| Abstract: |
| Continuous data assimilation concerns estimating the state of a dynamical system from partial measurements. In many applications, the state dynamics are unknown or too expensive to simulate at the desired resolution, leading to model error. Motivated by this challenge and the increasing use of machine learning surrogates in data assimilation, this talk presents a unified analysis of nudging algorithms that employ learned surrogate models of the dynamics. We first establish general conditions on the dynamics and measurements under which nudging with the true dynamics model achieves accurate tracking, both in the noise-free setting and under noisy measurements. 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 measurement noise. Finally, we analyze surrogate models constructed by learning either the vector field governing the dynamics or the system's solution map over a short time step. Our results quantify the amount of training data required for accurate nudging with these learned surrogate models. Numerical experiments illustrate and support the theory. |
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