimations in computational models
can lead to rapid divergence and substantial forecast errors
1University College London, London, United Kingdom
2CEREA, ENPC and EDF R&D, Institut Polytechnique de
Paris, Paris, France.
Correspondence to: Yiming Yang <zc-
ahyy1...