Continuous Data Assimilation with Learned Surrogate Dynamics
For practitioners of data assimilation, this work provides theoretical guarantees for using machine learning surrogates in nudging algorithms, addressing model error when true dynamics are unknown or expensive.
This paper develops a unified finite-dimensional analysis of nudging algorithms that use learned surrogate models for continuous data assimilation, establishing conditions for exponential convergence and quantifying error floors due to surrogate approximation and observation noise. Numerical experiments support the theory.
Continuous data assimilation seeks to estimate the state of a dynamical system from partial observations. In many applications, however, the state dynamics are unknown or prohibitively expensive to simulate at the required resolution, leading to model error. Motivated by this challenge and the increasing adoption of machine learning surrogates in data assimilation, this paper develops a unified finite-dimensional analysis of nudging algorithms that employ learned surrogate models of the dynamics. We first establish general conditions on the dynamics and observations that guarantee accurate tracking for nudging with the true dynamics model, both in the noise-free and noisy settings. We then show that nudging algorithms that employ surrogate models retain exponential convergence up to an explicit error floor that quantifies the effects of surrogate approximation error and observation noise. Finally, we analyze surrogate models obtained by learning either the vector field or the short-time solution map of the system, and quantify the amount of training data needed to ensure accurate nudging in the noise-free setting. Numerical experiments support the theory.